Class Notes, 31415 RF-Communication Circuits

Chapter II

RF-CIRCUITS

Jens Vidkjær

NB230 ii Contents

II RF-Circuits, Concepts and Methods ...... 1 II-1 Parallel Resonance Circuits ...... 2 Frequency Response ...... 3 Poles and Zeros ...... 6 Transient Response ...... 8 II-2 Series Resonance Circuit Summary ...... 11 II-3 Narrowband Approximations ...... 14 II-4 Series-to-Parallel Conversions ...... 19 Example II-4-1 ( impedance matching ) ...... 20 Conversions in Narrowband Applications ...... 23 Conversions in Broadband Modeling ...... 26 Example II-4-2 ( spiral inductor ) ...... 26 II-5 Tuned Amplifiers ...... 30 Synchronously Tuned Amplifiers ...... 32 Example II-5-1 ( synchronous tuning ) ...... 33 Butterworth Stagger Tuned Amplifiers ...... 36 Example II-5-2 ( Butterworth amplifier ) ...... 41 II-6 Transformers and Transformerlike Couplings ...... 45 Review of Mutual Inductances ...... 45 Equivalent Circuits for Two Coupled Inductors ...... 49 RF-Transformers ...... 53 Example II-6-1 ( practical RF transformers ) ...... 55 Tuned Transformers ...... 57 Example II-6-2 ( transformer-coupled tuned amplifier ) .... 58 Autotransformers ...... 61 Example II-6-3 ( autotransformers in a tuned amplifier ) . . . 62 Transformerlike Couplings ...... 63 Example II-6-4 ( uncoupled reactance transformers in tuned amplifiers ) ...... 64 Three-Winding Transformers ...... 67 Transformer Hybrids ...... 72 Example II-6-5 ( diode-ring mixer preliminaries ) ...... 78 II-7 Double-Tuned Circuits and Amplifiers ...... 80 Coupling between Identical Resonance Circuits ...... 81 Example II-7-1 ( Foster-Seely FM detector ) ...... 85 Example II-7-2 ( quadrature FM detector ) ...... 88 Double-Tuned Amplifier Stages ...... 94 Asymptotic Behavior ...... 98 II-8 Impedance Matching ...... 101

J.Vidkjær iii Lumped Element Impedance Matching using Smith Charts ...... 101 Example II-8-1 ( double and standard Smith charts ) ...... 104 Example II-8-2 ( bandwidth estimations ) ...... 108 APPENDIX II-A, Power Calculation and Power Matching ...... 113 APPENDIX II-B Signal Flow Graphs ...... 117 Elementary Signal Flow Graphs ...... 117 Mason’s Direct Rule ...... 121 Problems ...... 125 References and Further Reading ...... 131

Index ...... 133

iv J.Vidkjær 1

II RF-Circuits, Concepts and Methods

In RF-communication system, control of frequency bands and impedance matching conditions between functional blocks or amplifier stages are problems that constantly face a circuit designer. The tasks are so frequent that many analytical techniques and approximation methods especially suited for high-frequency circuits have evolved and strongly influenced the jargon of RF engineering. Below we shall introduce the most important basic concepts and methods that are required to

- understand data sheets and literature, - make simpler design decisions or calculations, and - prepare and interpret simulation data.

The selection of topics and examples have furthermore been conducted to suit the needs in the following chapters, which are still in preparation.

Scanning contents, the chapter starts summarizing basic properties of resonance circuits. Although significant by themselves, the importance of acquiring familiarity with ideal resonance circuits is the fact, that any narrowbanded resonance circuit may be approximated by the ideal ones around resonance frequencies. This property reduces significantly the efforts that are required to understand and explore operations of tuned bandpass circuits, which are frequently used in RF-communication systems. Foundations of the simplifications are dealt with in sections concerning narrowband approximations and series-to-parallel transformations. The tuned amplifiers are introduced in ideal form concentrating on simple frequency charac- teristics. Coupling techniques using transformers and coupled resonance circuits are still highly useful methods in RF-designs, so they are considered in some details here. Finally, the very general method of constructing lumped element matching networks using a Smith chart is exemplified.

Power matching is fundamental for designing and understanding many RF circuits. Although this concept is mandatory in basic circuit theory curriculums, it is repeated for convenience in an appendix. Also the method of illustrating and solving network equations by the signal flow graph method is summarized in an appendix.

J.Vidkjær 2

II-1 Parallel Resonance Circuits

(1)

Fig.1 Parallel resonance circuit

A basic parallel resonance circuit is shown in Fig.1. Besides component values the combinations, which are summarized by Eq.(1), are frequently used. The resonance frequency is the frequency where the capacitive and the inductive susceptances are equal in magnitude as indicated in Fig.2a. When an external steady state sinusoidal voltage-source of frequency ω 0 is applied to the resonance circuit, the two opposite currents through the capacitor and the inductor balance each other, and only the resistor current flows through the terminal. This situation is sketched in Fig.2b, which also shows how the quality factor Q indicates the magnitude ratio of the internal reactive currents over the resistive terminal current at reso- nance.

Fig.2 Parallel resonance. (a) Susceptance composition as function of frequency. (b) ω Current and voltage phasors at the resonance frequency 0.

Another view upon resonance and the quality factor concerns the energy in the circuit under steady state conditions. At instants where the two phasors iC and iL are perpendicular to the real axis, no currents flow into the capacitor or inductor, but the capacitor hold maximum energy

(2)

J.Vidkjær II-1 Parallel Resonance Circuits 3

The first equation is the usual electrostatic energy expression. The second takes into account that rms values - indicated by small letters - are conventionally used when dealing with steady state linear circuits. A quarter of a period later the current phasors project in full onto the real axis while the voltage is zero. The capacitor holds no energy, but the inductor energy peaks with the same maximum that formerly was held in the capacitor,

(3)

Thus, a constant amount of energy laps between the capacitor and the inductor at resonance, and the quality factor may be expressed

(4)

π ω where the loss is calculated as the resistor power times the resonance period T0=2 / 0. This interpretation of resonance is often useful in the construction of lumped circuit equivalents for the variety of electromagnetic and mechanical resonators that are used in RF-circuits.

Frequency Response

Expressed through circuit element values, the impedance function for the parallel circuit in Fig.1 is

(5)

ω ω Introducing 0 and Q from Eq.(1), the impedance expressed as a function of frequency s=j becomes

(6)

The frequency dependency of the impedance is kept in the quantity β(ω), which is zero at the ω resonance frequency 0. Here the denominator of Eq.(6) gets its smallest size and the ω impedance has maximum Rp, the parallel resistance. The magnitude and phase of Zp(j ) are

(7)

The two functions are shown in Fig.3(a) while Fig.3(b) shows the corresponding admittance characteristics,

J.Vidkjær 4 RF-Circuits, Concepts and Methods

Fig.3 Impedance (a) and admittance (b) magnitudes and phases of the parallel resonance ω circuit in Fig.1. The curves are symmetric around o due to the logarithmic fre- quency scales.

(8)

Upper and lower bounds of the 3dB bandwidth intervals W3dB, which are indicated in Fig.3 , correspond to a denominator size equal to 2 in Eq.(7). The bounds are found setting the imaginary part of the denominator equal in magnitude to the real part, i.e.

(9)

Both negative and positive frequencies are contained in the conditions. We call the largest ω ω valued solutions, where the two terms in b have equal signs, the upper bounds ± bu. The ω lower bounds ± bl are obtained with terms of opposite signs. Fig.4 summarizes how the different solutions are formed. By definition, the 3dB bandwidth is taken to be the distance between positive or zero-valued 3dB frequency bounds, and we get the result that was incorporated in Fig.3,

(10)

ω It follows from the solutions in Eq.(9) that the resonance frequency 0 is not centered between the 3dB bounds but is the geometrical mean of the bounds,

J.Vidkjær II-1 Parallel Resonance Circuits 5

Fig.4 Upper and lower 3dB bound positions from Eq.(9) . Note, in linear frequency scale ω →∞ the 3dB bands are not symmetric around the resonances at ± 0 unless Q .

(11)

(12)

so in logarithmic frequency scale the upper and lower 3dB frequency bounds are symmetric ω ω ω with respect to log 0. However, any other pair of frequencies, u, l that has the resonance ω frequency as geometrical mean maps symmetrically around log 0. Since both frequencies provide the same absolute |β|, i.e.

(13)

the impedance or admittance magnitude characteristics of the types in Fig.3 have even symmetry with respect to the resonance frequency in a logarithmic frequency scale. Corre- spondingly, the phase characteristics show odd symmetry because tan-1(-Qβ)=-tan-1(Qβ). At β π the 3dB boundaries where |Q |=1, the phase angles of impedance Zp become ±¼ . Fig.5 shows plots of the impedance function with various Q-factors. The normalization in magnitude corresponds to keeping the inductor and capacitor fixed while letting the parallel resistance follows Q according to Eq.(1). The asymptotic behavior of the impedance approximating the inductor reactance below and the capacitor reactance above resonance respectively are readily observed. Calculating magnitudes, it may suffice to use the inductor or capacitor alone at frequencies that differ more than a factor of three from resonance.

Summarizing the frequency characteristics, we have seen that the greater Q, the smaller bandwidth and in turn, the steeper phase characteristics around the resonance frequen- ω cy 0. Moreover, the frequency characteristics were symmetric in logarithmic frequency scale. Phase steepness is an important property when a resonance circuit is employed in an oscillator and we shall return to this question later. Also, the symmetry property will be reconsidered. Important classes of signal handling expect linear symmetry that may be approached with high

J.Vidkjær 6 RF-Circuits, Concepts and Methods

Fig.5 Normalized magnitudes and phases in the impedance of parallel tuned circuits with varying Q-factors.

Q circuits too.1 For obvious reasons such circuits are also called narrowbanded.

Poles and Zeros

Pole and zero positions are useful for investigating responses of frequency selective networks that include parallel tuned circuits. Using parameters from Eq.(1), the impedance Zp of Eq.(5) is rewritten,

(14)

Solving for the s-values, which set the numerator and the denominator equal to zero respec- tively, gives the zero and the poles of the impedance function. Once poles and zeros are known, the impedance may be cast in the form that suits the analysis of composite networks,

(15)

1 ) See section VI-1 on frequency stability in oscillators and section I-4 on transmission of narrowband signals.

J.Vidkjær II-1 Parallel Resonance Circuits 7

While there is a tradition of using Q for characterizing frequency responses of resonant RF circuits, the conceptually equivalent damping ratio ζ is often seen in pole-zero and especially transient response calculations, where it leads to more compact expressions. We proceed here with both forms and get,

(16)

Up to this point no attention was given to actual parameter values. The last result requires a distinction between circuits having ζ≥1 and ζ<1 ( Q≤½ and Q>½ ). In first case the square roots of Eq.(16) are real and the poles are on the real axis. In second case the poles move from the real axis and make a complex conjugated pair. To emphasize this property we change the last part of Eq.(16) to read

(17)

Fig.6 Position of poles s1,s2, and the zero s0 in the impedance ζ Zp(s) of the parallel resonance circuit with <1 or Q>½.

The square roots in (a) are real valued if the poles are complex. The approximations in the following lines apply to lightly damped circuits with higher and higher Q´s, where the

J.Vidkjær 8 RF-Circuits, Concepts and Methods expressions in Eq.(17)(b) are based on the estimate

(18)

Fig.6 sketches the geometry of complex pole and zero positions. Starting from damping ζ=1 ω ( Q=½) the poles are by Eq.(17)(a) constrained to move along a circle of radius 0 from the real towards the imaginary axis with declining damping or growing Q. Once the poles and zeros are known, the frequency characteristics of Zp may be calculated from geometrical considerations as sketched in Fig.7 and Eqs.(19) to (21).

(19)

(20)

(21)

ω Fig.7 Calculation of Zp( ) from poles and zeros

Transient Response

Fig.8 Charging of capacitor C to voltage VC0 =QC0/C.

The projection of the poles on the imaginary axis determines the oscillatory modes in the transient responses of the circuit. To see this, we consider the decay of circuit energy when the circuit is left alone once the capacitor has been charged to a voltage of VC0 with the inductor current initialized to zero. The initial charging of the capacitor is equivalent to forcing a pulse current of strength

(22)

J.Vidkjær II-1 Parallel Resonance Circuits 9

2 through the circuit as indicated by Fig.8. The corresponding transient voltage decay Vdcy(t) is the impulse response of the impedance function, which is given by its inverse Laplace transform, i.e.

(23)

The last rewriting prepares for use of standard Laplace transform tables, from which we get

(24)

where the angular frequency is recognized as the size of the imaginary pole component if ζ<1. Introducing the auxiliary phases and the identities

(25)

(26)

(27) the time domain responses may be condensed to read

(28)

2 ) Note, the delta function δ(t) has dimension [sec-1] to comply with the requirement

J.Vidkjær 10 RF-Circuits, Concepts and Methods

Fig.9 Transients from Eq.(28). Examples in (a) are heavily damped and (b) is lightly π ω damped. Time scales are in units of the resonance period T0=2 / 0.

Fig.9 shows examples of the responses with different dampings. The case ζ=1, where the two poles coincide on the real axis, sets the border between aperiodic and oscillatory solutions. In the latter case, the leading exponential factor shapes the envelope to the solution

(29)

The corresponding time constant,

(30)

is sometimes called the logarithmic decrement of the circuit. Observe that the result is in agreement with the previous equivalent baseband considerations in example I-4-2. In practical terms we notice that the significant number of cycles through the decay approximates Q if Q ≈ ζ≈ φ ≈ > 5so < 0.1 and cos 1 1. It follows from the fact that at t=QT0, the exponential has fallen from one to

(31) so more than 99.8% of the initial energy is lost in the resistor at that instant.

J.Vidkjær 11

II-2 Series Resonance Circuit Summary

(32)

Fig.10 Series resonance circuit

A series resonance circuit is connected as shown by Fig.10 and Eq.(32) summarizes its common parameter combinations. The circuit admittance and impedance are,

(33)

Compared to the parallel circuit impedance and admittance in Eq.(5) it is seen, that the numerator and denominator are similar in structure with respect to s, but the coefficients are different. Connected to a source, the two types of resonance circuits behave in duality. This means that the reader could kindly be asked to repeat the preceding section exchanging terms, voltage and current, parallel and series, impedance and admittance, inductance and capaci- tance, resistance and conductance, and then we were done. To avoid confusion in future references, however, the main concept of the series resonance is summarized below in its own terms, but without detailed derivations.

Fig.11 Series resonance. (a) Reactance composition as function of frequency. (b) Voltage ω and current phasors at the resonance frequency 0.

At resonance in a series circuit the voltages across the inductive and the capacitive reactances are equal in magnitudes but opposed in phases. With the same current flowing through all components the requirement is that the two reactances balance each other at the

J.Vidkjær 12 RF-Circuits, Concepts and Methods resonance frequency as indicated by Fig.11(a). The phasor diagram in Fig.11(b) shows that the quality factor now represents the ratio of the reactance voltage magnitudes over the voltage vR across the series resistor Rs. At resonance this voltage equals the terminal voltage v. With one terminal grounded, the potential at the interconnection between the inductor and the capacitor becomes Q times as high as the potential at the driving terminal. This fact may significantly influence the practical realization of high Q series circuits.

Like the parallel circuit in steady state resonance, a constant amount of energy is exchanged between the capacitor and the inductor in the series circuit. In terms of energy there are no differences between the two types of resonance circuits regarding the quality factor, but the loss calculation must now be detailed as a series loss,

(34)

Using the resonance frequency and the quality factor from Eq.(32) the impedance of the series circuit is expressed

(35)

The frequency is again accounted for through β(ω), so the impedance and admittance func- tions become symmetric in logarithmic frequency scales as showed in Fig.12.

Fig.12 Impedance (a) and admittance (b) magnitudes and phases of the series resonance ω circuit in Fig.10. The curves are symmetric around o due to the logarithmic fre- quency scales.

Impedance and admittance magnitudes and phases in Fig.12(a) and (b) are given by

J.Vidkjær II-2 Series Resonance Circuit Summary 13

Fig.13 Normalized magnitude and phases for the impedance of series resonance circuits with varying Q-factors.

(36)

(37)

The equations are equivalent to Eqs.(8) and (7), so all results concerning bandwidth and symmetry of bounds and characteristics may directly be overtaken from the forgoing section. Fig.13 holds impedance characteristics with different Q-factors and shows the asymptotes set by the capacitor and inductor below and above resonance respectively.

Poles and zeros for the series circuits are based on the expressions

(38)

Comparison to Eq.(14) reveals that the zeros of Zs(s) must follow the pattern of the poles in the parallel circuit while the pole of Zs in origo corresponds to the zero of Zp there. In terms of poles and zeros, the series circuit impedance is

(39)

where the geometrical properties of s0,s1, and s2 are the same as in Fig.6.

J.Vidkjær 14

II-3 Narrowband Approximations

Frequency dependency of the impedance in the parallel resonance circuits was expressed through the nonlinear function β(ω). In consequence, we had to solve 2nd order equations to find prescribed bandlimits. If the circuit is narrowbanded, a linear approximation to the frequency relationship may suffice for design considerations around the resonance frequency. A Taylor expansion provides

(40)

(41)

Following Eq.(9), the 3dB bounds are the frequencies where |Qβ(ω)|=1. With the approxima- ω tion, bandlimits are placed symmetrically around 0, and the 3dB bandwidth becomes

(42)

The same result was obtained in Eq.(10) from the nonlinear β(ω) expression, so the linear approximation gives the right bandwidth. However, Fig.14 reveals that the approximation places the 3dB interval below the correct one. But it is also seen in the figure that the greater Q, the smaller spacing between the 1/Q and -1/Q lines, and the smaller is the error introduced by approximating to symmetrical 3dB bounds through Eq.(41). To quantify this point

Fig.14 Comparison of true and approximate 3dB bandlimit and bandwidth calculation in ω simple resonance circuits. The expression for the true middle frequency m is taken from Fig.4.

J.Vidkjær II-3 Narrowband Approximations 15 we calculate the relative difference between the approximated and the true value of the bandlimits. They are equal to the relative difference between the corresponding middle ω ω frequencies. It is 0 in the linearization while the true middle frequency m depends on Q as indicated by Fig.4. The relative error becomes

(43)

The figures show that the accuracy of the approximation is admirable for most applications in circuits where Q is five or more. With Q exceeding one the accuracy may even suffice for initial design estimations.

Inserting the linear expansion of β, the impedance of the parallel resonance imped- ance circuit is approximated

(44)

ω ω In terms of frequency deviation from resonance, - 0, the expression is as simple as a first order lowpass characteristic, for instance the impedance of a resistor and a capacitor in parallel as sketched in Fig.15. Thus, instead of the resonance curves in Fig.3 or Fig.5, we may take

Fig.15 Impedance of parallel RC circuit. the impedance from a normalized first order lowpass characteristics like the one given in Fig.16. Note, however, that compared to the lowpass case of Fig.15, the frequency deviation in the last denominator of Eq.(44) is normalized with respect to half the 3dB bandwidth, ω because there are 3dB limits on either side of 0. The lower frequency bound in lowpass is fixed to zero and not found from an expression. A standardized characteristic as the one in Fig.16 has several interpretations, including

(45)

J.Vidkjær 16 RF-Circuits, Concepts and Methods

Fig.16 Normalized first order lowpass characteristics. The sign of Ω must precede the phase in bandpass interpretations, either complete or narrowband approximated.

