Elements of Passive Filter Design

J. Sebek July 26, 2001

1 Introduction

The design of arbitrary ¯lters is a complex subject about which much has been written [1][2][3]. It is not the purpose of this note to repeat all of that information. Rather, this note concentrates on the few elements that are required to design and synthesize some elementary ¯lters that are commonly used. With the information presented here, one should, using only some simple numerical calculations, be able to design and build these common ¯lters. This note will concentrate on the design of lowpass ¯lters. It will start with a discussion of several types of ¯lter designs, noting their strengths and weaknesses, and give the equations necessary to design the ¯lter transfer functions. From this viewpoint, the transfer functions describe the transmission of signals from the source through the network to the load. Next the synthesis of simple ladder circuits will be reviewed. This approach views the network and load as an impedance. Standard two port network theory gives the relationship between the transmission and impedance representations. The elements of this theory necessary to make this relation will be reviewed. Using this knowledge, the lowpass ¯lters can be synthesized as common ladder circuits. Finally the lowpass to bandpass transformation will be presented. This transformation can be used to design most band-pass ¯lters of interest.

2 Filter Types

The purpose of a ¯lter is to condition an input signal in a way that preserves the signal characteristics of interest while eliminating the undesirable components. The characteristics can be speci¯ed in the frequency domain, the time domain, or a combination of both. However, ¯lter design is most easily accomplished in the frequency domain, so the ¯rst step in any of the designs discussed here will be to make the speci¯cation in that domain.

2.1 Transfer Function The variable of interest in the frequency domain is the radian frequency !. The frequency domain ¯lter speci¯cation, the transfer function, can be denoted

1 in several ways. Since this function a®ects both the amplitude and phase of the input signal as a function of frequency it is convenient to represent it in complex notation. To emphasize the complex nature of the function and the way in which the frequency enters into it, the transfer function is often denoted H (i!), where i = p 1. For notational purposes, a variable s = i! is often introduced so that th¡e transfer function can also be written as H (s). Transfer functions realizable with discrete components are polynomials. The complex conjugate of such a transfer function satis¯es the Schwarz re°ection principle [4]

H¤ (i!) = H ( i!) : ¡ The complex transfer function can be separated into its amplitude and phase

H (s) = H (s) eiÁ(s) j j When approximating the phase of H (s), one can work with the transfer function itself. But when dealing with the amplitude, it is often more convenient to work, not with the magnitude itself, but with the square of the magnitude, since this can be represented by a polynomial fraction. The square can be represented either as

H (i!) 2 = H (i!) H ( i!) j j ¡ or

H (s) 2 = H (s) H ( s) : j j ¡ In general, the ¯lter approximations are ideal; they cannot be represented by polynomials of ¯nite order. The approximations to the ideal improve with the complexity and number of components of the ¯lter. A lumped element ¯lter, the kind discussed in this note, in general requires one reactive component for each order of the polynomial in H (s). One of the choices in the ¯lter design process is to determine the level of ¯lter complexity required to provide the desired level of signal processing. In order for the transfer function to be realizable, all of its poles must lie in the left half complex plane, that is, Re s 0. Once the square of the magnitude function is determined from othefr cgo·nsiderations, the realizable transfer function will be determined by this condition on the location of the poles.

2.2 Butterworth Approximation 2.2.1 Butterworth Transfer Function Perhaps the most common ¯lter approximation type is the Butterworth ¯lter. Its characteristic is that its low-pass ¯lter is \maximally °at" at the origin. The perfect Butterworth ¯lter would be perfectly °at at the origin; that is, all

2 derivatives of H (s) with respect to s would vanish there. For a ¯nite Butter- worth approximation of order n, the best polynomial fraction approximation is given by 1 H (s) 2 = j j 1 + C ( s2)n ¡ 1 = 1 + C ( 1)n s2n ¡ 1 = (1) 1 + ( 1)n s2n ¡ where C is a constant that sets the frequency scale of the ¯lter. In the last line, the scaling constant has been incorporated into the frequency variable s. In all subsequent discussions, unless otherwise noted, such scaling constants will be assumed incorporated into s. If one needs to calculate a ¯lter function for a particular frequency, one can start with a normalized function such as the one above, and in the end replace s with the appropriately scaled value s=s0, where s0 is the scale of the physical problem of interest. An analytic solution for the pole locations of the Butterworth ¯lter can be easily found. The 2n possible locations are the solutions of the equation

i¼ 1+n s = e 2n i¼ i¼ = e 2 e 2n and the realizable poles are the n in the left half of the complex plane. One sees that these poles uniformly populate the unit circle. If n is even, the poles are complex pairs centered about the real axis; if n is odd, there is one real pole on the negative real axis, and the rest form pairs centered around that axis.

2.2.2 Butterworth Characteristics Although ¯lters are generally characterized by their transfer functions in the frequency domain, their e®ect on signals may be best described in either the frequency of time domain. All lowpass ¯lters pass DC and, above some frequency, attenuate the signals with an attenuation that grows with frequency. For a polynomial ¯lter as described above, the high frequency behavior is determined by the order of the ¯lter. The di®erence in the response of the di®erent ¯lter families occurs in the intermediate frequencies. The Butterworth ¯lter is the \vanilla" ¯lter. Its frequency response is monotonic; its attenuation increases with frequency. It passes signal pulses reasonably well and its step response has some overshoot, but not an excessive amount. A half-power (3 dB) point can easily be calculated for this ¯lter. From equation 1, this corner frequency is easily seen to be s = 1. The Butterworth design is often the default ¯lter of choice. In applications in which the system needs a sharper transition between the passband and stopband, or in which the system needs a more faithful response to a pulse, however, other ¯lter families have better performance.

3 2.3 Chebyshev Approximation 2.3.1 Chebyshev Transfer Function Another common desired ¯lter response is one with \equal ripple" throughout the ¯lter passband. The magnitude squared of a transfer function satisfying this requirement is 1 H (i!) 2 = (2) 2 2 j j 1 + ± Tn (!) where ± is a measure of the passband ripple and Tn (!) is the Chebyshev polynomial of order n. The Tn (!) are de¯ned by the equation

cos (n arccos !) ! 1 T (!) = : (3) n cosh (n arccosh !) j!j · 1 ½ j j ¸ Straightforward algebraic manipulation results in the recursion relation

T0 (!) = 1

T1 (!) = !

Tn+1 (!) = 2!Tn (!) Tn 1 (!) ¡ ¡ which demonstrates that these functions are, in fact, polynomials. The locations of the poles of the Chebyshev transfer function can also be analytically calculated. The poles of equation 2 satisfy i T (!) = : n §± Since s = i!, this can also be written s i T = : (4) n i §± ³ ´ Setting

w = x + iy and

s = i cosh w = i cosh (x + iy) = i cosh x cos y sinh x sin y (5) ¡ and substituting in equation 3, equation 4 can be rewritten as i cosh nx cos ny + i sinh nx sin ny = : §±

4 Equating the real and imaginary parts of the this equation gives the possible values of x and y for the poles 2k 1 ¼ y = ¡ n 2 1 1 x = arcsinh : n ± Substituting these values back into equation 5 gives the pole locations of the transfer function in the left half complex plane sk = ¾k + i!k 1 1 2k 1 ¼ 1 1 2k 1 ¼ = sinh arcsinh sin ¡ + i cosh arcsinh cos ¡ : ¡ n ± n 2 n ± n 2 µ ¶ µ ¶ µ ¶ µ ¶ Geometrically, these poles are seen to lie on an ellipse in the complex plane described by the equation 2 2 ¾k !k 2 1 1 + 2 1 1 = 1: sinh n arcsinh ± cosh n arcsinh ± ¡ ¢ ¡ ¢ 2.3.2 Chebyshev Characteristics The frequency response of a Chebyshev lowpass ¯lter ripples in the passband. These ripples have equal magnitude, hence the term \equal ripple". The last ripple of the ¯lter always points toward greater attenuation, so that the transition between the passband and the stopband is sharper for the Chebyshev than for the Butterworth; the Chebyshev is a better approximation to a \brick wall" ¯lter. This sharp transition is responsible for increased ringing in the time domain responses of this ¯lter family, an often undesireable property. This ¯lter can also be described by a cuto® frequency. At the edge of the ripple band, the magnitude squared of the transfer function is 1 H (1) 2 = : j j 1 + ±2 Only if ± = 1 has the amplitude decreased by 3 dB at s = 1. If ± < 1, the 3 dB frequency can be calculated from equation 3 1 cosh (n arccosh ! ) = 3dB ± resulting in 1 1 ! = cosh arccosh : 3dB n ± µ ¶ Normalized to these values the pole locations are sinh 1 arcsinh 1 2k 1 ¼ cosh 1 arcsinh 1 2k 1 ¼ s3dB = n ± sin ¡ + i n ± cos ¡ : k ¡cosh 1 arccosh 1 n 2 cosh 1 arccosh 1 n 2 ¡ n ±¢ µ ¶ ¡n ± ¢ µ ¶ ¡ ¢ ¡ ¢