ω It must still be kept in mind that the narrowband approximation is useful around 0, but gives wrong results far from this region. For instance, the true and the approximated bandpass expressions in Eq.(45) are seen to give different results using ω =0orω→∞.

Fig.17 Contributions from poles s1,s2 and zero s0 to Zp in narrowband approximation where (b) is an expanded view of the encircled region in (a).

Another approach to narrowband approximations takes an outset in the pole and zero constellation. With high Q-values, the poles are close to the imaginary frequency axis. If we limit our scope of investigation to the region around the upper pole, for instance the encircled region in Fig.17, the distances to the other pole s2 and the zero s0 are long. As indicated by ω ω the figure, their contributions to the impedance are taken constant, 2j 0 and j 0 respectively, so the impedance becomes,

(46)

J.Vidkjær II-3 Narrowband Approximations 17

Using the pole position estimates from Eq.(17)(c) and inserting the Q -factor from Eq.(1), the approximation along the imaginary axis s=jω it calculated to yield

(47)

As seen, this is the same impedance approximation that was obtained in Eq.(44) on basis of the Taylor expansion of β(ω). So it is concordant to approximate the impedance function by the last parts Eq.(46) in s-plane calculations or by Eq.(44) along the jω axis.

Simplifications by the narrowband approximations for resonance circuits are beneficial in many computations. Before leaving the subject we shall, however, expand the scope beyond resonance circuits. The frequency dependency of a network is determined through the reactances of capacitors and inductors. Suppose we know a network and its corresponding frequency response H(s). This response may be transformed to another frequency range by mapping frequencies while preserving individual reactances everywhere in the circuit, a technique that is fundamental to filter design [1],[2],[3]. One such mapping - the only one we consider - is the lowpass to bandpass transformation given by

(48)

Here subscripts "lp" and "bp" are introduced to distinguish between lowpass and bandpass ω ω frequency planes. Along the imaginary axes where slp=j lp and sbp=j bp, the transformation agrees with the lowpass and bandpass relations in (45). Applying the transformation directly to circuit components maps inductors and capacitors in lowpass networks to series and parallel resonance circuits in bandpass, where we get

(49)

(50)

Fig.18 Lowpass-to-bandpass trans- formation of inductors and capacitors.

J.Vidkjær 18 RF-Circuits, Concepts and Methods

If the lowpass response of a network is known in form of a transfer or immitance3 function H(s)=H(slp), the corresponding bandpass expressions, which accounts for the components mappings, is the one obtained replacing slp by sbp. If the lowpass response is characterized by poles, zeros, or other characteristic complex frequencies, we must solve for sbp in terms of slp to get the corresponding bandpass properties, i.e.

(51)

A major reason to consider the lowpass to bandpass transformation is the fact that many common, renowned frequency characteristics4 are described as lowpass prototypes. To use them in bandpass circuits, we must use the lowpass to bandpass transformation. If the transformation is based on pole-zero patterns, Eq.(51) must be employed. A center frequency in bandpass, which is large compared to the bandwidth in translation, imply the simplified solution

(52)

If the approximation applies, we have narrowband conditions. Here, a lowpass pole-zero pattern around origo described by positions slp in the lowpass s-plane transforms to bandpass ω ω by copying two linearly scaled versions of the patterns, one centering at j 0 and one at -j 0. This is clearly much simpler than solving for the correct poles and zeros through Eq.(51). Example II-5-2 in the next section discusses the question further. Like other narrowband considerations the simplifications remain valid with poles and zeros close to the lowpass origin or the bandpass centers. Far apart the true solution must still be employed, in particular with respect to the zeros at infinity that are inherent properties of the lowpass characteristic. Zeros at infinity means that the power of slp in the transfer function denominator is higher than the power in the numerator. By Eq.(51) the lowpass plane point of infinity maps to two zeros in bandpass, one at infinity and one in origo. The latter are not encompassed by the narrowband approximation. It must be separately accounted for if the approximation method is used to realize bandpass circuits.

3 ) Immitance is a collective name for impedance and admittance.

4 ) Names like Butterworth, Chebyshev, Legendre, and Cauer or features like elliptic or equal-ripple are examples.

J.Vidkjær 19

II-4 Series-to-Parallel Conversions

Many practical situations include resonant circuits that are not ideal parallel or series circuits. Clearly, it is always possible to elaborate impedance or transfer function expressions in full details for the particular circuits in question. At single frequencies or under narrow- band conditions, however, a technique known as series-to-parallel conversion may greatly simplify the efforts to get useful results while keeping major insight on the circuit perfor- mance. Due to the frequent - but often tacit - use of the method in technical literature and data sheets, its foundation is presented here in some details.

Fig.19 and Fig.20 show a series connection and a parallel connection of a resistance and a reactance. The resultant impedances and admittances are given in Eqs.(53),(54) and (55),(56) respectively.

(53)

(54)

Fig.19

(55)

(56) Fig.20

Forcing agreement between the two admittances gives the series-to-parallel conversions, i.e. resistance and reactance conditions by which the parallel connection in Fig.20 may replace the series connection in Fig.19.

(57)

Similarly, equating impedances gives the parallel-to-series conversions,

J.Vidkjær 20 RF-Circuits, Concepts and Methods

(58)

If reactance is the dominating contribution to the impedance in consideration, that is either Rs |Xs| in series or Rp |Xp| in parallel connections, the two types of conversion simplify to the approximations

(59)

The last expressions are the ones most commonly associated with the concept of parallel-to- series conversion. In this form they are particularly easy to memorize due to the symmetry of converting back and forth between series and parallel representations. The following example illustrates the technique of using the series-to-parallel method directly in design.

Example II-4-1 ( impedance matching )

Fig.21 A power transistor of known input impedance should match a 50Ω generator at 470 MHz using the circuit in Fig.21, where biasing components are left out. Inductor L is fixed and C1, C2 are trimmer capacitors. Find the trimmer settings and estimate the half power bandwidth of transfer to the transistor.

Fig.22

To solve the problem we consider the equivalent circuit in Fig.22. A basic requirement for matching is Rp =Rg, so the mapping of Rin to parallel form through the combined series reactance Xs must equal Rg. When the required Xs is found, capacitor C1 is adjusted to tune

J.Vidkjær II-4 Series-to-Parallel Conversions 21

out the corresponding parallel reactance Xp by setting X1=-Xp. Using terms from the figure and the simplified conversions from Eq.(59), where Xp=Xs ,weget

(60)

(61)

(62)

(63)

The value of Xs in Eq.(60) may seem marginal with respect to the prerequisites of the simplified method. Without any assumptions about sizes of impedances, Eqs.(57),(58) provide correspondingly

(64)

(65)

(66)

As seen only small changes follow from the more elaborate but correct conversions. To estimate the bandwidth for power transfer, we consider the network as a parallel resonance circuit and divide it into a capacitive and an inductive side as sketched in Fig.23. At the center frequency the whole available power from the generator is transferred to the transistor since there are no resistive losses in between. Therefore, the network determined above implies conjugated matching across the cut. The parallel resistance Rpp, obtained by converting Rin to parallel form through inductances only, must be equal to the parallel resistance, which

J.Vidkjær 22 RF-Circuits, Concepts and Methods

Fig.23 originates from Rg and converts through the capacitive side. In a parallel resonance circuit the Q-factor is the ratio of the total parallel resistance, here ½Rpp, over either the inductive or the capacitive reactances. Continuing the simplified approach from Eqs.(60) to (63), which here implies Xpp =XL +Xin,weget

(67)

Due to losslessness, the absolute level of the voltage transfer function at the center frequency is most easily calculated letting Rin consume the available power, i.e.

(68)

This quantity wasn’t actually asked for, but now it possible to compare our results with simulations as it is done in Fig.24. The fully drawn curve is calculated using capacitances from the simplified method from Eqs.(62),(63) while the dotted curve is based on Eqs.(65), (66). The observable consequence of the simplified method is a slight displacement of the

Fig.24

J.Vidkjær II-4 Series-to-Parallel Conversions 23 center frequency where matching is obtained. However, the differences in capacitance values between the two situations are easily compensated with trimmer capacitors.

One final point should be observed concerning the bandwidth estimation. The transistor impedance to be matched was taken from data-sheets, and the positive reactance part was treated by Eq.(66) like an inductor for bandwidth estimations. Here it is a tacit assumption that the input reactance does not change faster than an inductor across our frequency band, i.e. without sensible resonances from the transistor or its mounts. If this occurs, we have no means for estimating bandwidth, but the center frequency matching procedure remains valid, if the transistor data are reliable.

Example II-4-1 end

Conversions in Narrowband Applications

Fig.25 Conversion of a small inductor series resistance RLs to a parallel resistance RLp . ω Circuit (b) is a narrowband approximation to (a) around resonance frequency 0.

Reactances of inductors and capacitors depend on frequency so - strictly speaking - a series-parallel conversion applies to a single frequency. However, we could estimate the bandwidth of the resultant circuit in the example above, a consequence of the fact that in narrowband circuits, component values obtained by series-parallel conversions at the center frequency are useable throughout the passband. To see this from an analytical point of view we consider the example in Fig.25. Inductor L is employed in a parallel resonance circuit, but it has a small series resistance RLs, so the resonance circuit is no longer ideal. To deal with this circuit easily, RLs is converted to the parallel resistance RLp and parallel resonance methods are used on the circuit in Fig.25(b). It is supposed that the series resistance is much ω smaller than the reactance Xs =L 0, so the simplified relations from Eq.(68) are used including the fact, that the inductor reactance remains unaffected of the conversion. Therefore, ω resonance frequency 0 is figured out the usual way and the corresponding Q-factor and band- width follows from the sequence,

(69)

J.Vidkjær 24 RF-Circuits, Concepts and Methods

Notice that the contributions from the two resistors to the resultant Q-factor are distinguishable through a "parallelling" relationship

(70)

(71)

It is not self-evident that the circuit in Fig.25(b) with frequency independent RLp may substitute the original circuit across a passband. To convince, the two impedance functions must be compared. The ideal parallel circuit is given by Eq.(14),

(72)

Casting the impedance of the original circuit in Fig.25(a) into a comparable form we get

(73)

Identification of Q-factor through the first order denominator term gives

(74)

which agrees with the subdivision of Qtot in Eq.(71), provided that the resonance frequencies ω ω m0 and 0 are equal. But this is not so unless the resistors meet the condition

(75)

Implicitly the inequalities express a narrowband assumption. The result of a parallel connec- tion cannot exceed any of its components. If Qm=Qtot 1 we must therefore require

J.Vidkjær II-4 Series-to-Parallel Conversions 25

(76)

Under narrowband conditions the two circuits in Fig.25 will have the same poles because the impedance functions get the same denominators. Regarding numerators Eq.(74) shows that Zm(s) has a zero on the negative real axis compared to the zero in origo for Zp(s). Therefore, the impedance Zm reduces correctly to the parallel combination RLs Rxl at dc while the inductor short-circuits in the narrowband approximation.

(77)

Observe, however, that the difference in zeros between Zm and Zp is an asymptotic deviation outside the scope of the narrowband assumption. If the response is dominated by poles, which ω is the case in the passband, the parallel circuit obtained by series-to-parallel conversion at 0 is usable to the same degree of confidence that accompanied other types of narrowband approximations.

Parallelling of Q-factor contributions has wider implication than just being a vehicle in the previous discussion. Imperfections in reactive components that cause loss of power may be specified in data sheets or measurements by associating a quality factor directly to the component. In agreement with Eq.(71), the Q-factor is either the ratio of the reactance over a series loss resistance or the ratio of a parallel loss resistance over the reactance. With higher Q´s, say three or more, it makes no difference for narrowband computations whether the physical loss originates in series, parallel, or combined connection, we convert to the form most suited for the problem at hand through the approximate conversion from Eq.(59),

(78)

The combination of Q-factors is sometimes expressed in a terminology of loaded, unloaded, and external Q´s, which follows the paralleling expression, cf.[4] sec.7.1,

(79)

Subscript "unloaded" refers to unavoidable loss components in the reactances that make the resonator and "external" to all other resistors. The resultant "loaded" Q determines the bandwidth around resonance and is always smaller than any component Q-factor. In this terminology the example from Fig.25 and Eq.(71) should read

(80)

J.Vidkjær 26 RF-Circuits, Concepts and Methods

Conversions in Broadband Modeling

In narrowband applications of reactive components it suffices to specify its reactance and Q-factor as a function of frequency. Then we may use Eq.(78) and the series-parallel conversion around the center frequency to make design calculations. On the contrary charac- terization of reactive components for broadband application, including modelling by frequency independent circuit elements, requires detailed equivalent circuits. To find the models we may still benefit from the series-parallel relationships. The following example illustrates this aspect.

Example II-4-2 ( spiral inductor )

Fig.26 Spiral inductor forward y-parameters. Reverse y12 and y22 are similar. Data are converted from s-parameter measurements. g21´s are uncertain at high frequencies.

Data from an inductor in a GaAs microwave integrated circuit are shown in Fig.26. The layout is a planar spiral of the type in Fig.27 and the measurements are made with the inductor connecting an input and an output port. Due to small dimensions, thin metal and isolation layers, the inductor is far from being ideal. We wish to find an equivalent circuit based on the measurements. Like other integrated passive component a Π-structured model is expected, so it is natural to translate the experiments to y-parameters. The definitions in forward measurements take form of input and transfer admittances with shorted output as summarized by Fig.28.

Fig.27 Spiral inductor layout example. Fig.28 Definition of y-parameters y11,y21.

J.Vidkjær II-4 Series-to-Parallel Conversions 27

Fig.29 Spiral inductor equivalent circuit where (a) suffices below the resonances that are included by (b). The shorts at the outputs are required to interpret y-parameters.

Fairly below 10 GHz the data show nearly ideal asymptotic behavior, where imaginary and real parts are inversely proportional to frequency or squared frequency respectively. The latter indicates that the dominant inductor loss is a series loss. To see this we convert the simple equivalent circuit in Fig.29(a) to y-parameter parallel form through

(81)

The real conductance component shows the observed second order relationship. Extrapolating the declining asymptotes to the frame of the figures at 31.62 GHz gives

(82)

The series resistance is high and troubles many designs using integrated spiral inductors. At 5GHz, for instance, the experimental data ( dots in Fig.26 ) are close to the maximum Q- factor, which develops

(83)

Above 10 GHz the data indicate resonances in both y-parameters. The susceptances of the inductance are here canceled by stray capacitors, at f01=11.52 GHz in y11 and at f02=18.16 GHz in y21. To include resonances the model is enhanced to Fig.29(b), which now gives

(84)

J.Vidkjær 28 RF-Circuits, Concepts and Methods

(85)

(86)

A final look at the measurements shows a distinct asymptote in the real part of y11 above resonance f01. The asymptote increases in proportion to the squared frequency, which might be the effect of a small resistance in series with a capacitor. It is therefore tempting to make

Fig.30 Complete y11 model and the elements that are sensed above resonance. one more step in the modeling. Considering y21, the real parts of the experimental data are too noisy and uncertain for further identifications, and we concentrate on y11 as shown in Fig.30. Now the asymptote for g11 at the frame of the data implies

(87)

Including the last extension, the equivalent circuit for the spiral inductor takes the shape of Fig.31. The y-parameters, which this model accounts for, are shown in Fig.32. The only observable discrepancies to the measurements are in g21 above resonance, where the experi-

Fig.31 Complete equivalent for the spiral inductor measurements.

J.Vidkjær II-4 Series-to-Parallel Conversions 29 mental data are badly conditioned. In view of literature on GaAs IC design, for instance [5], the model we have constructed from nothing but basic knowledge on series-parallel transformations and resonance circuits is rather complete.

Fig.32 Simulated verification of the complete spiral inductor equivalent circuit from Fig.31. The corresponding experimental data were shown in Fig.26.

Example II-4-2 end

J.Vidkjær 30

II-5 Tuned Amplifiers

Fig.33 Functional and simplified equivalent circuit of a single-tuned amplifier.

Resonance circuits are used to shape the frequency response of frequency selective amplifiers. Fig.33. shows an example where a bipolar junction transistor is loaded by a parallel resonance circuit. A simple transistor equivalent circuit is employed to keep the amplification function uncomplicated. Using Eq.(45) the voltage gain v2/v1 is expressed

(88)

The transistor output capacitance and conductance add to the external tuning circuit compo- nents to give center frequency amplification A0 and bandwidth,

(89)

Note, the input resistance and capacitance in the transistor model have no direct effects on these expressions, but play the role of loads if more stages are cascaded. The frequency ω response, which shapes like the parallel circuit impedance function Ztot(j ), may be calculated either fully correct or narrowband approximated by the frequency expressions

(90)

Recall from the discussion on the narrowband approximation in section II-3 that the two forms differ with respect to bandlimits but not bandwidths.

To compare frequency selective amplifiers, two figures of merits are the gain-band- width product and the gain-bandwidth factor. The product is defined as the center frequency voltage gain times the 3dB bandwidth,

(91)

J.Vidkjær II-5 Tuned Amplifiers 31

With more stages the GW-product commonly refers to an average gain per stage, i.e. 5

(92) where N is the number of tuned stages. The single stage amplifier above has

(93)

This product is used as a reference for the gain-bandwidth factor, GBF, which is defined by

(94)

It is supposed, that the transconductance and the total loading capacitance are the same in both numerator and denominator. Transconductance gm is a property of the transistor. In the lower limit Ctot holds the transistor output capacitance and other unavoidable contributions from mounts and loads. Therefore, GW is often used as a goal for optimizing or comparing transistor and IC performances. The normalized GBF is suited for comparing amplifier structures without detailed regards to the devices that give the gain.

There exists a series of fundamental theorems concerning GBF’s of interstage connections between amplifier stages, first introduced by Bode [6]. Their derivations and detailed interpretations are beyond the present scope and needs. Briefly, when one amplifier stage is connected to the next, it is stated that GBF cannot exceed two with a passive one-port paralleled across the connection. A passive two-port interstage coupling has a theoretical maxi- mum of four. Networks representing the maxima are complicated and of little practical use with the high gain RF transistors of today. However, it is still informative to use GBF in comparisons between different amplifier structures even when they fall considerably below the theoretical limits.

An amplifier stage holding only one resonance circuit is called a single-tuned amplifier. With more resonance circuits separated by transistors there are many possibilities on how to organize their center frequencies and Q-factors. The simplest case is that all tuning circuits have identical resonance properties. This well-defined situation is called synchronous tuning, as opposed to a more diversified group of so-called stagger tuned amplifiers.

5 ) There are many competing definitions and notations around. Bandwidth in units of radians per second is denoted GW here, while GBW is used if bandwidths are in units of Hz. The term MGBW - "M" for mean - may be seen instead of GWav.

J.Vidkjær 32 RF-Circuits, Concepts and Methods

Synchronously Tuned Amplifiers

Cascading N equally tuned amplifier stages of the type from Fig.33 gives a voltage amplification of form,

(95)

Ω The center frequency amplification AN0 at = 0 is the product of the center frequency amplifications in each stage. In normalized frequency a single stage amplifier has 3dB bandlimits corresponding to Ω=±1. With N equally tuned stages we must solve for the Ω’s that reduces the absolute amplification by a factor of 2, that is

(96)

Without normalization with respect to frequency, the result gives the bandwidth of N stages in terms of one stage through

(97)

where the gain-bandwidth factor GBFN in this particular application also is called the bandwidth reduction or bandwidth shrinkage factor. Examples of the reduction process are

Fig.34 Normalized amplitude and phase characteristics in synchronous tuning. Gain-band- width factors GBF2, GBF3 give the bandwidth reductions for 2 and 3 stages.

J.Vidkjær II-5 Tuned Amplifiers 33 seen in the characteristics of Fig.34. The amplitude of N equally tuned stages rolls off approaching an asymptote of -N×20dB/decade. The stronger the bending of the corresponding curve, the smaller becomes the bandwidth. Since 21/N → 1 with growing N , the reduction factor may be approximated from the following estimations,

(98)

True and estimated bandwidth reductions are compared below in Table I. Whenever more stages are cascaded, the approximation gives reasonable results.

Table I True and approximated Gain-Bandwidth Factors ( Bandwidth Reduction Factors ) for N Synchronously tuned amplifier stages. ∼ ∼ N GBFN GBFN N GBFN GBFN 2 0.6436 0.5893 5 0.3856 0.3727 3 0.5098 0.4811 6 0.3499 0.3407 4 0.4350 0.4167 7 0.3226 0.3150

Example II-5-1 ( synchronous tuning )

Fig.35 Amplifier principle, (a), and transistor equivalent circuit, (b).

Ω The amplifier in Fig.35(a) should operate with Rg=RL=75 having a total bandwidth in synchronous tuning of BWamp=55 MHz around f0=460 MHz. Transistor data are (99)

Find the external components C0,L0,C1,L1, and the center frequency gain v2/Eg.

J.Vidkjær 34 RF-Circuits, Concepts and Methods

Fig.36

Including the transistor model, a complete functional equivalent circuit for the amplifier becomes the one shown in Fig.36 where the generator is changed to the Norton equivalent.

The two tuned circuits have equal bandwidths BW0 and quality factors Q0. Compensating for bandwidth reduction, we get

(100)

At the input side, the parallel resistance R00 determines the impedance level and in turns give the external components,

(101)

The corresponding calculations at the output side of the transistor read,

(102)

The inductances we have found here approaches the lower borderline for practical lumped inductors. A guideline is that unwounded wires like component leads have inductances about 0.5nH/mm to 1nH/mm, a result that is verified in chap.4 of ref.[7]. The smallest inductor available as a commercial component for surface mounting on PC-boards is presently 2nH.

J.Vidkjær II-5 Tuned Amplifiers 35

The impedances of the tuned circuits are equal to the parallel resistances at the center frequen- cy, so the voltage gain becomes

(103)

The frequency characteristic of the amplifier is shown later in Fig.43.

Example II-5-1 end

The phase characteristic in synchronous tuning with N stages may be written

(104) where the last equation introduces the Taylor expansion of tan-1(Ω). Instead of using Ω, which is normalized with respect to the 3dB bandwidth of a single stage, we normalize frequency with respect to the approximate 3dB bandwidth from Eq.(98) for the N stages in consideration. The new frequency variable becomes

(105)

Ω Inserting N into the phase characteristic above gives

(106)

As N raises, the nonlinear third, fifth, and higher order terms loose significance compared to Ω the linear term. With more stages, the phase characteristic in the 3dB bandwidth | N|<1 becomes more linear or - equivalently - the group delay stays more constant. The correspond- ing amplitude characteristic approaches a Gaussian curve in the passband. To see this we rewrite the amplitude expression using the fact that, cf.[8] p.228,

(107)

As bandwidth in measures of Ω reduces, the limit value is suited for large N, where we get

J.Vidkjær 36 RF-Circuits, Concepts and Methods

(108)

The exponential gives a Gaussian characteristic due to the squaring of the frequency variable, and the last version shows readily that the scaling from Eq.(105) gives 3dB limits with Ω N=±1.

Butterworth Stagger Tuned Amplifiers

Each stage in a chain of amplifier stages provides a pole zero pattern corresponding to its load impedance. Synchronous tuning let the poles and zeros from all stages coincide. Staggering the pole-zero patterns gives provisions for a variety of amplification characteristics. We shall demonstrate the concept of stagger tuning by picking a simple but common case, the family of Butterworth characteristics of various order. They are defined by requiring maximal flatness in magnitude, which means that derivatives with respect to frequency up to the order 2N-1 are zero in the center at Ω=0 6. The normalized N-th order Butterworth function is

(109)

Examples of the magnitude characteristics are given in the upper part of Fig.37. The 3dB bandlimits corresponding to Ω=±1 are independent of order, but as N increases the roll-off from the passband becomes more and more abrupt.

To find the pole patterns that give the normalized Butterworth characteristics, we start considering the squared magnitude,

(110)

Tracing back from jΩ along the imaginary axis to the corresponding normalized complex frequency variable S implies the substitutions

(111)

(112)

6 ) These and related concepts are part of the approximation problem in filter design. Consult ref´s [1], [2], or [3] for further details including the multitude of cases that are left out in this text.

J.Vidkjær II-5 Tuned Amplifiers 37

Fig.37 Normalized Butterworth characteristics of orders up to N=5. The phases are deter- mined from pole positions like Fig.39.

Solving for the S values that set the denominator equal to zero provides poles for the squared magnitude

(113)

The poles are confined to the unit circle where they are equally spaced with an angle of π/N. When index k runs from 0 to 2N-1, all poles positions are covered once in the sequence sketched by Fig.38.

2 Fig.38 Pole patterns in N-th order Butterworth squared magnitude |BN(S)| .

J.Vidkjær 38 RF-Circuits, Concepts and Methods

The Butterworth amplitude characteristic applies to lowpass as well as bandpass amplifiers. Both types may be realized using tuned LC circuits as interstage networks. The lowpass case ease the discussion because we transform from the normalized frequency Ω to ω the lowpass simply by multiplying with the bandwidth W3dB. Thus, to represent a causal lowpass system, the basic requirements of having even real and odd imaginary parts along the imaginary frequency axis are met by imposing the condition

(114) so the squared amplitude function may be rewritten

(115)

2 By the last factorization, poles of |BN| having positive real parts can be ascribed to BN(-S). This assures that the transfer function BN(S), which is targeted in circuit design, fulfills a necessary requirement of stability, as it only collects the N poles in the left half-plane. They were given by the first N indices in Eq.(113), in summary

(116)

Pole positions in the second, third, and fourth order normalized Butterworth functions - also called prototype functions - are detailed in Fig.39.

Fig.39 Examples of pole positions on the unit circle in normalized prototype Butterworth functions BN(S).

ω Poles sp,k in a bandpass amplifier of center frequency 0 and bandwidth W3dB are related to the prototypes in Eq.(116) through the lowpass to bandpass transformation intro- duced in section II-3. Without any assumptions they provide

(117)

J.Vidkjær II-5 Tuned Amplifiers 39

ω Under narrowband conditions, where W3dB/ 0 1, the square root is dominated by its second term and the poles are approximated

(118)

Thus, in narrowband we scale the prototype to half the desired bandwidth and copy the pole ω pattern from its lowpass center in origo to the bandpass centers at s=±j 0. This process is demonstrated for a third order function in Fig.40.

Fig.40 Mapping of Butterworth prototype poles (a) to poles in a narrowbanded bandpass amplifier (b). In (c) the poles are paired for stagger tuning that also requires three zeros in origo.

To realize the third order Butterworth characteristics exemplified by Fig.40(b), we need three basic parallel tuned stages. The idea of stagger tuning is to adjust the load of each stage to a required pole pair as indicated by Fig.40(c) and Fig.41. Fortunately, the parallel resonance circuits insert the necessary zeros at origo. As discussed in section II-3 they are not automatically encompassed by narrowband techniques. The sequence of pole pairs is arbitrary, but if we choose to let stage k realize the pole pair define by sp,k in Eq.(118), its resonance ω frequency 0k and quality factor Qk are given by,

Fig.41 Three stage tuned amplifier. By stagger tuning the pole pair of each resonance circuit corresponds to at pole pair in the transfer function like the example in Fig.40(c).

J.Vidkjær 40 RF-Circuits, Concepts and Methods

(119)

Pole positions determine the shape of the frequency response but not the absolute amplifi- ω cation level. The quantities 0k,Qk constrain the tuning components by

(120)

It is a design decision to set the absolute impedance level. If the total amplifier is narrow- ω banded in the sense of W3dB/ 0 1, each stage must be narrowbanded. By the approximation from Eq.(46) the gain of stage k, which realizes pole pair sp,k becomes

(121)

The last rewriting is based on Eq.(118) and refers back to the prototype function poles on the ω ω unit circle. At the center frequency = 0, stage k has the voltage gain

(122)

In a complete chain of stages for a Butterworth characteristic, there is a real, negative center frequency factor if the order is odd. The Sp,k’s in the remaining stages appear in complex conjugated pairs, so the center frequency gain of a N-th order stagger tuned Butterworth amplifier is given by

(123)

To elaborate further, the number of stages and an impedance strategy must be known. Consider as an example a third order amplifier where all transistor transconductances and load resistances are equal, gm0=gm1=gm2=gm and R0=R1=R2=Rp respectively. We observe from the third order prototype in Fig.39 that the k=1 stage tunes to the center frequency. Using Eq.(119) we get

(124)

The real values of poles with k=0 and k=2 are half the size of the k=1 pole, which gives

(125)

J.Vidkjær II-5 Tuned Amplifiers 41

Knowing all capacitances, Eq.(123) provides the center frequency gain in this particular setup

(126)

To estimate gain-bandwidth factors if loading capacitances differ among the stages, the one with the smallest capacitance, i.e. the greatest single-stage GW, is commonly chosen as the reference. In the present amplifier this is clearly the C1 stage, so we get

(127)

Example II-5-2 ( Butterworth amplifier )

By this example we change the components of the synchronously tuned amplifier from Example II-5-1. Keeping bandwidth, center frequency, and impedance level specifications, the frequency characteristic should now be of 2nd order Butterworth type. The task is to get the new component values and calculate the center frequency gain.

First we find the required pole-positions. Assuming narrowband condition we consider the pattern in the upper half-plane, Fig.42, where simple geometrical reflections give the coordi- nates of the poles.

Fig.42

Associating the upper pole with the input circuit, the Q-factors of both tuned circuit are

(128)

Using resistance figures from Example II-5-1, the tuning components now become

J.Vidkjær 42 RF-Circuits, Concepts and Methods

(129)

(130)

Recalling the discussion in the previous example, we are very close to practical bound on small inductances. Note, however, that Example II-6-4 shows one way to transform so compo- nent values stay practical. To find the center frequency gain Eq.(123) is used directly,

(131)

Compared to synchronous tuning, Eq.(103), maximal flatness halves the voltage gain.

The synchronously tuned amplifier from Example II-5-1 is shown with the present stagger tuned Butterworth amplifier in Fig.43. The curves are simulated results from the equivalent

Fig.43 Simulated voltage gain magnitude, phase, and group delay for the synchronously and stagger tuned amplifiers in Examples II-5-1 and 2. Narrowband approximated circuit data are used.

J.Vidkjær II-5 Tuned Amplifiers 43 circuit in Fig.36 without further assumptions. It is therefore worth noticing how close we come to the specifications regarding gain and bandlimits, although all underlying component calculations were based on narrowband assumptions. The most visible consequence of the simplifications is the lack of complete flatness in stagger tuning. Had the poles been correctly found by solving Eq.(117) instead of the simpler scaling and copying approach in Eq.(118), the result improves as will be seen below. From a design point of view it is dubious to go further analytically. Practical component values need fine-tuning, either physically or by circuit optimization, to compensate other design simplifications, for instance the employment of uncomplicated transistor models or the ignorance of parasitic elements in the lay-out. The simple narrowband methods give a good starting point for this process. However, in the present context we shall enlighten the assumptions and approximations behind narrowband methods whenever possible and therefore compare the previous result with their true counter- parts. Applying prototype poles to Eq.(117) gives

(132)

Realize that by this calculation we give up the narrowband assumptions in the complete ω ω Butterworth amplifier characteristics. To identify Q00, 00 or Q11, 11 from the pole positions, narrowband assumptions about the individual stages are maintained. With Q-factors over 10, Eq.(43) shows, that this is still a very satisfactory assumption. Having established the Q-factor and resonance frequencies for the two stages, computations similar to Eqs.(129), (130) lead to new components values,

Fig.44 Simulated voltage gain magnitude, phase, and group delay for stagger tuned amplifier in Examples II-5-2. True pole positions are used to calculate circuit data.

J.Vidkjær 44 RF-Circuits, Concepts and Methods

(133)

Fig.44 shows the simulated frequency response obtained by these data, and clearly the flatness in magnitude has improved. The reason why group delay no longer gets symmetric appearance is the fact, that the Butterworth characteristic is symmetric in the normalized frequency Ω, not in ω with respect to which, the phase was differentiated.

Example II-5-2 end

J.Vidkjær 45

II-6 Transformers and Transformerlike Couplings

Needs for transforming signal and impedance levels in RF circuits are both frequent and diversified, so a multitude of approaches and techniques are available for solving that sort of problems. Among them are the conventional magnetically coupled transformer, which is a highly useful component at frequencies up to approximately 3 GHz. We shall consider transformers and circuits that behave similarly in some depth in this section. The scopes are to gain basic understanding of advantages and limitations of transformer couplings, and to provide a background for setting up simulator models. Some simulation programs are sparsely equipped with the coupled inductor and transformer models that are needed in RF-design, so they must be build from basic circuit functions and components.

Review of Mutual Inductances

Two or more inductors have mutual inductance if they interact through their magnetic fields. To fix ideas we start considering two inductors where - as shown in Fig.45 - currents are separately applied. Superposition of the two conditions gives the following expressions for Φ Φ terminal voltages and the total magnetic fluxes 1 and 2 through the inductors,

(134)

Fig.45 Inductors with mutual inductance. Surfaces for the flux integrals are bounded by the coils and lines through the terminals. Dots indicate orientation. Current entering a dotted terminal support flux in the other coil.

The flux integrals condense in the inductances L1,L2, and the mutual inductances M12,M21. They are factors of proportionality between inductor currents and the different flux contributions. The factors are the elements of an inductance matrix, which finally gives the two-port impedance matrix for two coupled inductors,

(135)

J.Vidkjær 46 RF-Circuits, Concepts and Methods

Including more than two inductors, the equations are generalized to

(136)

Here v, Φ, and i are vectors holding the port voltages, fluxes, and currents respectively, while L and Z are the inductance and impedance matrices. The inductance matrix is assumed to be symmetric and positive definite or semidefinite. The first property expresses reciprocity, which apply if the inductors have isotropic surroundings. It evolves from the renowned Lorentz reciprocity theorem, [4] sec.2.12,4.5 , and causes the total magnetic energy to depend only upon the instant current vector, no matter how it behaved in the past. The second property is a passivity requirement ensuring that starting from zero initial conditions, the total stored magnetic energy will always be positive or zero, [9] chap.15. In mathematical terms, the requirements are that all subdeterminants of L, which can be taken symmetrically around the diagonal, must be positive or zero, cf.[10] sec.7.2.

With two coupled inductors, the reciprocity and passivity conditions become

(137)

(138)

The limit cases of L1=0 or L2=0 implying M=0 are of no practical interest. Mutual inductanc- es are often expressed by the coupling coefficient k,

(139)

In circuit schematics, mutual inductances may be represented as shown in Fig.46(a,b). If we start fixing the orientations of the two ports, the sign of M and k is implicitly determined. Applying a positive current to port one, M and k are positive if both port voltages are in phase, negative if they are 180° out of phase. To emphasize phases rather than the more arbitrary port orientation, the dot-convention, which defined flux directions in Fig.45, will also tag a set of terminals that gives in-phase port voltages. Using port orientations opposite the dot indication implies negative mutual inductances or couplings as shown in the figure.

Fig.46 Inductors with (a) positive and (b) negative mutual inductances following the dot- convention from Fig.45. The signal flow-graph represents the z-parameter matrix.

J.Vidkjær II-6 Transformers and Transformerlike Couplings 47

The borderline to passivity, |k|=1, is called tight or close coupling and is one goal aimed upon when mutual inductances are employed in transformers. With k=±1 the two voltage equations that stem from Eq.(135) become

(140)

However, the inductance matrix is singular in this case, so the two equations are linearly related, and the lower equation follows from the upper one

(141)

The square root of the inductance ratio is called the winding or turns ratio N, because the inductances of typical transformer coils are proportional to the squared number of windings, cf.[7] sec.4.8. Equation (141) expresses the transformer voltage relationship

(142)

To share and confine the magnetic field, the inductances are often wound on a core of high permeability material, for instance a toroide like Fig.47. The phrases of tight or close cou- plings refer to the fact, that the two inductors must encompass the same magnetic flux when |k|=1. Suppose the two coils are similar so they have the same ratio AL between inductance and the squared winding count,

With k=±1, the flux per winding in L2 that originates from current i1 equals in size the flux per winding in L1,

(144)

Fig.47 Toroidal transformer where both inductors share the same magnetic field.

J.Vidkjær 48 RF-Circuits, Concepts and Methods

This result means that the two coils encompass the same magnetic flux when driven from the primary port (no.1). Due to reciprocity this will also be the case if the current is applied to the secondary port. With k less than one, part of the flux from L1 leaks outside L2 and vice versa so the pictures tend towards Fig.45. We shall see later that flux leakage deteriorates the transformer frequency response.

Two tightly coupled inductors are sometimes called a perfect transformer, cf.[10] pp.44. It is a step towards the more familiar ideal transformer, and should not be confused with it. From the lower equation in (135) we have

(145)

Letting the two inductances simultaneously rise towards infinity - keeping the winding ratio constant and the voltage time integral limited - provides

(146)

(147) which is the ideal transformer current condition. It is more restrictive than Eq.(142), as it requires infinitely large inductors, while the voltage condition only assumes tight coupling. In combination the voltage and current conditions define the ideal transformer. It has only algebraic constraints between the terminal currents and voltages, and it is solely specified by the winding ratio N. The ideal transformer is symbolized as shown in Fig.48(b), where the two lines between the inductors indicate a hypothetical high permeability core that would be necessary to approximate an ideal transformer in practice. Be aware that there is no conven- tion on the side taken as unit reference for N although we often use port one, the primary side, as reference.

Fig.48 Progressing form tightly coupled inductors in (a) to the ideal transformer in (b) if the inductors approach infinity with a constant winding ratio N.