5 2.4 Bessel Approximation 2.4.1 Bessel Transfer Function Often one is more interested in the characteristics of the ¯lter in the time domain, particularly when one deals with pulsed, rather than continuous, wave- forms. In such cases the desired e®ect of the ¯lter is to preserve, as well as possible, the shapes of the pulses. The ¯lter then approximates a linear phase system. The \maximally °at" delay function is also known as a Bessel approximation, since it can be expressed in terms of Bessel polynomials. The ideal linear phase transfer function is the unit delay function

s H (s) = e¡ :

Since we are working here with the phase and not the amplitude, we just need to approximate H (s), and not the square of its magnitude. By expressing this function cleverly [5], the origin of the Bessel response can be seen. 1 H (s) = es 1 = cosh s + sinh s 1= sinh s = : 1 + coth s coth s can be expressed by a continued fraction 1 1 coth s = + : s 3 1 s + 5 + 1 s 7 +::: s This fraction is then truncated to desired level of the polynomial expansion and the numerator and denominator of this expansion are identi¯ed with cosh s and sinh s, respectively. To calculate the fourth order expansion, for example, 1 1 coth s + 3 1 ¼ s + 5 1 s s + 7 s 1 1 + 3 1 ¼ s + 5 s s s + 7 1 1 + 3 7s ¼ s s + s2+35 1 s3 + 35s + ¼ s 10s2 + 105 s4 + 45s2 + 105 ¼ 10s3 + 105s

6 so that

sinh s 10s3 + 105s ¼ cosh s s4 + 45s2 + 105 ¼ and 105 H (s) = s4 + 10s3 + 45s2 + 105s + 105 where the numerator of H (s) has been set to give a unity gain at s = 0. The denominator is related to a class of Bessel polynomials for which there exists a recursion relation that generates these polynomials

B0 (s) = 1

B1 (s) = s + 1 2 Bn (s) = (2n 1) Bn 1 (s) + s Bn 2 (s) ¡ ¡ ¡ 2.4.2 Bessel Characteristics As can be seen from the coe±cients calculated for the fourth order Bessel transfer function, the high frequency terms in the denominator are relatively less important here than in the Butterworth and Chebyshev functions. This shows that the increase in attenuation is less steep for the Bessel than for the other families. This is to be expected since the Bessel was designed to preserve the time domain characteristics, which requires that a large fraction of the signal frequency content be preserved. The coe±cients for the transfer function derived above were normalized for the time domain. They provided for a unit delay and unity gain at ! = 0; they are not normalized to a 3 dB attenuation at s = 1. Tabulated coe±cients given in references[1] may di®er from the above as they may be calculated for the 3 dB condition. In such a case, scaling of the frequency will recover the coe±cients given above.

2.5 Gaussian Approximation 2.5.1 Gaussian Transfer Function The other style of ¯lter often used for preserving the time domain characteristics of a signal is the Gaussian ¯lter. The property that the Fourier transform of a Gaussian is itself a Gaussian explains the symmetry between the frequency and time domain characteristics of this ¯lter. Here again one starts with the magnitude squared of the ideal transfer function

2 H (s) 2 = es j j 1 = s2 e¡

7 and then truncates the polynomial expansion at order n 1 H (s) 2 = : 1 ( )n j j 1 s2 + s4 + ::: + ¡ s2n ¡ 2! n! Again, the poles of H (s) 2 which lie in the left hand complex plane are the poles of the transfer jfunctijon.

2.5.2 Gaussian Characteristics The response of the Gaussian transfer function is also good at preserving the pulse shape of the signal. The Bessel ¯lter gives a more uniform passband delay and greater stopband attenuation, while the amplitude of the overshoot of a Gaussian ¯lter is smaller.