J.Vidkjær II-6 Transformers and Transformerlike Couplings 49

Equivalent Circuits for Two Coupled Inductors

Instead of a circuit diagram that indicates mutual inductance, it is often more informative to represent a pair of coupled inductors by an equivalent circuit, where the inductors are uncoupled. The impedance matrix from Eq.(135) directly suggests the T-form in Fig.49(a). However, this circuit has no DC separation between the two inductors. If it is important to portray this property in a circuit diagram, the enhancement by an ideal 1:1 trans- former as shown in Fig.49(b) is straightforward.

Fig.49 T-equivalent modeling two coupled inductors by three uncoupled inductors. Dia- gram (a) suffices for dynamic responses. The ideal transformer in (b) emphasizes DC separation between the inductors.

Including an ideal transformer, the circuit in Fig.49(b) is one particular choice among all possible T-equivalent circuits for two coupled inductors. To see this we consider the T- circuit and the corresponding z-parameter signal flow-graph in Fig.50.

Fig.50 T-equivalent circuit and a signal flow-graph for calculating z-parameters. Forcing the graph to provide the original z-parameters from Fig.46(c) yields

(148)

Here, three equations constrain four unknowns, Lx,Ly,Lz, and the effective winding ratio n, so one condition must be fixed initially. Above in Fig.49(b) n was set to one. To account for

J.Vidkjær 50 RF-Circuits, Concepts and Methods nearly perfect transformers it is more natural to get an n value close to the physical winding ratio, which is achieved choosing Lx=0 or Ly=0. To solve for the first requirement we use

(149)

and the resultant circuit becomes the one in Fig.51(a). The circuit in Fig.51(b) has Ly=0, which gives

(150)

In both cases n comes close to the physical winding ratio, which follows the square root of the inductance ratio, when the coupling tightens towards k=±1. In the limit, furthermore, the series inductances from Lx and Ly approach zero leaving alone the primary inductance L1 across the input port. Therefore, the T-type of equivalent circuit is especially suited for tight coupling calculations in transformer designs.

Fig.51 Equivalent circuits suitable for tightly coupled inductances. For |k| → 1, the effec- tive winding ratio n approaches the physical winding ratio and, simultaneously, the series spreading inductances disappear.

Fig.52 Equivalent circuits suitable for tightly coupled inductances. The circuits are similar to the cases in Fig.51, but now the ideal transformer is moved to the primary side.

J.Vidkjær II-6 Transformers and Transformerlike Couplings 51

Had we from the outset placed the ideal transformer on the primary side, the resultant equivalent circuits would be the ones in Fig.52. The parameters follow from mirroring previous results by exchanging terms and inverting winding ratios.

An alternative to the T-equivalent circuit is the Π diagram in Fig.53. This circuit structure is most conveniently described by y-parameters, so the component values are calculated from term by term comparisons, inverting the original z-parameters to an admit- tance matrix, i.e.

(151)

(152)

The last equation shows that the Π-diagram precludes tight coupling, because the denominator determinant becomes zero if k=±1. For |k|<1, component values are calculated,

(153)

(154)

(155)

Fig.53 Π-equivalent for two coupled inductors where the DC separation is emphasized in (b). Eqs.(153) to (155) give components in terms of original parameters.

J.Vidkjær 52 RF-Circuits, Concepts and Methods

The results are reasonable as in the limit |k|→0, the two parallel inductors go towards the → → inductances prior to coupling, La L1,Lb L2, while the connecting inductor Lc and its pertinent impedance rise towards infinity and uncouples the two sides. Therefore, the Π-type equivalent circuits are often preferable when dealing with loosely coupled inductors.

Fig.54 Π-equivalent circuit and a signal flow-graph for calculating z-parameters.

Like the T-circuit, the diagram in Fig.53 is one particular choice where the winding ratio n is set to one and the remaining three unknowns La,Lb, and Lc have been solved for. To get expressions for other possibilities, the easiest way is now to repeat the process from the T-circuit, i.e. make a z-parameter signal flow-graph and identify terms. The graph in Fig.54(b) holds two loops that give a denominator D different from one. We get

(156)

A particular solution, which is useful for considering the impedance conversions through loosely coupled inductors, emerges if the two parallel inductors are set equal to each other. Here a solution process leading to the result in Fig.55 is

(157)

J.Vidkjær II-6 Transformers and Transformerlike Couplings 53

Fig.55 Π-equivalent circuits with equal parallel inductances. The circuit is suitable for loosely coupled inductances.

Observe, that the effective winding ratio in the ideal transformer is independent of the coupling coefficient k and equals the physical winding ratio N of the two coils. Even with loose coupling it is easy to incorporate the effect of secondary side loading in computations. The coupling coefficient in terms of the Π-equivalent circuit follows directly from Eq.(156),

(158)

The last equivalent circuit for two inductors with mutual inductance we consider is shown in Fig.56(a). Here the coupling is accounted for by two voltage controlled voltage sources. The method is verified by identifying paths in the signal flow-graph compared to the z-parameters in Fig.46(b). This diagram may be useful in simulation programs that supports controlled sources without including dedicated models for coupled inductors.

Fig.56 Equivalent circuit for inductors where the coupling is simulated by voltage con- trolled voltage sources.

RF-Transformers

The most prominent use of two coupled inductors is the transformer, where the aim usually is to get a coupling coefficient as close as possible to unity. If, first, the reactances of the inductors are high compared to any connected impedance, the transformer approximates the ideal one. Besides the current and voltage constraints from Eqs.(142),(147), impedances are transformed as indicated by Fig.57,

J.Vidkjær 54 RF-Circuits, Concepts and Methods

Fig.57 Ideal transformer with terminal voltage and current conditions and the correspond- ing input and output impedances.

(159)

An ideal transformer was included in nearly all the equivalent circuits above, so these relations are constantly used.

Ideal transformers are independent of frequency, but there are both upper and lower frequency bounds in practical transformers. Direct causes are the finite inductances and less than unit coupling, which were accounted for in all the equivalent circuits of Fig.51 or Fig.52. If a transformer is supposed to match load resistances RL to generator resistance Rg, the winding ratio must be chosen to transform RL to RL´=Rg, when it is seen from the generator side of the transformer. Using the equivalent circuit from Fig.51(b), the frequency response is described by the diagram in Fig.58(a). The low frequency 3dB limit fl is determined by the shunting effect of inductor Lp across RL´. The high frequency bound, fu, is due to the voltage division between the leakage inductance Lx and the ohmic generator and load impedances. If the coupling coefficient is still close to one in magnitude, the two frequencies may be estimated directly from the equivalent circuit if Lx is of no significance at frequency fl and Lp of no significance at frequency fu, i.e.

Fig.58 Effects of parallel and leakage inductances in non-ideal transformer. The response is the insertion loss, i.e. the deviation of v2´from the nominal output Eg/2. Note the scale direction.

J.Vidkjær II-6 Transformers and Transformerlike Couplings 55

(160)

(161)

Below fl and above fu the transfer characteristic rolls off towards 20 dB/decade asymptotes as seen in the figure. To get a good high-frequency response, the coupling should be as tight as possible, and to get a good low-frequency response, the inductance should be as large as possible. It is difficult to satisfy the requirements simultaneously taking into account other practical obstacles like ohmic losses or winding capacitances, which we have disregarded here. The following example makes a brief account on the problems and demonstrates practical transformer data.

Example II-6-1 ( practical RF transformers )

Fig.59 Inclusion of other transformer imperfections than leakage. Cw accounts for inter- winding capacitances, Rws for winding series resistances, and Rcp for core losses.

Besides finite inductances and non-zero leakage contributions, a series of other secondary effects limits transformer performances. Fig.59 shows a common equivalent circuit holding first order approximations to an imperfect transformer. At RF-frequencies the additional circuit components are briefly explained as follows.

J.Vidkjær 56 RF-Circuits, Concepts and Methods

Fig.60 Extracts from Varil FP-518/FP-530 data sheets

Series resistance Rws comes from the inductor coils. At RF frequencies the skin effect dominates the resistance, so currents are concentrated to the conductor region from the surface down to one skin depth, [7] sec.3.16. Wound in coils the effect is worsened by the so-called proximity effect, where the total flux causes unsymmetrical current distributions around the conductor. The skin depth decreases with the square root of frequency, so resistance grows with the square root of frequency. Measures against skin depth series resistances are to use thick, sometimes silver plated wires; at lower frequencies also use of parallel isolated wires, known as litze wires, which have a high surface to volume ratio.

The parallel resistance represent losses from the core in case a high permeability material is used to improve coupling and raise inductances. At RF-frequencies it is eddy currents that dominate losses. They are counteracted using fragmented materials, for instance iron powder in an isolating binder.

A first order approximation to interwinding capacitances in the transformer places a single, lumped capacitor across the inner inductor. The capacitance may limit the high frequency 3dB cut-off and cause non-ideal roll-off characteristics. Interwinding capacitances

J.Vidkjær II-6 Transformers and Transformerlike Couplings 57 are reduced with more spacing between the turns,7 which on the other hand could cause more leakage, unless the flux is trapped by a high permeability core. Toroidal cores are especially efficient in that respect.

The list of secondary effects above is not exhaustive, ref’s.[11] chap.2 and [12] chap.2 give more comprehensive ones, but it suffices to show that patience and experience are required to make good RF-transformers. Alternatively, a wide selection of RF- transformers is commercially available. One example is given by the data sheet extracts in Fig.60. There are two versions of this 6.3×6.3×3.2mm transformer, the most broadbanded has winding ratio 1:2 (50Ω to 200Ω), the other one 1:3 (50Ω to 450Ω). Besides curves of the insertion loss like Fig.58, impedance matching is specified by the VSWR curve ( voltage standing wave ratio, cf.[7] sec.5.8 ). Be aware of the unusual frequency scaling in the upper curves, which is introduced to emphasize roll-off details.

Example II-6-1 end

Tuned Transformers

Fig.61 Transformation of a secondary side load to primary side of a tightly coupled transformer. Its equivalent circuit comes from Fig.51 using |k|=1.

In broadband applications the inductances of the transformer are supposed to be large compared to the impedances that are connected. Besides ideal performance the transformer should be transparent. In bandpass applications the inductances of the transformer may be a part of the tuning circuits. Using the tight coupling idealization |k|=1 in the equivalent circuits of Fig.51 or Fig.52, the impedance seen from at either side of the transformer now includes the corresponding inductance in parallel to the load provided through the ideal transformer. Fig.61 shows as an example that a secondary side load seen from the primary side is shunted by the reactance of the primary side inductance L1. Here, the secondary side inductor mani- fests itself solely through the winding ratio N of the ideal transformer. Seen from secondary side the parallelling reactance corresponds to L2 as displayed by Fig.62, which shows how a generator in Norton equivalent form transforms to the secondary side.

7 ) So-called transmission line transformers exploit interwinding capacitances. The windings are made conductors in a transmission line that is wound around the core. The method facilitates broad-band transformers of simple ratios, see [12] chap.5.

J.Vidkjær 58 RF-Circuits, Concepts and Methods

Fig.62 Transformation of a generator from primary side to secondary side of a tightly coupled transformer equivalenced by Fig.52 with |k|=1.

Commonly the easiest way to figure out tuning conditions with transformer couplings is to refer all components to one side of the transformer before making design calculations. The following amplifier example illustrates computational technique when dealing with tuned transformers.

Example II-6-2 ( transformer-coupled tuned amplifier )

Fig.63

The task is to find components values and the voltage gain v2/Eg in the amplifier above, which is required to operate at the following conditions:

1. Center frequency, fo = 120 MHz. 2. Bandwidth, BW3dB = 10 MHz, with frequency characteristic corresponding to two synchronously tuned resonance circuits. Ω 3. Generator and load impedances Rg=RL=50 , with center-frequency impedance matching at the input side. Ω 4. Transistor data: Rπ=305 ,Cπ=19pF, gm=200mS, and Co=1.3pF. The internal transistor feed-back is ignorable, if the following conditions are met8. 5. The total collector load at the center frequency should not exceed 88Ω to ensure robustness of the frequency response with respect to parameter tolerances. Ω 6. If Rg,RL, or both are absent stability is secured using Rp=220 or less.

8 ) Here we simply adopt the requirements. The theory behind is developed in the "Stabilizing Active Two-Ports" paragraph in section III-1.

J.Vidkjær II-6 Transformers and Transformerlike Couplings 59

Fig.64

Inserting a transistor model and converting the generator to Norton form provides the equiva- lent circuit in Fig.64. Going to Fig.65, all components at the input side of the amplifiers are collected at the primary side of the input transformer following the pattern from Fig.61. Correspondingly, all components on the output side are referred to the secondary side of the output transformer. The scaling of the transistor transconductance follows Fig.62 and the scaling in control voltages from the input transformer.

Fig.65

The winding ratios of the two transformers are found directly from the basic requirements of input matching and loading for stability,

(162)

(163)

Due to matching, the total resistance at the input port is ½Rg. At the output port we have

(164)

The bandwidth requirement pertains to the complete amplifier. To get the Q-factor for each resonance circuit, the two stage bandwidth reduction, GBF2 in Table I, must be invoked

(165)

J.Vidkjær 60 RF-Circuits, Concepts and Methods

Knowing the Q-factor, the remaining input circuit components in Fig.65 follow from basic resonance circuit relationships from Eq.(1) applied to the primary side,

(166)

The primary side inductance is determined form the resonance condition and the secondary side inductance follows from the winding ratio, i.e.

(167)

At the transistor output side we get correspondingly

(168)

Since we here refer all impedances to the secondary side, the tuning conditions give the secondary inductor and the primary inductor is determined by the winding ratio,

(169)

Input matching implies v1=½Eg. By the transformed transconductance shown in Fig.65 it leads to the voltage gain

(170)

The results are checked by frequency response simulations below in Fig.74.

Example II-6-2 end

J.Vidkjær II-6 Transformers and Transformerlike Couplings 61

Autotransformers

Fig.66 Autotransformer in separate inductor form (a), as a reconnection of the conventional transformer (b), and as a tapped inductor (c).

Fig.67 Signal flow-graphs for autotransformers, (d) corresponds to circuits (a) and (b) in Fig.66. Flow-graph (e) to the tapered inductor in Fig.66(c).

Series connecting the two inductors in a transformer as shown by Fig.66(a) makes a so-called autotransformer. To ease comparisons to previous results, the two coupled inductors are connected to the new terminal conditions in Fig.66(b). The corresponding embedding of the original z-parameters is indicated by the signal flow-graph in Fig.67(d). Concentrating the graph to terminal nodes gives z-parameters of the autotransformer in terms of the two series-connected, coupled inductors. The new parameters correspond to an inductor with a tap as shown by Fig.66(c). This is a common way of making autotransformers. Identifying similar paths through the flow-graphs provides

(171)

(172)

Observe that k=1 implies kA=1. Following the equivalent circuit of Fig.51(b), the effective winding ration nA becomes

(173)

J.Vidkjær 62 RF-Circuits, Concepts and Methods

The last tight coupling limit expression supports the fact, that the winding numbers are propor- tional to the square roots of the inductances, and that the primary side in the present orienta- tion has two parts separated by the secondary side tap.

Example II-6-3 ( autotransformers in a tuned amplifier )

Fig.68

The transformers in the amplifier from the previous Example II-6-2 could be autotransformers as sketched in Fig.68, where the inductors to be tapped correspond to the largest inductors from the conventional transformers. The placement of the tap with respect to the ground terminal corresponds to the smallest inductor value in the original transformers. Tight couplings with few windings due to the relatively small inductances, suggests employment of iron-powder toroidal cores, where reasonably high inductor Q-factors are obtainable. An example is the T25-0 core from Amidon, which has an outer diameter of 5mm, and which give inductor Q-factors above 100 in the 100 MHz frequency range. Compared to the Q´s of size 7.73 in the tunings, the inductor losses are insignificant. The core is specified to give an inductance of 4.5μH with 100 windings, the so-called AL,100 value. Assuming proportionality between inductance and the squared winding count, the number of windings for an arbitrary inductance becomes

(174)

With this core, the inductances that were calculated in Example II-6-2 correspond to the following number of turns,

(175)

Fig.69

J.Vidkjær II-6 Transformers and Transformerlike Couplings 63

To make an autotransformer, we start as sketched in Fig.69 winding a number of turns - say n1 - corresponding to the smallest inductor, then make a kink for the tap, and continue up to the winding count n2 of the greatest inductance.

Note that the input tuning capacitor in the example of Fig.68 has moved to the transistor side of the transformer compared to Fig.63. At this side it gets the value

(176)

Example II-6-3 end

Transformerlike Couplings

Fig.70 Transformation through uncoupled reactances. R |X2| and Rs |X1+X2| are assumed in the parallel to series to parallel sequence. Note, the sequence may be reversed.

Commercial RF-transformers are available with simple winding ratios like 1: 2, 1:2, 1: 3 etc. To make transformers with non-standardized data in quantities beyond laboratory scale may be tedious and costly. An alternative solutions to a transformation problem in bandpass applications could be employment of uncoupled reactances. If the conditions for simple parallel-to-series conversions following Eq.(59) are met, the conversions are applied in a parallel-to-series-to-parallel sequence as shown in Fig.70. The resultant transformation becomes,

(177)

If both reactances are of same signs, fulfillment of the assumption in the first part of (177) automatically implies that the requirement in the lower part is met. This is the case with either two capacitors or two uncoupled inductors.

J.Vidkjær 64 RF-Circuits, Concepts and Methods

Fig.71 Uptransformation of resistor through uncoupled capacitances or inductances. The ω ω expressions assume R 1/ C2 and R L2 respectively.

The capacitive utilization of (177) is called a capacitive transformer. It is shown in Fig.71(a). The equivalent winding ratio is expressed through sizes of the reactances,

(178)

With two uncoupled inductors, the sequence in (177) gives the result in Fig.71(b). The equivalent winding ratio is expressed,

(179)

Note the difference to the autotransformer in Eq.(173), where it was the square roots of the inductances that gave the winding ratio.

Example II-6-4 ( uncoupled reactance transformers in tuned amplifiers )

By this example we consider again the transformer-coupled tuned amplifier that was started using conventional transformers in example II-6-2, changed to autotransformers in example II-6-3. Below it is modified to make the transformations by either uncoupled inductors or capacitors. The original requirements from page 58 shall still be met, so the quality factors and by that the total reactances on either side of the transistor are the same as before. The transformations by uncoupled inductances shown in Fig.72 require

(180)

J.Vidkjær II-6 Transformers and Transformerlike Couplings 65

Fig.72 Similarly, the output side gives

(181)

The capacitances in the tuning are unaltered from Example II-6-2, i.e. C1=26.5 pF and C2=75.9pF. To check assumptions, we get at the center frequency 120 MHz,

(182)

Both comparisons are above the factor of three that may be taken as a practical lower bound where the simplifications behind Eq.(177) apply. We may go that low because it is actually the square of the ratio that determines the approximations, cf. Eqs.(57) to (59).

In the capacitive transformer coupling of Fig.73, the input winding ratio defined by Eq.(178), and the requirement that the total capacitance seen from the secondary side must equal C1, provide

(183)

Fig.73

J.Vidkjær 66 RF-Circuits, Concepts and Methods

At the output side we get correspondingly,

(184)

The tuning inductors are here equal to the transformer inductors towards the transistor, i.e.

(185)

Before using the results, we should again check the assumptions. Now we get

(186)

so the method is still usable although the input condition is not as good as it was in Eq.(182). The reason for the difference is that the whole inductive reactance is engaged in the inductive transformations, but only a part of the capacitive reactance is available for transformation because a fraction belongs to the transistor.

The agreements between the transformation methods are illuminated by Fig.74. It compares the simulated responses of the transformer-coupled amplifier from II-6-2, heavy lines, with the responses of the two configurations above. As seen, the simplifying assumptions manifest themselves mostly by a slight displacement in phases with the inductive coupling lagging behind and the capacitive leading the transformer-coupled amplifier.

Fig.74

Example II-6-4 end

J.Vidkjær II-6 Transformers and Transformerlike Couplings 67

If the requirements for simple series-to-parallel conversions do not apply, it is possible to calculate components in the transformations without any assumptions from more elaborates relations in Eqs.(57) and (58). Under such circumstances, where also the reactive components are influenced, there is no benefit of referring to transformer concepts as the problem behaves more likes that of establishing impedance transformation through a lossless network. Smith chart constructions, which are considered in section II-8, are here alternatives to calculations by the two formulas mentioned above.

Three-Winding Transformers

Transformers with three windings are often encountered in RF-circuits to change signals back and forth between unbalanced and balanced forms, or to isolate parts of the circuits from each other. Setting up general purpose equivalent representations based on uncoupled inductors and ideal transformers in the three-winding case is both tedious and confusing, so we shall limit ourself to consider idealized situations that closely follow the intentions behind the use of the transformers. Before doing this, however, an equivalent representation suitable for computer simulations of less ideal cases is briefly discussed.

(187)

The inductance matrix of three coupled inductors is given by Eq.(187). The corre- sponding three port circuit may be equivalenced by the network in Fig.75, which is the three port counterpart to Fig.56. The last version of the inductance matrix includes the assumption of reciprocity using only three different coupling coefficients because Mij=Mji. To ensure passivity the matrix must be positive semidefinite, and the diagonal elements satisfy

(188)

Zero-valued inductors are uninteresting and are disregarded below. Taking two-dimensional subdeterminants symmetrically around the diagonal shows that no coupling coefficient may exceed one in size, i.e.

(189)

The final three-dimensional determinant requirement relates the three coupling coefficients

(190)

J.Vidkjær 68 RF-Circuits, Concepts and Methods

Fig.75 Equivalent circuit for three inductors where the couplings are simulated by voltage controlled voltage sources. The signal flow-graph shows z-parameters.

To illuminate the last requirement, suppose that inductor number one and two are tightly coupled, k12=±1. Then the size of the coupling from inductor number three must be the same to both number one and two. The condition develops

(191)

Taking negative value of a squared number cannot exceed zero, so only the equality condition is in effect making the two remaining couplings equal in size. Signs are adjusted so there are either none or two negative coupling coefficients. The result is the limit case of the require- ment from Eq.(190), which in a less stringent form is phrased that if two inductors are strongly coupled, a third one cannot be loosely coupled to one of them and strongly to the other.

Sign conventions with three windings are more confusing than the two winding case, because it is now possible both to exchange terminal orientations, equivalently winding senses, and to alter the magnetic flux paths from a serial to a parallel one. In transformer idealizations without leakage the two last alternatives are sketched in Fig.76, where the flux orientations correspond to currents entering the positive terminals. With the toroidal core, where the flux path follows the inductors in series, it is possible to organize port orientations to let all fluxes support each other, so here dot tagging remains meaningful. Without leakage all coupling coefficients are simultaneously of unit size with either none or two being negative as we saw in Eq.(191). Following the dots all couplings are positive, so turning the direction of one coil lets its flux opposes the fluxes from the other two, thus changing signs of two coupling coefficients.

J.Vidkjær II-6 Transformers and Transformerlike Couplings 69

Fig.76 Three-winding flux-paths. Couplings are positive between coils with supporting fluxes, negative on opposing fluxes. In series paths (a) there may be none or two, in parallel (b) one or three negative couplings.

In parallel flux paths, like Fig.76(b), there will always be at least one path opposing the others, so either one or three coupling coefficients are negative. Here, a single dot indication is meaningless because it is no longer possible to establish an unambiguous refer- ence direction for fluxes, so it seems better to resort to signed mutual inductance or coupling coefficient indications in that case. Moreover, the flux from either port divides between the two others so none of the coupling coefficients may approach one in size if all inductors should be coupled together. The requirement was foreseeable from the passivity condition in Eq.(190). If all the k’s are negative, all k-terms in the last factor are negative and so the sizes of the couplings are severely restricted. Suppose the fluxes from either coil in Fig.76(b) distributes evenly between the two others. Then we have k12=½, k13=½, and k23=-½, which is the borderline for the passivity criterion.

RF-applications of three winding transformers are dominated by the series flux method in Fig.76(a), so the reason for considering the parallel case is mainly to show how easily passivity criteria are violated in a simulation, for instance by setting all couplings close to one in size but let one of them be negative. Transformers that are not passive may supply power and cause unrealistic results or introduce instability in computations.

The impedance matrix for three coupled inductors, which is based on the inductance matrix in Eq.(187), leads to the requirement,

(192)

A winding scheme like Fig.76(a) is assumed, so all coupling coefficients are taken positive and port orientations follow the dot marking. Tightly coupled, Eq.(192) implies

(193)

J.Vidkjær 70 RF-Circuits, Concepts and Methods

Fig.77 Ideal three-winding transformer and a flow-graph of its terminal constraints.

Taking the first inductor as reference, the three-winding transformer voltage relations become

(194)

N12 and N13 hold winding ratios of inductor no.2 over no.1 and inductor no.3 over no.1 respectively. The ideal voltage conditions require tight coupling only. If the inductances are raised towards infinity keeping unaltered winding ratios, Eq.(193) gives the current relation for an ideal three-winding transformer,

(195)

Fig.77 shows the transformer symbol and a flow-graph representation of the ideal, algebraic current and voltage constraints.

The three equations in Eq.(192) may be written

(196)

These equations have form of the ideal transformer current relation using the current sets

(197)

Each hatted component is adjusted for the current through an uncoupled inductor. To account for finite inductances in the three-winding case, we may use the same approach that formerly applied to two windings, namely to include one of the inductors across the corresponding port of an otherwise ideal transformer. The three possibilities are shown in Fig.78.

J.Vidkjær II-6 Transformers and Transformerlike Couplings 71

Fig.78 The three possibilities of taking into account finite inductances in a three-winding transformer with tight couplings.

The input impedance of a loaded three-winding transformer is calculated as shown by Fig.79. There are no loops in the flow-graph, which directly gives

(198)

If the reactances of the transformer inductances compare to the load impedances, the effect of finite transformer inductances may be accounted for by connecting inductor L1 across the input-port as indicated in part (b) of the figure.

Fig.79 Determination of the input impedance of a loaded, three-winding transformer. The flow-graph assumes ideality. L1 may be added to account for finite inductances.

To get the generator equivalent circuit at a transformer port, when the generator is connected to another port and the third port is loaded, we consider Fig.80. The loop provides

(199)

By that, the EMF and generator impedance at port number two becomes

(200)

J.Vidkjær 72 RF-Circuits, Concepts and Methods

Fig.80 Signal flow-graph for determining a generator equivalent circuit at port 2.

(201)

This result is interpreted as shown by Fig.81, where it is seen that the third port loads the input generator before it is transformed to port 2. Again, one inductor may be included to account for finite transformer inductances.

Fig.81 Interpretation of Eqs.(200),(201) where (a) holds the generator components that are transformed to (b). L2 may account for finite transformer inductances.

Transformer Hybrids

A coupling method based on the three winding transformer that has great practical importance is the so-called hybrid in Fig.82. It is used to combine or isolate signals from different sources. To achieve such goals, winding ratios and load resistances must be realized accurately. Failures to meet requirements set the performance limits, so we shall start analyz- ing the circuit without any assumptions, and then simplify to see how the desired properties evolve. As indicated in the diagram, there is an impedance and an EMF connected to each port. We consider the corresponding port voltages - the vpj’s - as the responses when the EMF’s are successively applied. With the transformer conditions from Eqs.(194),(195), the connections in Fig.82 imply

(202)

J.Vidkjær II-6 Transformers and Transformerlike Couplings 73

Fig.82 Three-winding transformer coupled as a hybrid. On proper conditions there are no transmission from E1 to vp4 , from E2 to vp3 , and vice versa.

The port currents and voltages are constrained by the branch relations

(203)

The two last sets of relationships are drawn together in the signal flow-graph of Fig.83. There are six loops and two sets of non-touching loop products in the graph yielding the common transfer function denominator

(204)

The two sets of ports between which it is possible to get isolation are 1,4 and 2,3 respectively. For the first set the flow-graph provides,

Fig.83 Signal flow-graph for the hybrid in Fig.82. The insets show six loops, where the upper ones contribute loop-loop and loop-transmission terms in gain computations.

J.Vidkjær 74 RF-Circuits, Concepts and Methods

(205)

A simple criterion where these transmissions become zero is symmetry,

(206)

Transmissions between the second pair of potentially isolated ports give

(207)

The joint conditions for isolation between the two port pairs are called double balance, where

(208)

Under double balance conditions the denominator reduces to

(209)

Conditions at port 1 and 4 are now expressed by

(210)

(211)

Fig.84 Conjugated matching.

J.Vidkjær II-6 Transformers and Transformerlike Couplings 75

Fig.84 and Eq.(211) summarize conjugated matching condition at the j’th port including the port voltage, vpj, the power delivered to the port Ppj, and the available power at the port, Pav,j. Applied to the last expression in Eq.(209), conjugated matching or shortly just matching is obtained if

(212)

In case impedances are ohmic, the voltage ratios are simple and equal to one half,

(213)

Obtaining matching conditions at the two other ports is a little lengthy, because there are four transmission paths in the flow-graph, two including non-touching loop products. We find

(214)

Imposing the conditions from Fig.84 on the second but last expression above shows that conjugated matching at port 2 and 3 resorts to the requirement from port 1 and 4 in Eq.(212). Again, ohmic conditions simplify to halving the port voltage compared to the generator EMF. The transfer functions that now remain are all the desired signal-branchings. From port 1 we get

(215)

Note the sign-shift in the transmission to port 3, which was required to bring the two transmis- sions on final common forms. To admire the matching result we show that the power Pp2

J.Vidkjær 76 RF-Circuits, Concepts and Methods delivered to port 2 is half the available power from the source (the other half goes to port 3). Incorporating the balancing and matching conditions from Eqs.(212),(213) we get

(216)

Transmission from port 4 becomes

(217)

Observe here, that none of the paths required sign-shift to reach the common forms. The power branching when transmitting from this side of the transformer is expressed

(218)

To economize writings we take the remaining transmission in output port order, i.e. to the same sides of the transformer. Then we get

(219)

where the lower ratio has changed sign like its opposite direction counterpart in Eq.(215). Due to the similarities in the expressions, power transmission from port 2 and 3 to port 1 are the same, for instance

(220)

J.Vidkjær II-6 Transformers and Transformerlike Couplings 77

Finally, the transmissions to port 4 are

(221)

where the power balance with a source on port 2 or equivalently on port 3 is,

(222)

A summary of the possible transmissions between port voltages is given in Fig.85. It assumes ohmic matching conditions, where the voltage across any driving port is half the EMF. Due to the sign-shifts between port 1 and port 3, the coupling is referred to as a 180° hybrid and it has the property of being broadbanded. With tuned transformers or in transmis- sion line techniques it is also possible to make 90° hybrids, cf.[4] sec.6.5.

Fig.85 Port-voltage relations summary in ohmic matched trans- former hybrid, classified functionally as a 180° hybrid.

One reason for working out details of the three-winding transformer hybrid is its widespread use in mixer circuits. Mixers are often the most critical component in communica- tion systems, and derivations above are helpful tools for exploring their performance limits. As a brief introduction, the following example shows a diode-ring mixer that includes two three-winding transformers, one operating as a hybrid.

J.Vidkjær 78 RF-Circuits, Concepts and Methods

Example II-6-5 ( diode-ring mixer preliminaries )

Fig.86 Diode-ring mixer. A large local oscillator signal switches the two diode branches instantly and reverts sign of transmission from input to load on positive (b) and negative (c) VLO’s.

The basic function of the diode-ring mixed in Fig.86 is most easily explained if the mixer is employed to generated a double sideband AM signal with suppressed carrier, the so- called DSB-SC modulation. For simplicity it is supposed that both transformers have winding ratios equal to one and we let a low-frequency sinusoidal represent the baseband signal, VBB. It is applied to the isolated winding port of the input three-winding transformer, T1. The carrier signal is inserted by the local oscillator signal VLO at the isolated port in transformer T2. Proper operation of the mixer requires that this signal be large compared to both VBB - Fig.87 is not drawn to scales - and the voltage across two conducting diodes. The role of the local oscillator is merely to switch between the two branches in the diode-ring as showed by inserts (b) and (c) in Fig.86. The switching changes direction of transmission from the baseband input to the modulated RF output VRF across the load. This process corresponds to multiplying VBB by function fsw that switches evenly between 1 and -1. It has no DC term and a fundamental frequency equal to the carrier frequency of VLO. Proper low-pass or bandpass filtering of the VRF signal in the figure gives therefore a DSB-SC waveform.

No sophisticated balancing and transmission conditions are required to explain this basic function. In practical operation, however, a series of secondary phenomena related to unavoidable nonlinearities produces spurious components. They are unwanted signals at frequencies that might disturb further processing or other communications. One measure to prevent their build-up is to keep signal sources isolated from each other. In the present case, it is required that the two secondaries in T1 have exactly equal winding ratios and that the

J.Vidkjær II-6 Transformers and Transformerlike Couplings 79 diode impedances in the two switching directions are the same, to prevent a baseband signal component in the output. Both in Fig.86(b) and in (c) the output load resistor and one of T1´s secondary sides make the isolated port of T2 when seen from the local oscillator. Therefore, the accuracy of the double balance determines how good the carrier is suppressed in the output signal. Furthermore, the balance prevents baseband signals directly to enter the oscillator circuitry. It is for the quantitative analysis of this type of problem, we need details of the three-winding transformer and its use as a hybrid.

Fig.87 Diode-ring mixer waveform example. Baseband signal VBB is chopped at the local oscillator frequency. Proper filtering of the resultant VRF signal gives a DSB-SC output.

Example II-6-5 end

J.Vidkjær 80

II-7 Double-Tuned Circuits and Amplifiers

Two resonance circuits coupled so loosely that they nearly maintain their individual resonance properties constitute a double-tuned circuit. It can be build with both parallel and series resonance circuits, but only coupled parallel circuits are considered. This is the type that is most commonly encountered in RF circuit designs, but the concepts and results below translate to double-tuned series circuits like the translation between single tuned parallel and series circuits we have seen before.

Parallel resonance circuits are in principle coupled by a connecting element as illuminated by Fig.88(a) and (b), which show the important cases of purely capacitive and inductive couplings. To make the coupling loose while keeping transmission between the ports, the admittances of Cc or Lc must be comparable in magnitude to the admittances of the parallel resonance circuits. We shall be more precise about this matter below. Comparing the inductive case with Fig.53 in section II-6 shows that inductors La,Lb, and Lc could be the equivalent circuit of two coils with mutual inductances.

Fig.88 Double-tuned circuits based on two capacitively (a) and inductively (b) coupled parallel resonance circuits Ca,Ra,La and Cb,Rb,Lb.

Fig.89 Circuit structure and signal flow-graph for calculating transfer and input impedances of the double-tuned circuit.

The structure of the circuit to be considered is shown in Fig.89. The transfer and input impedances are derived directly from the diagram or guided by the flow-graph to yield

(223)

J.Vidkjær II-7 Double-Tuned Circuits and Amplifiers 81

(224)

To cast the expressions in symmetric forms we define a sum and a difference admittance,

(225)

Now the last parts of Eqs.(223) and (224) can be rewritten,

(226)

(227)

Coupling between Identical Resonance Circuits

The advantage of Eqs.(226) and (227) are, that they expose symmetry in the resultant impedances from the beginning even if the two coupled circuits are different. But we start considering the complete symmetrical situation, where the two resonance circuits are identical and we have

(228)

The sum admittance depends on the type of coupling and takes one of the forms,

(229)

Both expressions represent parallel resonance circuits, which include the coupling element, either Cc or Lc. The resonance circuit described by YΣ appears across either port in the two- ports of Fig.88 when the opposite port is short-circuited. To express both coupling methods equivalently, we denote

(230)

J.Vidkjær 82 RF-Circuits, Concepts and Methods

By that the resonance frequency and quality factor for YΣ are expressed by the usual relations

(231)

To obtain a unified description of both coupling methods, the coupling coefficient k is introduced by the definitions

(232)

Observe that the definition with respect to inductive coupling agrees with the k definition for two inductors having mutual inductances, cf. Eq.(158) and the equivalent circuit in Fig.55. The Q-value of the resonance circuit in YΣ is supposed to be high, so the narrowband approxi- mation from Eq.(46) applies. We may therefore approximate

(233)

ω in vicinity of the upper half-plane pole s1, particularly on the imaginary axis around 0.In the present symmetrical case Eqs.(226) and (227) reduce to

(234)

(235)

Before inserting terms into the equation for z21, narrowband considerations are applied once more. The resonance circuit admittance YΣ(s) exhibits its strong frequency dependency around s1 because it contains two counteracting reactive components, inductor La and capacitor Ca. Contrarily, Yc(s) is designated by a single component and shows no significant variations around s1. In a narrowband approximation it is therefore considered constant,

(236)

J.Vidkjær II-7 Double-Tuned Circuits and Amplifiers 83

These expressions are used jointly with Eq.(233) in Eqs.(234),(235) to give the narrowband approximation to the transfer and input impedances of the symmetrical double-tuned circuit,

(237)

(238)

The sign in the numerator of z21 is determined by the method of coupling, positive if it is capacitive and negative if it is inductive. Note, however, that k may be either positive or negative if two inductively coupled coils are employed. Coupling two identical resonance

Fig.90 Double-tuning. Encircling around s1 in (a) is expanded to show narrowband poles and zeros in z21 (b) or z11 (c). See page 99 about the number of zeros in origo. circuits together has the effect of splitting pole positions symmetrically apart from s1 in direc- tions parallel to the imaginary axis as indicated by Fig.90. The coupling does not affect the zero of z11 that remains at s1. Observe that to stay consistent with the narrowband assump- tions, the poles are still confined to the region around s1. Besides the basic high Q require- ment, k must be small. In summary upper half-plane poles are

(239)

Inserting the poles and s=jω into Eq.(237), using Eq.(231), gives the frequency response by the narrowband approximated transfer impedance

J.Vidkjær 84 RF-Circuits, Concepts and Methods

Fig.91 Double-tuned circuit normalized magnitudes and phases (inductive coupling) with various kQ settings. Part (a) shows transfer impedances z21 and (b) input imped- ances z11.