3 Ladder Filter Synthesis 3.1 Ladder Filter Algorithm Once the desired transfer function is known, the ¯lter must be realized. Of course, there are many di®erent choices of components that will synthesize the ¯lter. Given here is an algorithm to realize a given lowpass transfer function as a ladder ¯lter of inductors and capacitors.

Filter Network

ZS Z2

U Y1 Y3 ZL

Figure 1: Generic ladder ¯lter topology

A straightforward algorithm synthesizes this network as an impedance viewed from the source. The algorithm starts at the load and works upstream toward the source. The impedance of the load is rL. The combination of the last element of the network in ¯gure 1, the admittance Y3 and rL is the admittance

8 Y + 1 . Continuing in the same manner gives the total impedance of the ¯lter 3 rL network and load. As an example, the three pole ¯lter in ¯gure 1 with two shunt and one series element has an impedance 1 Z = 1 : Y1 + 1 Z2+ 1 Y3+ rL

In the standard lowpass ¯lter application ZL is a pure resistance, all shunt legs are capacitances, Yi = sCi, and all series legs are inductances, Zi = sLi. The impedance is then represented as the fraction 1 Z = 1 sC1 + 1 sL2+ 1 sC3+ rL 2 s L2C3rL + sL2 + rL = 3 2 s C1L2C3rL + s C1L2 + sC1rL + sC3rL + 1 1 s2 + 1 s + 1 = C1 C1C3rL C1L2C3 : s3 + 1 s2 + 1 C1+C3 s + 1 C3rL L2 C1C3 C1L2C3rL ³ ´ Once the impedance seen by the source is known as a polynomial fraction in s, the ¯lter components can be determined by synthetic division, the inverse operation of the above construction.

3.2 Scattering Parameters We now have the tools to both calculate the transfer function once the desired ¯lter response is known and to determine the values of the ¯lter elements once the ¯lter impedance is known. What remains is to learn the method to connect the impedance with the transfer function. This method requires some knowledge of scattering parameters. The ¯lter will be considered as a lossless two port network, and the scattering matrix properties of such a system will lead to the desired relationship.

3.2.1 Formalism A general two port network has two independent parameters and can be characterized in several ways. In the scattering matrix representation, the network is characterized by the incident and re°ected voltages at each port. The incident voltages are the independent parameters and the re°ected voltages are the de- pendent parameters. The corresponding currents are obtained from Ohm's law. Although this representation is commonly used with electromagnetic waves, with the proper identi¯cation of the incident and re°ected variables, it is also a well-de¯ned and useful representation for lumped element networks.

9 + + + + (V1 ,I1 ) (V2 ,I2 ) - - - - (V1 ,I1 ) (V2 ,I2 )

Z1 Z2

U1 Two Port Network U2

Figure 2: Two port network showing incident and re°ected voltages and currents at the ports.

At each port, the source voltage, port voltage, and port current are related by Ohm's law

Un = Vn + ZnIn: (6)

+ + The incident voltage and current, (Vn ; In ), are de¯ned as the voltage and current when the port is matched by the network impedance. Therefore, by de¯nition,

+ + Vn = ZnIn :

The re°ected voltage and current, (Vn¡; In¡), are de¯ned as the di®erence between the actual and matched voltage and current

+ Vn¡ = Vn Vn + ¡ I¡ = I I : n n ¡ n with the standard relation between the re°ected variables

Vn¡ = ZnIn¡:

The incident and re°ected variables can be represented as a symmetrized combination of the total port voltage and current 1 V + = (V + Z I ) n 2 n n n 1 V ¡ = (V Z I ) n 2 n ¡ n n

10 In the cases of interest for this note, the external impedances, Zn, will all be resistive and will be hence denoted by rn. It is more convenient to normalize these equations by the external resistances in each port

1 1 1 1 r¡ 2 V + = r¡ 2 V + r 2 I n n 2 n n n n 1 ³ 1 1 ´ 2 1 2 2 rn¡ V ¡ = rn¡ V rn I n 2 n ¡ n ³ ´ This linear system can be written in a way to exploit simpli¯ed matrix notation. The explicit matrices for the incident variables

+ + V1 r1 0 I1 + = + V 0 r2 I µ 2 ¶ µ ¶ µ 2 ¶ and the corresponding ones for the re°ected variables can be written as

V+ = rI+

V¡ = rI¡:

De¯ning

1 2 1 § r1 0 r§ 2 1 ´ § 2 Ã 0 r2 ! the matrix notation for the normalized variables becomes

1 + 1 1 1 r¡ 2 V = r¡ 2 V + r 2 I 2 1 1 ³ 1 1 ´ r¡ 2 V¡ = r¡ 2 V r 2 I : 2 ¡ ³ ´ In general, the two port network is not matched, so that the relation between the total port voltage and current is

V = ZocI where Zoc is the open circuit impedance matrix of the two port. The equations for the incident and re°ected voltages are 1 V§ = (Z r) I (7) 2 oc§ Algebraic manipulations give the matrix relations between the incident and re°ected voltages. De¯ning the normalized variables

1 + a = r¡ 2 V (8) 1 b = r¡ 2 V¡ (9)

11 and the matrix

1 1 1 S r¡ 2 (Z r) (Z +r)¡ r 2 ´ oc¡ oc one obtains the desired scattering matrix relation

b = Sa:

Introducing a normalized impedance Z

1 1 Z = r¡ 2 Zocr¡ 2 (10) the S matrix reduces to a normalized relation

1 S = (Z r) (Z + r)¡ : ¡ 3.2.2 Re°ection and Transmission Coe±cients In this representation the independent variables are in the vector a, which is the combination of the port voltage and current de¯ned above. The re°ection and transmission coe±cients of the network are de¯ned as expected

b1 S11 = a1 ¯a2=0 V2= r2I2 ¯ ) ¡ b2 ¯ S21 = ¯ a1 ¯a2=0 V2= r2I2 ¯ ) ¡ ¯ with similar de¯nitions for port 2. ¯ The evaluation of these parameters is straightforward and results in

Z1 1 S11 = ¡ Z1 + 1 r V S = 2 1 2 21 r U r 2 1 where Z1 is the normalized impedance of port 1. Z1 is also the re°ection coe±cient of the system and can be expressed in terms of S11 1 + S Z = 11 ; 1 1 S ¡ 11 showing how the impedance is calculated from the knowledge of S11. S21 is a forward voltage gain, known as a forward transducer voltage ratio. When the load impedance is matched to the source impedance, maximum power

2 U1 PL max = j j 4r1

12 is delivered to the load. In general, the power at the load is

2 V2 PL = j j : r2

Therefore the magnitude square of S21 is the ratio of these two powers

2 PL S21 = j j PL max and is also called the transducer power gain !2 = s2 . This is the magnitude square of what we called the ¯lterGtransfer fGun¡ction in the earlier discussion. ¡ ¢ ¡ ¢ Now we have found the relationship between the impedance and S11 and the transfer function and S21. All that is left is to ¯nd the relationship between these two elements of the S matrix.

3.2.3 Power into Network The power delivered into the network is the sum, including sign, of the power input at each port. The power is also represented as a complex quantity. The real part of the power is a real °ow of power into the network from that port. (Complex power is interpreted as oscillating energy that is stored in the network, but since its voltage is out of phase with the current, there is no °ow of energy associated with it.) This power is

VyI = aya byb + bya ayb ¡ ¡ ³ ´ ¡ ¢ where Vy refers to the Hermitian conjugate of the voltage vector, V. What is of interest is the real part of the power into the two port

Re VyI = aya byb ¡ © ª = aya (Sa)y (Sa) ¡ = ay I SyS a: ¡ ³ ´ For a passive system

Re VyI 0 ¸ where the equality holds for a lossle©ss sysªtem. The desired ideal ¯lter network is a lossless system, so that for this network S is unitary

SyS = I:

13 Carrying out the multiplication for the two port system

S1¤1 S2¤1 S11 S12 SyS = S¤ S¤ S21 S22 µ 12 22 ¶ µ ¶ S S + S S S S + S S = 1¤1 11 2¤1 21 1¤1 12 2¤1 22 S¤ S11 + S¤ S21 S¤ S12 + S¤ S22 µ 12 22 12 22 ¶ 2 2 S11 + S21 S1¤1S12 + S2¤1S22 = j j j j 2 2 S¤ S + S¤ S S + S µ 12 11 22 21 j 12j j 22j ¶ 1 0 = 0 1 µ ¶ results in four equations that are satis¯ed for the lossless two port system. If S = 0, one can divide S in both sides of the equation for I and obtain 11 6 11 21 S21 S1¤2 = S2¤2 ¡S11 so that