(240)

It is the same frequency normalization that formerly was introduced for a single tuned circuit by Eq.(45), which is applied here. Magnitude and phase curves of z21 using different kQ products are shown in Fig.91(a). When kQ is below one, the amplitude characteristic shapes likes single tuned responses. If the kQ products are greater than one, the two poles are so widely separated that they are sensed individually to produce two maxima in the amplitude curves. In between is the so-called critical coupling, where kQ equals one. It has the widest

J.Vidkjær II-7 Double-Tuned Circuits and Amplifiers 85 bandwidth that can be achieved without ripples in the amplitude characteristics. The coupling admittance, which by Eqs.(231) and (236) is expressed

(241) becomes equal in size to the admittances of either parallel circuit at resonance, if the coupling is critical. The sign still follows the coupling method being positive with capacitive coupling.

A narrowband approximation for the input impedance z11 is obtained by a procedure similar to the one above based on Eqs.(231),(233), and (238). It gives

(242)

Curves of the impedance are shown in Fig.91(b). Due to an intervening zero compared to the transfer impedance, cf.Fig.90(c), the impedance magnitude gets a minimum at the center frequency with critical coupling, kQ=1. Flatness in z11 requires that the two poles moves close to compensate the effect of the zero. As seen, this happens if kQ=0.5.

Example II-7-1 ( Foster-Seely FM detector )

In frequency modulation, FM, signals of constant amplitude are transmitted with an instanta- Δω ω neous frequency that differs from a fixed carrier frequency c in proportion to an informa- tion bearing low-frequency signal. Recovering information in demodulation requires a circuit whose output is proportional to the frequency deviation between the instantaneous frequency

Fig.92 Foster-Seely FM detector. Besides the overcoupled signal v2, primary voltage v1 adds as a common signal to the secondary side by coupling capacitor Cm.

J.Vidkjær 86 RF-Circuits, Concepts and Methods and the carrier frequency. A classic construction having this property is the Foster-Seely detector in Fig.92. Here, current source ig introduces the FM signal of instantaneous angular ω ω Δω Δω frequency inst= c+ . The detector outputs a LF voltage proportional to .

We shall consider how a double-tuned circuit is used to achieve the goal above with good linearity. Source ig drives the primary side of two inductively coupled parallel resonance circuits, each holding elements Ra, C, L. The resonance frequencies are equal to the carrier ω frequency c. Transimpedance z21 causes the secondary voltage v2, which is shown by two halfs on either side of a center tap in the secondary inductor. Left alone this point would have zero potential as it is connected to ground through the choke Lm, which is a high impedance inductor at the signal frequency. However, the primary voltage v1 is added through the coupling capacitor Cm as a common voltage on the secondary side. Due to the symmetry of grounding, the voltages presented at the envelope detector inputs, vA and vB respectively, get geometrical interpretations as shown by Fig.93. The detectors are supposed to provide no loading compared to Ra.

Fig.93 Phasors for secondary side voltages in the Foster-Seely detector. The instantaneous ω ω Δω ω frequency, inst= c+ , equal to, above, and below carrier frequency c. To realize the three phase conditions in the figure, we observe that the FM signal frequencies are related to our previous frequency variables by

(243)

θ The phase difference between v2 and v1 follows from Eqs.(240),(242), and becomes

(244)

where the last expression is based on a first order expansion of tan-1. We assume k>0. An Ω π angle difference of - s offsetted by -½ agrees with the situations in Fig.93, since there is Δω Ω Ω proportionality between and s from Eq.(243). At s=0, where v1 and v2 are perpendicu- lar to each other, the voltages are said to be in quadrature .

ω The two voltages vA and vB are AC voltages of instantaneous frequency inst. Lengths of their phasors, |vA| and |vB|, are envelopes to the AC signals, and these voltages are the assumed

J.Vidkjær II-7 Double-Tuned Circuits and Amplifiers 87 outputs from the ideal envelope detectors. The difference in envelope values makes up the Δω Ω detector output, which is expected to be proportional to or, equivalently, s. To investigate the question, we calculate again from Eqs.(240),(242)

(245)

(246)

(247)

Curves illustrating the last equation are plotted in Fig.94 for various values of the kQ product. As seen, kQ-products about three give the largest, linear output interval.

Fig.94 Foster-Seely FM detector characteristics for different kQ’s. The curve in heavy line, kQ=3, has rather good linear output span.

Before leaving the Foster-Seely detector, we shall make a few comments on its implementa- tion and use. First, practical envelope detectors are commonly based on diode rectification, so there is voltage loss in the detection. The detector circuit contributes nonlinearily to both the primary and secondary side load resistors, so a thorough analysis of this case becomes rather involved. Second, the detector is sensitive to input level, in addition to the frequency deviations. This is expressed here by the ig factor in Eq.(247). Therefore, a separate so-called limiter circuit, which ensures constancy of the ig amplitude, must precede a Foster-Seely detector.

Example II-7-1 end

J.Vidkjær 88 RF-Circuits, Concepts and Methods

Example II-7-2 ( quadrature FM detector )

Fig.95 Quadrature detector principles. Quadrature of v2 is made by voltage division between Ck and the impedance of a single-tuned (a), or double-tuned circuit (b).

Alternatives to the Foster-Seely detector are the quadrature detectors in Fig.95. Here the FM modulated input signal v1 and the quadrature component v2 are multiplied and filtered to recover the baseband signal, which is proportional to Δω. This last part of the process gives

(248)

where b is the multiplicator scaling factor. This result is based on the trigonometrical identity

(249) and the assumption, that the LP filter totally removes the high-frequency component at twice the carrier frequency. To see the significance of quadrature, observe that it is exact if θ=±½π, and around this angle we approximate

(250)

To detect FM, the divider circuit must provide a deviation angle φ from quadrature, which to a first order approximation must be proportional to the deviation Δω from the carrier frequen- ω cy c. More generally, the complete processing of the divider in combination with the nonlinear multiplication should possess linearity between frequency deviation and detector output voltage. The two detectors in Fig.95 have different spans of linearity where the more complicated solution with double-tuning is the better one. As a reference, however, the simpler single-tuned circuit in Fig.95(a) is considered first.

J.Vidkjær II-7 Double-Tuned Circuits and Amplifiers 89

Using narrowband techniques including the impedance Zp of the single tuned circuit, cf.Eq.(45), the quadrature voltage may be written

(251)

ω ω The resonance circuit is tuned to the carrier frequency of the FM signal, so 0= c. To assure Ω quadrature at s=0, the last term in the denominator of Fs must be insignificant, i.e.

(252)

In its first form, the inequality shows that the impedance of the divider capacitor Ck must be large compared to the peak-impedance of the resonance circuit. The second form quantifies the requirement in terms of the tuned circuit Q-factor. Including the assumptions we express

(253)

Factors at the right-hand side of Vout from Eq.(248) are now related through,

(254)

Expressed in either normalized or absolute frequency variables, the detector output becomes

(255)

It is clearly the frequency dependent term in the denominator that limits linearity in the output voltage versus frequency deviation detector characteristics. To improve the linear range , the input impedance of a double-tuned circuit may be employed instead of Zp. Then we get the division ratio, cf. Eq.(242),

(256)

J.Vidkjær 90 RF-Circuits, Concepts and Methods

The requirement of Eq.(252) must still apply, so the contribution from the quadrature compo- nent is calculated in analogy with the previous development, i.e.

(257)

(258) where the deviation from linearity is kept in the function U,

(259)

The detector output now becomes

(260) and corresponding detector characteristics are shown in Fig.96. To make easy comparisons the Ω outputs are scaled so all slopes in the center at s=0 are equal. The scaling is based on the U-function with arguments set to zero,

(261)

Fig.96 Normalized quadrature detector characteristics. The curves show that kQ=.45 is near-optimal with respect to linearity in double-tuning. The kQ=0 curve corresponds to single-tuning.

J.Vidkjær II-7 Double-Tuned Circuits and Amplifiers 91

Note, that kQ=0 gives the function that controls departures from linearity in the single tuned case from Eq.(255), so the figure includes both type of tunings. In double tuning, the curves confirm that a value kQ=0.45, which is often aimed upon in design, comes close to an optimum regarding linearity. The important consideration is here the span of normalized frequency where linearity is maintained, so double-tuning is approximately twice as good as single-tuning in that respect.

Quadrature detectors, compared to the Foster-Seely detector from the foregoing example, have the advantage that multiplication is better suited for integration than envelope detection. An example of a quadrature detector IC and its use is given by the data-sheet extracts9 in Fig.97, Fig.98, and Fig.100. The block diagram shows that the chip contains an "input amplifier", two coupling capacitors to the external tuned circuit, a demodulator that holds the multiplication function, and finally a video amplifier that includes lowpass filtering to a bandwidth of 10 MHz. The input amplifier terms are set in quotes above because it is not a conventional linear amplifier but a limiting one, which keeps its output amplitude constant across an input range from 10 to 300 mV rms. It is clear from the quadratic V1 amplitude dependency in Eq.(260),

Fig.97

9 ) GEC Plessey Semiconductors, Consumer IC Handbook, Sept.1991, pp.2-9,12

J.Vidkjær 92 RF-Circuits, Concepts and Methods

Fig.98 that the input to the detector must stay very constant, claimed as excellent threshold among the highlighted features in the data. The RC network around the input amplifier shows, that the inputs are biased internally.

Like other integrated circuits, the RF signal path is differential, i.e. the signal voltage is the difference between two identical processing chains. Difficulties of establishing good RF signal ground inside the IC are avoided this way, which benefits from the fact, that many basic IC constructions are differential anyway. A multiplicator that gives differential output as the product of two differential inputs - denoted demodulation in the block diagram - is a standard construction called a Gilbert cell [13],[14], or a double-balanced modulator (mixer) [15]. Thinking differential instead of single-sided is therefore easy in the present case, where the interface to the external tuning circuits is sketched in Fig.99. To get a v2 voltage here that equals v2 in Fig.95(b), for same v1´s, the voltage across each of two differential mode dividing capacitors, the Cd’s, must be half the difference voltage v1-v2. If all other

Fig.99 Details in differential mode connection of the double-tuned circuit to the block diagram in Fig.98.

J.Vidkjær II-7 Double-Tuned Circuits and Amplifiers 93 impedance levels are unaltered from the single-sided case, the differential mode capacitors should be twice the Ck capacitance to halve the corresponding impedances. We could repeat this arguing with respect to coupling capacitance Cc, so the balanced coupling in the figure requires two capacitors, each twice the size of the corresponding single-sided coupling capacitor. Strictly speaking it is unnecessary to balance the coupling capacitors in this application although it is shown in the data-sheets, where also the low number of external components required for this IC is praised. So here is a suggestion for further savings.

Fig.100

J.Vidkjær 94 RF-Circuits, Concepts and Methods

The use of the IC is illustrated by single and double tuned examples in Fig.100, which also gives examples of the detector characteristics for various Q values and, with double-tuning, the corresponding coupling capacitors. The strategy has been to keep the tuning components L and C fixed and let the resistor control the Q by the usual relationship, cf. Eq.(252). In agreement with Eqs.(255),(260) the resultant outputs get slopes proportional to Q2 -Ris proportional to Q - as seen in both types of detector characteristics. Comparing single and double tuned curves, it is also observed that the latter get the best span of linearity with a given center slope.

Example II-7-2 end

Double-Tuned Amplifier Stages

Fig.101 Bandpass amplifier with symmetrical, critically coupled, double-tuned output circuit. The transfer function is shown in Fig.102(b).

The flat amplitude response in critical coupling is a common reason for using double- tuned circuits in amplifiers. If the transistor is modeled by a transconductance, letting possible output impedance components contribute to the first resonance circuit, the voltage gain of the amplifier in Fig.101 can be expressed,

(262)

where the denominator D is taken from z21 in Eq.(240). Inserting kQ=1 for critical coupling, the denominator and its squared absolute value become

(263)

Ω To get upper and lower 3dB bandlimits we solve for the s´s where the squared denominator Ω has doubled compared to its center frequency value at s=0, i.e.

(264)

J.Vidkjær II-7 Double-Tuned Circuits and Amplifiers 95

Using Eq.(231), the corresponding unnormalized 3dB bandwidth is expressed

(265)

Considered separately, the two parallel resonance circuits have bandwidths corre- Ω sponding to s=±1. Broadening in bandwidth by a factor of 2 in z21 is readily seen in the comparison by Fig.102(b). The corresponding flatness of the amplitude response is maximal as the frequency characteristic with critical coupling is the Butterworth characteristic, which was discussed in section II-5, page 36. One way to realize this is by the pole positions in Fig.102(a). They equal the second order Butterworth pole patterns in Fig.39. Alternatively, the defining Butterworth magnitude equation, Eq.(109), follows directly from Eqs.(262), Ω (263), by introduction of a new frequency normalization btw, which has 3dB bandlimits at Ω btw=±1. We get

(266)

Taking the total capacitance CΣ in the first resonance circuit as reference, the gain- bandwidth factor of the amplifier with symmetrical double-tuned output circuit becomes,

(267) so this stage has a gain-bandwidth product below that of a single tuned stage.

To get flat response and simultaneously keep gain-bandwidth like a single tuned stage, symmetry must be given up so only the output circuit is resistively loaded .

Fig.102 Critical coupling. Poles of z21,(a), and amplitude response,(b). Scaling is chosen to compare the shape with a single tuned response of equal Q, i.e. with one pole in s1.

J.Vidkjær 96 RF-Circuits, Concepts and Methods

Fig.103 Bandpass amplifier with one-sided load of the double-tuned output circuit .

This is exemplified by the amplifier in Fig.103, where the secondary resonance circuit is resistively loaded. Both resonance circuits have identical reactive elements, and are therefore tuned to the same frequency. The quality factors are different being infinitely large at the transistor side and finite at the output side. To get the transfer impedance we use the technique developed by Eqs.(223) through (226). The sum admittance becomes

(268)

The reactive elements are defined to encompass both capacitive and inductive couplings by the definitions in Eq.(230). It is repeated here for convenience,

(269)

Following Eq.(268), the sum admittance YΣ has the resonance frequency and quality factor

(270)

where Qb is the quality factor of the output side resonance circuit alone. The narrowband approximation for YΣ is

(271)

With an unsymmetrical circuit we get a non-zero difference admittance

(272)

The coupling admittance is narrowband approximated as before in Eq.(236), where positive sign refers to capacitive coupling, negative to inductive coupling,

(273)

J.Vidkjær II-7 Double-Tuned Circuits and Amplifiers 97

All elements in the transfer impedance are now collected to yield

(274)

The last expression is a narrowband approximation around s1 in the upper half-plane. Casted into standard form the expression should read

(275)

so the poles s11 and s12 are easily identified. Using Eq.(271), they become

(276)

Suppose the coupling k is a parameter that can be independently adjusted. Starting from k=0, where the two resonators are isolated from each other, the two upper half-plane poles have ω ω imaginary part j 0 and real parts 0 and - 0/2Qb respectively as sketched in Fig.104(a). With

Fig.104 Poles of z21 in the unsymmetrical double-tuned circuit from Fig.103. (a) Poles at k=0 and directions of movements for increasing k. (b) Poles at kQb=1/ 2. growing k the poles attract horizontally until they met at s1 when the square root becomes zero for kQb=½. Further rise in k turns the square root imaginary and the poles split apart in parallel to the imaginary axis. A pole pattern like Fig.104(b) gives maximally flat transfer characteristics. It is reached when the square root is j and the kQb-product becomes

J.Vidkjær 98 RF-Circuits, Concepts and Methods

(277)

Inserting the radius r from the pole plot in Fig.104(b) into the denominator of Eq.(275) gives the center frequency gain and subsequently the gain-bandwidth product,

(278)

(279)

Hence, a double-tuned amplifier with one undamped resonator rise the gain-bandwidth product to the level of the single tuned amplifier and keeps the maximally flat transfer characteristic. The contrast to the symmetrical case may be understood taking the output side resistors as a load receiving power from the transistor. The intervening parts of the unsymmetrical double- tuned circuit are lossless, while the input side resistor causes an additional loss in the symmet- rical circuit.

Asymptotic Behavior

With exception of signs, neither qualitative nor quantitative distinction between capacitive and inductive couplings was required in the discussions above. They were all based on narrowband assumptions around the passband of the double-tuned circuits. Outside the passband, where the narrowband approximations are no longer adequate, dissimilarities show up as differences in the response asymptotes with frequency approaching either zero or infinity. We shall make a brief account on these questions.

Fig.105 Dominant contributions to asymptotic responses when the frequency is approaching zero, (a),(b) or infinity, (c), (d).

J.Vidkjær II-7 Double-Tuned Circuits and Amplifiers 99

When the frequency is approaching zero, Fig.105(a) and (b) indicate the circuit elements of the double-tuned circuit that are significant. The limit values are found using Eqs.(223),(224) to yield

(280)

(281)

ω Both input impedances z11 go towards zero like as does z21 with inductive coupling. In ω3 capacitive coupling z21 follows . The rate of decline, taken as the power in frequency that the response follows towards zero frequency, is equal to the number of zeros in origo.

In the high frequency limit the double-tuned circuit responds like the highlighted parts of Fig.105(c) and (d), where we get

(282)

(283)

Both input impedances z11 are approaching zero inversely proportional to frequency as is the transimpedance z21 if capacitive coupling is employed. The inductive coupling shows a z21 decline proportional to the third order frequency term in the denominator. It is common terminology to say that the impedance or transimpedance has a number of zeros at infinity that corresponds to the resultant power, by which the denominator goes towards infinity with frequency. Hence, capacitively coupled circuits have one and inductively three zeros at infinity.

J.Vidkjær 100 RF-Circuits, Concepts and Methods

Fig.106 Low and high frequency asymptotes of |z21| in double-tuned, critically coupled circuits using capacitive (a) or inductive (b) couplings.

The asymptotic behaviors of the input impedances follow the pattern we have already seen with single-tuned circuits in Fig.5. The results should not be surprising as they illuminate the fundamental lumped network property, that the driving point impedance of a passive network cannot approach zero faster than a single shunting inductor in the zero frequency limit and faster than a single shunting capacitor in the infinite frequency limit10. Trans- impedances, which have no similar restrictions, are summarized by Fig.106. The central parts in the figures are the regions encompassed by the narrowband approximation, where no distinctions between the two types of couplings were necessary. It is clear from the figure, that signal suppression requirements in the low or high-frequency range of a given application favors either a capacitive or an inductive solution. In practice, care must be exercised when realizing the selected coupling type in limits where the connecting impedance Zc is supposed to be large. At high-frequencies with inductive coupling, stray capacitances between the primary and secondary circuits will destroy the steep roll-off asymptote. A small mutual inductance between the coils would similarly change the low-frequency roll-off if capacitive coupling is employed. Placing coils perpendicular to each other is a first measure against inductive coupling.

10 ) See most books on circuit analysis and synthesis, for instance chap.5 in ref.[1].