2 2 S21 2 S12 = j j S22 : j j S 2 j j j 11j Using the relations in I11 and I22 one obtains

2 2 2 S21 2 S12 + S22 = j j2 + 1 S22 j j j j Ã S11 ! j j j j = 1 = S 2 + S 2 j 11j j 21j 2 S21 2 = j j2 + 1 S11 Ã S11 ! j j j j resulting in

S 2 = S 2 : j 11j j 22j

If S11 = 0,

S 2 + S 2 = 1 j 11j j 21j implies

S 2 = 1 j 21j and these relations, along with

S1¤1S12 + S2¤1S22 = 0

14 require

S22 = 0:

Therefore, it is always true that

S 2 = S 2 j 11j j 22j and

S 2 = 1 S 2 j 21j ¡ j 11j = 1 S 2 ¡ j 22j = S 2 j 12j This gives the desired relationship

S 2 = 1 S 2 j 11j ¡ j 21j and, along with the earlier relationships between the impedance and S11, and the transfer function and S21, provides the means to obtain the impedance from the transfer function.

4 Filter Design Algorithm

Now that all of the pieces have been derived, all that is left to do is to assemble them. The steps in the ¯lter design process are

1. Determine the desired transfer function H (s)

2. Compute the transducer power gain s2 = H (s) H ( s) = S (s) 2 G ¡ ¡ j 21 j 3. Compute the magnitude square S (s)¡ 2 =¢1 S (s) 2 j 11 j ¡ j 21 j 2 4. Partition S11 (s) = S11 (s) S11 ( s) so that all of the poles of S11 (s) are in the leftjhand sjide of the compl¡ex s plane. 5. Calculate Z (s) = (1 + S (s)) = (1 S (s)) 1 11 ¡ 11 6. Use synthetic division and the ladder expansion to calculate the values of the physical components.

7. Normalize Z1 to obtain the physical impedance (Zoc)1 = prsrlZ1 8. Scale the frequency as appropriate

15 4.1 Design Example: Fourth Order Gaussian Filter As an example that uses all of the parts of this note, we calculate a fourth order lowpass Gaussian ¯lter. The magnitude square transfer function is 1 H (s) 2 = j j 1 s2 + 1 s4 1 s6 + 1 s8 ¡ 2! ¡ 3! 4! 24 = s8 4s6 + 12s4 24s2 + 24 ¡ ¡ leading to the transfer function

p24 H (s) = (s + 1:1811 i1:0604) (s + 1:3554 i0:3279) § § 4:8990 = (11) s4 + 5:0729s3 + 10:8670s2 + 11:4226s + 4:8990

The magnitude square S (s) 2 is therefore j 11 j S (s) 2 = 1 S (s) 2 j 11 j ¡ j 21 j s8 4s6 + 12s4 24s2 = ¡ ¡ : s8 4s6 + 12s4 24s2 + 24 ¡ ¡ Taking the poles and zeros of the left hand complex plane and using the formula 1 + S Z = 11 1 1 S ¡ 11 where s (s + 1:6689) (s + 1:3309 i1:0789) S = § § 11 (s + 1:1811 i1:0604) (s + 1:3554 i0:3279) § § s4 + 4:3308s3 + 7:3779s2 + 4:8990s = §s4 + 5:0729s3 + 10:8670s2 + 11:4226s + 4:8990 results in 2s4 + 9:4037s3 + 18:2449s2 + 16:3215s + 4:8990 Z = ; 1 0:7421s3 + 3:4892s2 + 6:5236s + 4:8990 where the positive sign of S11 was chosen. The choice of the negative sign will give the dual relation for 1=Z1. The normalized values of the ladder ¯lter components needed to realize this ¯lter are obtained from the continued fraction expansion 1 Z1 = 2:6951s + 1 : 1:1189s + 1 0:6365s+ 0:2127s+1

16 1 2.6951 0.6365

U 1.1189 0.2127 1

(a)

1 1.1189 0.2127

U 2.6951 0.6365 1

(b)

Figure 3: Dual realizations of normalized four pole Gaussian lowpass ¯lter: (a) is series-shunt; (b) is shunt-series.