J.Vidkjær 101

II-8 Impedance Matching

Transformer and transformer-like couplings of the types considered before are not always the best suited solutions to impedance matching problems. Once build, traditional transformers have few provisions for adjustments. If the impedance to be matched has wide tolerances, the matching circuit should include trimming and tuning components, often two in number to compensate spreadings in both the real and the imaginary impedance compo- nents. The technique to be considered in this section has the advantage of being a direct, graphical approach to matching problems. It is not restricted to certain classes of circuit topologies, even not to certain types of circuit components, but we concentrate on lumped element networks here leaving mixed and pure transmission line methods to discussions in application and microwave literature, for instance ref’s [16], [12] or [4]. The new ap- proach has the drawback of being a single frequency method, which only admits a crude bandwidth estimation. However, if bandwidth control is in question we may get good starting points for adjusting to specifications by subsequent design optimization processes.

Lumped Element Impedance Matching using Smith Charts

Smith charts are commonly associated with problems of calculating impedances and designing matchings in transmission line networks, cf.[7] sec.5.9 or [4] chap.5. Staying solely with lumped elements - as we do in this section - the Smith chart is still a powerful tool in construction of lossless matching circuits. A given problem has often many solutions. With a Smith chart it is possible to overview and chose among different alternatives. Before exemplifying the method a few basic properties on reflection coefficients and the Smith chart are reviewed.

The reflection coefficient Γ of an impedance Z with respect to a reference impedance Z0 is defined

(284)

Although written like a complex impedance, the reference Z0 is purely resistive in all practical Ω Ω Ω Ω applications we consider, commonly Z0 is 50 or 75 , but lower values like 20 and 10 are sometimes used in transistor data. Once Z0 is known, it is convenient to normalize all impedances in a given problem with respect to Z0. As indicated by the last equation above, we let small letters refer to normalized impedances or admittances, while capitals stands for -1 -1 the absolute counterparts. Expressed by admittances Y = Z ,Y0 =Z0 , the reflection coeffi- cient is

(285)

J.Vidkjær 102 RF-Circuits, Concepts and Methods

Coming from normalized admittances, the mapping to reflections is seen to be mathematically equivalent to the mapping from normalized impedances, but turned 180° due to the leading minus sign. A few basic reflection coefficients, i.e. the shorted, open, and matched load are summarized in Fig.107(a), which also shows the following important properties,

(286)

i.e. a passive impedance maps inside or on the unit circle in the reflection plane. The circle itself corresponds to the lossless case of a pure reactance. Impedances were used in the demonstrations of Eq.(286) but admittances could be used as well. Fig.107(b) repeats the basic properties taking outset in admittances.

The Smith chart is an overlay to the reflection plane that holds the mapping of a Cartesian coordinate net from either normalized impedance or normalized admittance planes. Both forms are shown in Fig.108. They are contrasted by the 180° rotation implied by the sign difference of the transforming equations. Without details, the mapping given by the normalized parts of Eq.(284) or (285) are known as the bilinear or the linear fractional transformation, [17] chap.16. It is conformal, which implies angle conservation. A major consequence is that circles are mapped onto circles where straight lines are included as a limit case of infinite radius. For the present purpose the most important point is that constant r lines in the z-plane, i.e. lines parallel to the imaginary axis map into a system of constant r circles in the Smith chart as indicated by the upper part in Fig.108. When a reactance is series-connected to an impedance the impedance of the connection is to be found along the constant r circle through the original impedance. Likewise, constant g lines in the y-plane map to constant g circles in the Smith chart. Parallel connecting a susceptance let the resultant admittance move along the circle through the original admittance Y.

Fig.107 Mapping of short, open and matched component to reflection coefficients. Ohmic components are transferred to the real Γ axis and reactive components to the unit circle in the Γ-plane.

J.Vidkjær II-8 Impedance Matching 103

Fig.108 Mappings of impedance or admittance coordinate nets to Smith charts in the reflection plane. Series connected reactance or paralleled susceptances map onto constant r and g circles respectively.

Knowing the reflection coefficient of a one-port, practical translations to both normalized impedances and normalized admittances are easily made by reading the coordinates in the appropriate Smith chart. Covering a Γ-plane with two Smith charts, one in impedance and one in admittance orientation, provides a tool for making graphical inversions from impedance to admittance representations and vice-versa. This is the property that makes the Smith chart highly useful in lumped element matching problems. The technique to find a path that connects the impedance to be matched to the complex conjugated of the matching goal where we go along the constant r circles in impedance representation when a reactance is series-connected and along the constant g circles in admittance representation when a suscep- tance is parallel-connected. The following example illustrates the method step-by-step.

J.Vidkjær 104 RF-Circuits, Concepts and Methods

Example II-8-1 ( double and standard Smith charts )

The task in consideration is the design of a lossless network, which matches a power transistor Ω Ω Ω Ω of 6.92 -j16.8 input impedance at 150 Mhz to a 50 source. Using Z0=50 , the starting point, (1), and the target point, (4), are fixed in the Smith chart. Our job is to find a route

Fig.109 Impedance matching steps using a double Smith chart. The dots correspond to the coordinates in the 1,2,3,4 sequence that is detailed by Eq.(287). between the two points following constant r and g circles only. The choice with two interven- ing points determines the particular network topology shown in the figure. Corresponding components are calculated using z and y values from chart,

(287)

J.Vidkjær II-8 Impedance Matching 105

Fig.110 Example of a double Smith chart ( Analog Instruments Company, N.Y.)

The double Smith chart in Fig.109 is simplified for clarity. A more dense chart is required to reed the normalized impedances and admittances with sufficient accuracy. Fig.110 is an example of a printed double Smith chart. It is sometimes called an immitance chart. Unfortu- nately, the example is also one of the few double charts that are commercially available. Doing without a double chart, several mirrorings through the center are required to substitute the 180° rotations between impedance and admittance representations. The method is demon- strated by Fig.111, where the previous design is repeated. Clearly, the process is more tedious

Fig.111 Impedance matching using a single Smith chart. The steps are similar to the steps in Fig.109, but inversions between impedances and admittances require mirroring through the center.

J.Vidkjær 106 RF-Circuits, Concepts and Methods than before, and it is also more difficult to overview. One guide, however, is to observe that matching to Z0 always have a final step along either the constant r or the constant g circle through the center. In the step prior to the last one, we may aim upon the inverse of the final circle, so it is helpful to draw it into the Smith chart from the beginning as seen in the figure. Example II-8-1 end

Fig.112 Lossless matching. Zin absorbs the available power Pav at the design frequency f0.

The reflection coefficient indicates the impedance difference to Z0, and it is a direct measure for checking the performance of a design. Complete match, Γ=0, means that all the generator available power, Pav, is transferred to the load. Since the matching network is lossless, the only place to deposit power is the resistive part of the input impedance to be matched. The bandwidth of power transfer in the resultant matching network is , therefore, the 3dB limits around the design frequency f0 of the ratio

(288)

Eg is the EMF in the generator of internal impedance Z0.

Fig.113 Conjugated matching across a cut in the lossless matching network. To control bandwidth while constructing matching networks in the Smith chart we get only crude guidelines. The result has to be checked afterwards by a circuit analysis or simulation. Transfer of the available power at the design frequency requires conjugated matching at the generator terminal and the load terminal. In the design process we work through a series of cuts in the lossless matching network where conjugated matching is assumed to maintain maximum power transfer11. Each cut can be considered as shown by

11 ) A formal proof is given in section III-1, where the so-called mismatch factor is introduced and shown to be invariant across any lossless two-port.

J.Vidkjær II-8 Impedance Matching 107

Fig.113. Including all impedances or admittances across a cut, the two imaginary parts cancel and the two real parts add. Had the cut separated pure inductance from pure capacitance, the ratio of absolute reactance value over twice the resistance or the absolute susceptance value over twice the conductance would be the Q-factor of an ideal resonance circuit, either series or parallel. The ratio of the matching network design frequency f0 over a Q-factor, which is calculated from less ideal imaginary part, gives the so called inherent bandwidth at the junction in consideration. In the k’th step that is

(289)

(290)

The resultant bandwidth in a design, which has bandpass characteristics, will be less than any inherent bandwidths. The inherent Q´s are not independent from step to step, so this way of estimating bandwidths is much more crude than any method we have used before. Neverthe- less, inherent bandwidth may guide design decisions on where to adjust a matching network to met bandwidth specifications. Constant Q´s patterns in a Smith chart follow the impedance or admittance angles according to

(291)

A mapping of a polar z or y grid to the reflection plane give a system of constant angle circles as shown in Fig.114. The grid in the reflection plane is sometimes called a polar Smith chart. Translating to Q-values the chart shows curves of constant inherent bandwidths.

Fig.114 Mapping of a polar grid in the impedance plane to a polar Smith chart by the same bilinear transformation, that gave the usual Smith cart.

J.Vidkjær 108 RF-Circuits, Concepts and Methods

Example II-8-2 ( bandwidth estimations )

To exemplify bandwidth estimations, data from the previous example in the steps that have imaginary parts give

(292)

The smallest inherent bandwidth in step 2 clearly overestimates the corresponding value of 55MHz in Fig.115(b), which is based on Eq.(288).

Fig.115 Reflection and transmission through the matching network, which was constructed by Fig.109 and Eq.(287).

Fig.116 Impedance matching steps corresponding to Fig.109, but enforcing a higher Q and a correspondingly lower bandwidth.

J.Vidkjær II-8 Impedance Matching 109

To get a more narrowbanded design, the inherent bandwidth in step 2 should be decreased. Enlarging inductor L1 moves the joining point closer to the border of the Smith chart and enlarges the inherent Q. The design in Fig.116 is controlled by

(293)

Fig.117 Matching network from Fig.116 and Eq.(293) ( dotted curves from Fig.115 ). A simulated response of the design is given in Fig.117. As seen, the bandwidth has reduced to 31 MHz, which is close to the calculated inherent bandwidth in step 2 above. To get the frequency response curves it was assumed that the input impedance originates from a parallel connection of a resistor and a capacitor. It is the most probable configuration in a transistor. In the present case, cf. step 1 in Eq.(293),

(294)

J.Vidkjær 110 RF-Circuits, Concepts and Methods

To make bandwidth wider than the one in the first design, the inductance L1 should be reduced. The smallest inductance, and the smallest inherent Q in step 2, corresponds to the situation in Fig.118. Here the matching network simplifies to only two components.

Fig.118 Impedance matching to the same requirements as the circuit in Fig.109, now using a simpler, low-Q circuit.

(295)

The resultant matching network is no longer a bandpass circuit, so bandwidth estimation becomes somewhat misleading, although the simulated response in Fig.119 indicates a widen- ing. Matching to transistors, as this and the previous examples were supposed to do, com- monly require two tunable components to adjust for parameter spreadings in both the real and the imaginary parts of device impedances. Since trimmer capacitors are far the easiest

J.Vidkjær II-8 Impedance Matching 111

Fig.119 Matching network from Fig.118 and Eq.(295) ( dotted curves from Fig.115 ). component to adjust, and since a transistor matching network must often include an inductor, most matching circuits for this purpose commonly hold three or more components, so our last example is a limit case. A little afterthought reveals that using three components, it is possible to match from everywhere to everywhere in the Smith chart. Different networks are used to avoid impractical component values with distinct topologies and matching goals.

Example II-8-2 end

If the possibility of easy tuning is unimportant in an application, two element matchings may suffice. However, different configurations have different ranges of coverage. They are summarized by Fig.120. where the hatched areas are the impedances that can be matched to the center impedance in the four possible LC combinations. The regions are covered by the element closest to the unmatched impedance, when the component nearest to the generator in the final step follows the particular constant g or r half-circle, along which each component type and configuration can reach the center.

Fig.120 Matching to impedance Z0 in the Smith chart center by two element LC networks. Hatched areas show the admittance or impedance ranges that can be matched.

J.Vidkjær 112 RF-Circuits, Concepts and Methods

J.Vidkjær 113 APPENDIX II-A, Power Calculation and Power Matching

This appendix summarizes basic power relations and impedance matching properties. Consider first an load impedance ZL, which is driven by a single tone of frequency fo Hz. At a given time t, the instantaneous current and voltage are expressed

(296)

ϕ ϕ IL and VL are amplitude or peak values of current and voltage respectively while I and V are the corresponding phases. It is assumed that current and voltage in ZL are sensed for power consumption as indicated by Fig.121. The instantaneous power absorbed by the load at any time t, PL(t), is given by the product

(297)

Under stationary conditions, the average power given to the load is the mean value of the instantaneous power over a period, i.e.

(298)

In this calculation, the last cosine integral vanishes - it would even do so had we taken only half a period as the averaging interval. At frequency f0 average power consumption is positive and the impedance termed passive if the cosine is positive, which requires that the absolute phase difference between current and voltage is less than 90°. Expanding to complex notation the average power - in brief often just the power - may be expressed

Fig.121 Current and voltage orientations for power calculations in a load impedance.

J.Vidkjær 114 RF-Circuits, Concepts and Methods

(299)

In the rewritings above we have first introduced the scalar rms (root mean square) current and voltage. Taking current as the main example they are given by

(300)

Evaluation of the integral follows the pattern from Eq.(298). Complex current and voltage

(301) were subsequently substituted. Here and in the main text lower case i’s and v’s always represent complex currents and voltages in rms scale while capitals are more freely used. Introducing the complex impedance ZL with real and imaginary components RL and XL, i.e. (302) the average power is given by

(303)

Using the impedance in polar form

(304)

the power may also be expressed,

J.Vidkjær II-A - Appendix - Power Calculation and Power Matching 115

(305)

Driving a load impedance ZL from a generator with rms-scaled EMF (electromotive force) Eg and impedance Zg=Rg+jXg, as sketched in Fig.122(a), the load current and voltage become

(306)

The corresponding power in the load, using Eq.(299), becomes

(307)

For a given generator Zg,Eg, where the impedance is supposed to be passive, the concept of power matching corresponds to the situation where a passive load impedance is adjusted to absorb maximum power from the generator. Since reactances may be both positive and negative, one step towards optimum power is to let the last term in the denominator vanish using

(308)

In this case power given to the load impedance is

(309)

Passive load resistances are positive so the value that gives optimum power must be found by setting the differential coefficient of the load dependent factor above equal to zero. We get

Fig.122 Thévenin (a) and Norton (b) form of driving a load impedance ZL or admittance YL from a generator

J.Vidkjær 116 RF-Circuits, Concepts and Methods

(310)

Both resistances must be positive due to the passivity assumptions. The condition corresponds to an optimum as it is easily seen that the second order derivative is negative if the two resistances are equal. Combining the two requirements from Eqs.(308) and (310), optimum power is absorbed in the load, if the load impedance is set equal to the complex conjugated of the generator impedance. Under this condition, the power given to the load is the maximum that the generator can provide, and it is termed the available power. It is expressed through Eq.(307), which gives

(311)

Had we used the Norton form in Fig.122(b) instead of the Thévenin form from Fig.122(a) to describe the driving of the load from a generator, the load voltage and current are expressed

(312)

Compared to Eq.(306) these expressions have the same structure if we replace generator EMF by the short-circuit current Ig, generator impedance with generator admittance, and load impedance with load admittance. In a strict Thévenin to Norton transformation we should use

(313)

Due to the duality between the two equivalent representations, we get similar results with respect to the optimal power transfer from the generator to the load. The results are in summary

(314)

For obvious reasons, power matching is sometime referred to as conjugated matching or impedance matching.

J.Vidkjær 117 APPENDIX II-B Signal Flow Graphs

Signal flow graphs are tools for visualizing the relationships between the variables in linear systems like electronic circuits under small-signal excursions12. Furthermore, flow graphs make provisions for writing down transfer functions in symbolic form between pair of variables from inspection of the graph. The latter is an alternative to solving the pertinent system of equations by eliminating all superfluous variables. With small or sparse circuits the signal flow graph approach is often the easiest way of the two.

Elementary Signal Flow Graphs

Basic elements in flow graphs are nodes and directed branches that connect the nodes. The nodes represent variables, typically currents and voltages in electronic circuits. There is a number associated with each branch. It is the factor that is multiplied on the variable at the node, where the branch has its outset, to give its contribution to the variable at the node in the receiving end of the branch. The branch factor is sometimes called the gain of the branch regardless of its actual units. Nodes that have outgoing branches only are called source nodes, and they represent the independent variables in a problem. Nodes with at least one incoming branch hold a dependent variable. As examples the simple proportionality relationship in Ohm’s law from Eq.(315) may be visualized by either of the two flow graphs in Fig.123(b), the upper one using current as independent source variable and voltage as dependent variables, the lower one using voltage as source variable and current as dependent variable.

(315)

Fig.123 Simple impedance,(a), with two flow graph representations,(b).

The amplifier circuit in Fig.124(a), where gain and input current are given by Eq.(316), may be represented by the flow graph in Fig.124(b) that has two final, dependent variables, input current iin and output voltage v2. To express the gain A=v2/v1 we follow a so- called path in the flow graph from source node v1 through the gm branch to the dependent current variable ic and further on through the Ztot branch to the dependent output variable v2.

12 ) Signal flow graphs should not be confused with the network graphs that are used in circuit theory to represent Kirchhoff’s equations.

J.Vidkjær 118 RF-Circuits, Concepts and Methods

Comparing with Eq.(316), the resultant gain illustrates the rule, that the gain of two cascaded flow graph branches is the product of the two branch gains.

(316)

Fig.124 Amplifier equivalent circuit,(a), and a corre- sponding flow graph,(b).

Adding contributions from more variables to a dependent variable appears in a signal flow graph by a node that has more than one incoming branch. This is the case for the dependent terminal currents i1 and i2 in the signal flow graph representation of y-parameters for a two-port where the port voltages v1 and v2 are the independent variables. The relations are expressed by Eq.(317) and shown in Fig.125. The corresponding expressions and graphs for a two-port that is characterized by z-parameters, where voltages v1,v2 are the dependent variables and currents i1,i2 are the independent sources, are given in Eq.(318) and Fig.126.

(317)

Fig.125 Two-port y-parameters in block form,(a), or the corresponding signal flow graph,(b).

(318)

Fig.126 Two-port z-parameters in block form,(a), or the corresponding signal flow graph,(b). For larger circuits with more branches or more complicated problems than the demonstrations above, the corresponding signal flow graphs become more involved as do the underlying system of network equation, we have to solve. One way of solving large system of equations is by eliminating superfluous variables successively. In a signal flow graph

J.Vidkjær II-B - Appendix - Signal Flow Graphs 119

(319)

(320)

(321)

(322)

Fig.127 Elementary reduction operations with signal flow graphs. representation, this process may be visualized by repeated use of a few basic rules for signal flow graph reductions. Fig.127 and Eqs.(319) through (322) show how to eliminate a variable, x3, and how to simplify two parallel branches to one branch. In the process of simplifying and redrawing a signal flow graph, we may get self loops. This means that a dependent variable is included at both sides of an equation. To simplify here, we must divide all incoming branches, whether there is one or more, by one minus the loop gain as seen from Eqs.(323),(324) and the corresponding signal flow graphs in Fig.128.

(323)

(324)

Fig.128 Elimination of self-loops in signal flow graphs.

J.Vidkjær 120 RF-Circuits, Concepts and Methods

Fig.129 Equivalent circuit for an amplifier, (a), and signal flow graphs, (b) to (e), which illustrates the successive elimination approach to get the gain A=v2/Eg.