In the case for which this expansion is an impedance, the ¯rst element of the network is a series inductor; the ¯rst element of the dual admittance representation is a shunt capacitor. There still are two transformations required to realize the actual ¯lter. The source and load resistances scale the impedances, and the cuto® frequency scales the inductances and capacitance. To account for the resistances, the source and load resistances are brought to their actual values, and, using equation 10, the inductances are increased and the capacitances decreased by prsrl. The frequency scaling is accomplished by reducing both the inductance and capacitance by the actual radian frequency, !c, that corresponds to the value

17 of s = i in equation 11

1 r ! s 1 r ! l prsrlLn Ln ! !c Cn Cn : ! prsrl!c

5 Narrow Bandpass Filter Realization 5.1 Lowpass to Highpass Transformation The bandwidth of many bandpass ¯lters is signi¯cantly narrower than the center frequency of the ¯lter. For such cases, the realization of this ¯lter is obtained by a simple transformation from the lowpass characteristic calculated above 1 s ! s + 0 ; lp ! ° ! s µ 0 ¶ where !0 is the center frequency of the ¯lter and ° is a dimensionless parameter that characterizes the ¯lter width. With this rule the inductors and capacitors in the lowpass ladder transform as

L 1 !0L slpL s + ! °!0 s ° C 1 !0C slpC s + : ! °!0 s ° A series inductor L transforms into a series inductor-capacitor pair with inductance L= (°!0) and capacitance °= (!0L), and a shunt capacitor C transforms into a parallel inductor-capacitor pair with capacitance C= (°!0) and inductance °= (!0C). As expected, the series and shunt pairs have zero and in¯nite impedance at !0, just as do the series inductor and shunt capacitor in the lowpass circuit at zero frequency. In the narrow band approximation, the ¯lter center frequency !0 is the image of ! = 0 in the lowpass ¯lter. °!0=2 corresponds to the cuto® frequency !c in the lowpass case. This approximation is only approximately symmetric about !0, with the accuracy of the approximation improving as ° decreases. However, for most ¯lters of interest, this approximation is still very good.

5.2 Design Example: SPEAR 3 IF Bandpass Filter The machinery developed above can now be assembled to design a bandpass ¯l- ter appropriate for the SPEAR 3 beam position monitor intermediate frequency of 16:645 MHz. This ¯lter is designed so that the impulse response of a single

18 bunch rings out in 781 ns, which is one revolution period of SPEAR, enabling single bunch dynamics to be studied in the machine. The four pole lowpass Gaussian ¯lter prototype calculated above is transformed as described above with ° = :129. An impulse passing through a ¯lter with these parameters deposits 99:99% of its energy in a revolution period.

-5

-10

-15

-20

amplitude (dB) amplitude -25

-30

-35

-40 13 14 15 16 17 18 19 20 frequency (MHz)

-100

-200

-300

-400

phase (degrees) phase -500

-600

-700

-800 13 14 15 16 17 18 19 20 frequency (MHz) Bode plot of four pole Gaussian bandpass ¯lter: upper plot is magnitude response; lower plot is phase response.

19 6 x 10 6

0 amplitude

-2

-4

-6 0 0.2 0.4 0.6 0.8 1 time (ms) Impulse response of four pole Gaussian bandpass ¯lter.

The procedure outlined above gives two dual realizations of ladder circuits of this design.

20 50 W 9.99 mH 9.15 pF 2.36 mH 38.8 pF

U 1.66 nF 55.1 nH 315 pF 290 nH 50 W

(a)

50 W 4.15 mH 22.0 pF 788 nH 116 pF

U 4.00 nF 22.9 nH 944 pF 96.9 nH 50 W

(b)

Two realizations of the four pole Gaussian bandpass ¯lter: (a) series-shunt con¯guration; (b) shunt-series con¯guration.

References

[1] A. I. Zverev, Handbook of Filter Design, John Wiley and Sons, Inc., New York, 1967. [2] H. J. Blinchiko® and A. I. Zverev, Filtering in the Time and Frequency Domains, Robert E. Krieger Publishing Company, Inc., Malabar, FL, 1987. [3] N. Balabanian and T. A. Bickart, Electrical Network Theory, John Wiley and Sons, Inc., New York, 1969. [4] R. V. Churchill and J. W. Brown, Complex Variables and Applications, McGraw-Hill, Inc., New York, ¯fth edition, 1990. [5] L. Storch, Proc. IRE 42, 1666 (1954).