A brief illustration of the signal flow graph simplification approach is given in Fig.129, which shows how to incorporate the generator and input impedance interaction in the amplifier from Fig.124 in order to calculate the voltage gain v2/Eg. The extension to the signal flow graph in (b) above, compared to the previous one, expresses that the transistor input voltage v1 is the EMF of the generator minus the voltage across the generator impedance Zg. By successive eliminations we first remove the two current variables iin and ic to get the graph with a self loop in Fig.129(c). The loop is removed following the rule from Fig.128 and Eq.(323), and in the final step, the transistor input voltage v1 is eliminated. The voltage gain becomes the branch gain of the final branch in Fig.129(e), which directly connects the independent variable Eg to the dependent variable v2, i.e.

(325)

The elimination process here is included to emphasize that the signal flow graph is just another way of representing linear equations like network equations, and that it is possible to operate on the graphs like we otherwise should handle the corresponding system of equations. It is clear, however, that to conduct the successive reduction method on larger problems would require a series of redrawings. It is not evident, that this would be beneficial in many practical cases. However, the potential of the signal flow graph approach lies not in tedious simplification techniques. It stems from the fact, that any desired transfer function - like the amplifier gain above - can be formulated directly from an inspection of the first signal flow graph that we may draw for a given problem.

J.Vidkjær II-B - Appendix - Signal Flow Graphs 121

Mason’s Direct Rule

Masons direct rule provides us with a recipe for writing down transfer functions between any particular set of an independent and a dependent variable directly from inspection of the signal flow graph. Before phrasing the method we should be a little more strict about terms and definitions. We use

Path: Sequence of branches between two nodes, a starting node and an end node. All branches must be directed the same way and any node included in the path must be encountered only once (there are no loops).

Loop: Like a path but with coinciding start and end nodes.

Nth order Loop: A set of N nontouching loops (i.e. loops without common nodes).

Path Gain: The product of all branch gains in a path.

Loop Gain: The product of all branch gains in a loop (also higher ordered loops).

The transmission Tij from an independent variable xsj represented by a source node to a dependent variable xi represented any non-source node may be written in the form

(326)

where k sums over all distinct paths from node xsj to node xi, and where

Δ : Graph determinant, calculated by

(327)

( Note the alternating signs )

Pk: Path gain of the kth path from node xsj to node xi.

Δ Δ k : Sum of terms from including the leading "1" followed by contributions that come from loops, which don’t touch the kth path.

The proof for the approach is somewhat complicated and beyond the scope here, so we familiarize ourself on how to use the rule in practice below. Mason, who formulated the rule, gave an intuitive derivation in ref´s [18],[19]. However, it couldn’t satisfy mathemati- cians and more formal proofs may be found elsewhere, for instance in [20]. For obvious reasons, Mason’s direct rule is also known as "the non-touching loop rule".

J.Vidkjær 122 RF-Circuits, Concepts and Methods

Fig.130 Indication of the path P1 and the loop L1 in the signal flow graph from Fig.129. Our first demonstration is to show how Mason’s direct rule works with the example from Fig.129. The signal flow graph for the problem is repeated in Fig.130(a). It has one path, Δ P1, and one loop L1, which touches the path in the v1 node, so 1 to be used as a factor on P1 becomes one. We get

(328)

As seen, the resultant voltage gain equals the gain in Eq(325), which formerly was found by successive reductions.

Fig.131 Calculation of input admittance of a loaded two-port, (a), from the corresponding signal flow graph (b). The paths and the loop are indicated in (c).

The problem in Fig.131 is to find the input admittance of a loaded two-port, which is characterized by its y-parameter matrix. As indicated, there are two paths and one loop in Δ this problem. The first path, P1, does not touch the loop, so the 1 factor includes the contribution to the determinant from the loop ( in this simple, one-loop example the factor even equals the determinant ). From the signal flow graph we identify the terms

(329)

J.Vidkjær II-B - Appendix - Signal Flow Graphs 123

Fig.132 Calculation of voltage gain through a loaded twoport,(a), from the signal flow graph in (b), where (c) indicates paths and loops. Note the 2nd order loop L1L2.

By the final example we shall find the voltage gain v2/Eg through a loaded two-port - specified by y-parameters - which is driven from a generator with given impedance Zg. The signal flow graph for this problem is shown in Fig.132(b). It is seen that there are three loops in the graph, L1,L2, and L3. Furthermore, there is a second order loop made of L1 and L2, since these two loops are non-touching. There is only one path P1 from the independent Δ voltage Eg to v2 and it touches all loops, so the 1 factor simplifies to one. The gain becomes

(330)

The last result demonstrates how the signal flow graph approach using Mason´s direct rule - even in relatively small problems - greatly simplifies the efforts that are required to set up transfer functions in electrical networks. To get the result in Eq.(330) by traditional network calculations, for instance from node equations, would require much more work.

J.Vidkjær 124 RF-Circuits, Concepts and Methods

J.Vidkjær II Problems 125

Problems

P.II-1 Draw phasor diagrams like Fig.2b showing iC,iR,iL, i, and v at the frequencies ω ω ω ω ω ω ω ω =0.1 0, =0.5 0, =2 0, and =10 0. P.II-2

Fig.133

A simple receiver for the medium frequency band is shown in Fig.133(a). The oscillator frequency, fo, is controlled by a parallel resonance circuit as is the RF filtering after the antenna, which is tuned to the carrier frequency fRF between 550 kHz and 1.550 MHz. The mixer downconverts to the intermediate frequency fIF=fo- ≈ fRF 450kHz, so the frequency difference between the two resonance circuits should stay constant in tuning. Tuning elements are two identical capacitors, the Ct´s, which are adjustable from 6pF to 100pF. Resonance frequencies of the tuned circuits are nonlinear function of the capacitances, so exact IF frequency can be obtained at two input frequencies only. They are chosen to fRFl=617 kHz and fRFu=1.483 MHz.

Find the components Lrf,Crf,Lo,Co, and draw a curve showing the deviation from the nominal IF frequency of 450 kHz, the so-called tracking error. What is the maximum error?

The IF error may be substantially improved using a so-called padding capacitor, Cp in the oscillator circuit as shown in Fig.133(b). Recalculate all components using Cp = 118 pF, and draw a new curve of frequency deviations. What is the new maximum tracking error?

P.II-3

Fig.134

J.Vidkjær 126 RF-Circuits, Concepts and Methods

≥ Derive an expression for the transient decay voltage Vdcy(t), t 0, when the switch in Fig.134 breaks the dc-current IL0 at t=0. What is the maximum value of Vdcy if Ω L=100nH, C=47pF, Rp=3.3k , and IL0=15 mA ?

P.II-4

Fig.135 Π Ω The -circuit C1,L,C2 is inserted to match a transistor of input impedance 150 paralleled by 12.3 pF at 100 MHz. C2 is chosen to 67.7 pF. Use series-to-parallel techniques to find L and C1. Calculate the voltage ratio vin /Eg at the center frequen- cy and estimate the 3dB bandwidth of power transfer to the transistor.

P.II-5

Fig.136 Show that the amplifier above in Fig.136(a) with the rudimentary transistor model Fig.136(b) can realize a third order lowpass Butterworth frequency characteristic.

Find the components C0,C1, and L1 that give the amplifier 230MHz bandwidth when Ω gm=50mS and Rg=RL=300 . Why is it not possible to use a parallel tuned circuit at the output to realize the complex conjugated pole pair as it is done in bandpass stagger tuned amplifiers?

P.II-6

Fig.137

J.Vidkjær II Problems 127

Fig.138

A bipolar transistor is described by the equivalent circuit in Fig.137. It is used in a tuned amplifier, which has functional diagram as shown by Fig.138. The two induc- tors L1 and L2 are tapped and used as autotransformers where tight coupling is assumed. Design the amplifier, i.e. find L1,N1,C1,C2,L2,Rp, and N2 to meet the specifications: - center frequency = 110 MHz, - 3dB bandwidth of v2/Eg = 8 MHz in synchronous tuning, - impedance matching at the input side - impedance matching at the output side with a total collector load of 80Ω at the center frequency

Calculate - in dB - the center frequency voltage gain v2 /Eg and the operating power gain Pout /Pin.

P.II-7

Fig.139

A bipolar transistor, for which we use the equivalent circuit in Fig.139, is employed in a tuned amplifier that has the functional diagram shown in Fig.140.

Fig.140

The requirements to the amplifier are,

- center frequency, f0 = 50 MHz, - 3dB bandwidth, BW3dB = 4 MHz,

J.Vidkjær 128 RF-Circuits, Concepts and Methods

- 3rd order Butterworth frequency characteristics for v2 /Eg, and - center frequency impedance matching at the input. The Butterworth characteristic is achieved by tuning the input circuit to the center frequency and let the double-tuned output circuit be coupled overcritically. It is assumed that the two output resonance circuits operates identically, i.e. with same resonance frequencies and q-factors. To ensure stability - and usefulness of the Ω transistor equivalent circuit - R2 is set to 100 .

Find the components C11,C12,L1,L2,Ck,C21, and C22.

Calculate the operating power gain, Pout /Pin at the center frequency .

P.II-8

Fig.141

The amplifier stage in Fig.141 is loaded with a critically coupled double-tuned circuit, where the two sides are equal usingC=60pF.Thetransistor is modeled by a simple transconductance gm=18 mS. Find R, L, and k so the gain v2/v1 gets 3dB bandwidth equal to 300 kHz around a center frequency of 10.7 MHz. Two of the stages above are cascaded. What is the resultant 3dB bandwidth? Sketch the low and high frequency asymptotes of the voltage gain.

P.II-9

Fig.142

Explain the function of the FM detector in Fig.142. Calculate ( Matlab, Matematica, etc. ) detector characteristics and find the kQ that gives the best linearity.

J.Vidkjær II Problems 129

Compare performance with the quadrature demodulator ( could show why the principle in Fig.142 was never successful )

P.II-10

Fig.143

Find component values Ca,Cb,Cc,La, and Lb so the amplifier in Fig.143(a) gets a maximally flat gain characteristics, v2/v1, of bandwidth 8 MHz and center frequency 100 MHz. Fig.143(b) shows the transistor equivalent circuit. It is supposed that the two sides of the output double-tuned circuit have equal Q-factors and center frequen- cies but differs in component values due to the impedance transformation between Ω Ω the load RL=50 and the transistor output resistance, Ro=215 . Calculate the center frequency voltage gain using gm=70mS.

P.II-11

A transistor has the input impedance 87Ω-j82Ω at 160 MHz. Use a Smith chart to design network that matches the transistor to a 50Ω source with an inherent band- width not exceeding 80 MHz. The network should include two capacitors to make the network suitable for adjustments.

P.II-12

Fig.144

A matching network for a RF power MOS transistor is shown in Fig.143, Fig.144. The input impedance is the nominal value from the data sheets (Motorola, MRF136). The two capacitors are identical trimmers that are adjustable from 15pF to 115 pF.

J.Vidkjær 130 RF-Circuits, Concepts and Methods

What is the value of the inductor L if it is supposed that matching is achieved when C1 is set to 45 pF ? Show in a Smith chart the range of input impedances that can be matched using the full adjustment range of the two trimmers.

P.II-13 Repeat problem P.II-4 using a Smith chart (exclude gain computations)

J.Vidkjær 131

References and Further Reading

[1] L.Weinberg, Circuit Analysis and Synthesis, McGraw-Hill 1962

[2] G.C.Themes, S.K.Mitra ed´s, Modern Filter Theory and Design, Wiley 1973

[3] R.Shaumann, M.S.Ghausi, K.R.Laker, Design of Analog Filters, Passive, Active RC and Switched Capacitor, Prentice Hall 1990.

[4] R.E.Collin, Foundations for Microwave Engineering, 2nd ed. McGraw-Hill 1992.

[5] R.Goyal ed., High-Frequency Analog Integrated Circuit Design, Wiley N.Y., 1995.

[6] H.W.Bode, Network Analysis and Feedback Amplifier Design, D.Van Nostrand, 1945.

[7] S.Ramo, J.R.Whinnery, T.Van Duzer, Fields and Waves in Communication Elec- tronics , 3rd.ed., Wiley 1994.

[8] R.A.Adams, A Complete Course in Calculus, 3rd.ed., Addison-Wesley, 1995.

[9] R.A.Rohrer, Circuit Theory, An Introduction to the State Variable Approach, Mc- Graw-Hill 1970.

[10] N.Balabanian, T.A.Bickart, Electrical Network Theory, Wiley, 1969.

[11] F.E.Terman, Electronic and Radio Engineering, 4th.ed, McGraw-Hill 1955

[12] P.L.D.Abrie, The Design of Impedance-Matching Networks for Radio-Frequency and Microwave Amplifiers, Artech House, 1985.

[13] B.Gilbert,"A Precise Four-Quadrant Multiplier with Subnanosecond Response", IEEE.J. Solid-State Circuits, Vol. Sc-3,pp.365-373, Dec.68, reprinted in [15].

[14] P.R.Grey, R.G.Meyer, Analysis and Design of Analog Integrated Circuits, 3rd ed.,Wiley 1993.

[15] A.B.Grebene, ed., Analog Integrated Circuits, Part V: Multipliers and Modulators ( 5 papers incl. [13] ) , IEEE Press, 1978.

[16] B.Becciolini,"Impedance Matching Networks Applied to RF Power Transistors", AN721, Motorola RF-Application Reports, 1995 ( includes many more application notes on matching methods and problems ).

[17] E.Kreyzig, Advanced Engineering Matemathics, 7th ed., Wiley 1993.

[18] S.J.Mason,"Feedback Theory: Some Properties of Signal Flow Graphs", Proc.IRE, vol.41, pp.1144-1156, 1953

[19] S.J.Mason,"Feedback Theory: Further Properties of Signal Flow Graphs", Proc.IRE, vol.44, pp.920-926, 1956.

J.Vidkjær 132 RF-Circuits, Concepts and Methods

[20] K.Thulasiraman,"Signal Flow Graphs", Chapter 8 in W.K.Chen (editor) "The Circuits and Filters Handbook", CRC and IEEE Press, 1995.

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Index

β(ω)...... 3,12,14 identical resonance circuits ....81 A_L value (inductors) ...... 47,62 inductive coupling ...... 82 Autotransformer ...... 61 LF and HF asymptotes ...... 98 Bandwidth unsymmetrical load ...... 96 double tuning, kQ=1 ...... 94 Envelope Detector parallel circuit ...... 4 ideal ...... 87 series circuit ...... 13 External Q-factor ...... 25 Bandwidth Reduction Factor ...... 32 FM Detector Bandwidth Shrinkage Foster-Seely ...... 85 !see bandwidth reduction .....32 quadrature ...... 88 Broadband Modeling ...... 26 Foster-Seely FM detector ...... 85 Butterworth Characteristic ...... 36 Frequency Modulation, FM double-tuned circuit ...... 95 Foster-Seely detector ...... 85 Capacitive Coupling quadrature detector ...... 88 double-tuned circuits ...... 82 Gain-Bandwidth factor, GBF ...... 31 Capacitive Transformer ...... 64 double-tuning ...... 95,98 Close Coupling ...... 47 maximum ...... 31 Conjugated Matching ...... 116 Gain-Bandwidth Product, GW ...... 31 Coupled Inductors ...... 49 Gaussian frequency characteristic ....35 PI-equivalent circuits ...... 51 Gilbert Cell T-equivalent circuits ...... 49 quadrature FM detector ...... 92 Coupled Resonance Circuits Hybrid ! see Double-Tuned Circuit ....80 transformer ...... 72 Coupling Immitance ...... 18 tight or close inductive ...... 47 Immitance Chart ...... 105 Coupling Coefficient ...... 46,82 Impedance Matching ...... 101, 116 autotransformer ...... 61 lumped elements ...... 101 capacitive ...... 82 two element LC ...... 111 inductive ...... 46,82 Inductive Coupling ...... 45 PI equivalent circuit ...... 53 double-tuned circuit ...... 82 Critical Coupling ...... 84 Inherent Bandwidth ...... 107 Damping Ratio Loaded Q-factor ...... 25 parallel circuit ...... 2 Logarithmic decrement ...... 10 series circuit ...... 11 Lowpass to Bandpass Transfor- Diode-Ring Mixer ...... 78 mation ...... 17 Double-Balanced Modulator Mason’s Direct Rule ...... 121 quadrature FM detector ...... 92 Mixer Double-Sideband Suppressed Carrier, diode-ring ...... 78 DSC-SC Mutual Inductances ...... 45 modulator ...... 78 Narrowband Approximation ...... 14 Double-Tuned Circuits ...... 80 _beta(omega) ...... 14 capacitive coupling ...... 82 accuracy ...... 15 critical coupling ...... 84 double-tuning ...... 82,96

J.Vidkjær 134 RF-Circuits, Concepts and Methods

pole-zero conditions ...... 16 polar ...... 107 resonance circuit ...... 15 Spiral Inductor ...... 26 Padding Capacitor ...... 125 Stagger Tuned Amplifier Parallel Resonance ...... 2 Butterworth ...... 36 bandwidth ...... 4 Tapped Inductor damping ratio ...... 2 !see autotransformer ...... 61 inductor series loss ...... 23 Three-Winding Transformer ...... 67 poles and zeros ...... 6 passivity ...... 67 quality factor ...... 2 Tight Coupling ...... 47 resonance frequency ...... 2 Transformer ...... 45 transient response ...... 8 autotransformer ...... 61 Parallel-to-Series Conversion capacitive ...... 64 !see series-to-parallel conver- hybrid ...... 72 sion ...... 19 ideal ...... 48 Passivity perfect ...... 48 coupled inductors (2) ...... 46 RF...... 53 coupled inductors (3) ...... 67 three-winding ...... 67 Poles and Zeros tuned ...... 57 Butterworth amplifier ...... 39 uncoupled inductors ...... 64 double-tuned circuits ..... 83,97 Transient Response parallel circuit ...... 6 parallel circuit ...... 8 series circuit ...... 13 Tuned Amplifier ...... 30 Power autotransformer coupling .....62 calculation ...... 113 Butterworth characteristic .....36 matching ...... 113 double-tuning ...... 94 Q-factor Gaussian characteristic ...... 35 see quality factor ...... 2 stagger tuning ...... 36 Quadrature Detector, FM ...... 88 synchronous tuning ...... 31 Quality Factor transformer-coupled ...... 58 inherent ...... 107 Tuned Transformer ...... 57 loaded, unloaded, external .....25 Unloaded Q-factor ...... 25 parallel circuit ...... 2 Winding Ratio ...... 47 reactive components ...... 25 capacitive transformer ...... 63 series circuit ...... 11 effective inductive ...... 49 Reflection Coefficient ...... 101 ideal, inductive ...... 47 RF-Transformer ...... 53 physical ...... 48 frequency response ...... 54 uncoupled inductors ...... 64 Series Resonance ...... 11 damping ratio ...... 11 poles and zeros ...... 13 quality factor ...... 11 resonance frequency ...... 11 Series-to-Parallel Conversion ...... 19 Signal Flow Graphs ...... 117 Smith Chart ...... 102

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