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Negative Resistance Oscillators

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Negative Resistance Oscillators Negative Resistance Oscillators Ceramic Oscillator (CRO) Dielectric Oscillator (DRO) 1 Index 1. Scattering-Matrix Definition and Properties 2. Flow Graph Theory 3. Negative Resistance Oscillator Theory • Oscillator Structure • Steady State Conditions • Triggering Conditions • Nyquist Method for Oscillators 4. Negative Resistance Oscillators Realization 5. Ceramic Resonator Oscillator (CRO) • Ceramic Resonator • CRO Design 6. Dielectric Resonator Oscillator (DRO) • Dielectric Resonator • DRO Design Scattering-Matrix Definition and Properties 2 Voltage waves a1 a2 1 2 a = incident waves N-Port b1 i b Network 2 bi = reflected waves n 3 Vi = ai + bi Ii = (ai – bi)/Z0i Voltage waves Vi I iZ0i Vi I iZ0i a bi i 2 2 Z0i = Transmission Line Characteristic Impedance Tipically Z0i = 50 3 Power waves 1 Vi Vi VVii Z 0i a i bi I i I i IIii a ibi Z0i 1 V 1 V i b i I Z a i I i Z0i i i 0i 2 Z 2 Z 0i 0i Scattering matrix for a N-Port Network b1 S11a1 S12a2 ... S1nan For linear networks b S a S a ... S a holds the 2 21 1 22 2 2n n superposition principle .... .......... .......... .......... ........ bn Sn1a1 Sn2a2 ... Snnan b1 S11 S12 ..... S1n a1 [b] = [S] [a] b S S ..... S a 2 21 22 2n 2 .. .... .... ..... .... b b S S ...... S a S i n n1 n2 nn n i j a j ak 0 con kj 4 Matched Load 퐼1 푉1 Z0 푉1 =−푍0퐼1 푉 +푍 퐼 푎 = 1 0 1 = 0 1 2 When a network is closed on an matched load (50 ) the wave reflected by the load and incident on the port (a1) is equal to zero Why scattering parameters ? • Scattering parameters can be measured at microwave frequencies as they require the network to be closed on 50 Ohm and this kind of loads are "relatively" easy to realize 5 Why scattering parameters ? • At microwave frequencies, voltages and currents are hard to measure and the open and short circuits necessary to measure impedance and admittance parameters can cause the destruction of active devices • The scattering parameters are defined for all two-port networks while the admittance and impedance parameters are not defined in some cases. For example, for a series impedance the impedance parameters are not defined because the currents at the two ports are not independent Scattering matrix for a 1 Port Network a1 E = Incoming I1 I = Incident V1 R = Reflected b1 * 1 * 1 a1 b1 1 2 1 2 P1 PE Re V1 I1 Re a1 b1 a1 b1 2 2 Z 2Z 2Z 0 0 0 2 푃푅 푏1 푃1 = 푃퐸 = 푃퐼 - 푃푅 = 푃퐼 1 − = 푃퐼 1 − 2 푃퐼 푎1 2 2 2 푏1 푃푅 the square modules of the scattering 푆 = 퐼 = 2 = 푎1 푃퐼 Parameters are ratios between powers 6 Scattering matrix for a 2 Port Network a2 b1 b2 a1 * 1 * 1 b2 a2 1 2 1 2 P2 Re V2 (I2 ) Re a2 b2 a2 b2 2 2 Z 2Z 2Z 0 0 0 1 2 1 2 2 2 2 PO ifse a2 0 P2 PO b2 S21 a1 PI S21 S21 2Z0 2Z0 PI 2 PR the square modules of the scattering ifse a2 0 I S11 S11 PI Parameters are ratios between powers Reciprocal two port network (Absence of saturated ferrites or controlled generators in the component under test) Z ZT S ST always true if [S] is defined if [S] is defined starting starting from power waves from voltage waves, this is true if all the Z0i are equal s21 = s12 7 Symmetrical two port network A two port network is symmetrical if its input impedance is equal to its output impedance Symmetrical networks are also physically symmetrical S11 = S22 Lossless two port network s s T 11 12 T s11 s21 S S 1 S S s21 s22 s12 s22 2 2 2 2 s11 s12 1 s21 s22 1 s11 s21 s12 s22 j1121 j1222 s11 s21 e s12 s22 e 0 1121 12 22 2 │S21│ = │S12│ │S11│ = │S22│ s12 1 s11 8 Flow Graph Theory Given a network with incident and reflected waves defined at the ports a node (symbol •) is a graphical symbol associated with each of these waves A branch (symbol •-->--•) is an oriented line which connects two nodes. This line is oriented in the direction of the power flow Each branch is linked to a value which gives the multiplicative factor that correlates the two waves at the ends of the branch (nodes) 9 A Path is a continuous set of branches all equally oriented that touch the single nodes only once. The value of the path is equal to the product of the values of the single branches Loops of the 1st order are closed paths formed by branches all oriented in the same direction that touch the nodes only once. The value of the loop is equal to the product of the values of the single branches Loops of the Second order are those formed by two loops of the 1st order with no node in common; the value of the 2nd order loop is equal to the product of the values of the two loops of the 1st order Similarly for subsequent orders Mason formula P 1 1L1 1L2 1L3...P 12L1 2L2...P ... T 1 2 3 12 1 L1 L2 L3... with Pi = value of the i-th path between the two nodes under examination ΣL(i) = sum of all the possible loops of the i-th order Σ(k)L(i) = sum of all possible loops of the i-th order without points in common with the k-th path 10 Example ZL 푏1 = 푆11푎1 + 푆21푎2 푎2 푏1 퐿 = 퐼푁 = 푏2 푎1 푏2 = 푆21푎1 + 푆22푎2 L Paths Loop 푃 = 푆 푆 퐿 = 푆 푃1 = 푆11 2 21 퐿 12 1 퐿 22 푆11 1 −퐿푆22 +푆21퐿푆12 푆12푆21퐿 Γ퐼푁= = 푆11 + 1−퐿푆22 1−퐿푆22 푆12푆21퐿 퐼푁 = 푆11 + 1−퐿푆22 11 Negative Resistance Oscillator Theory Oscillator structure Oscillators convert energy from continuous to alternating, and are generally constituted by the three following subsystems: • A non-linear active component (ZD) • A resonant structure that fixes the frequency of oscillation • A transition for transferring the signal to the load (ZC) resonantcircuito componenteactive Transitiontransizione ZD risonante carico ZC componentattivo circuit load Figure 1 12 The active element of an oscillator can be seen as a negative resistance component When two-terminal devices, such as Gunn or IMPATT diodes, are used the negative resistance behavior is achieved simply by polarizing the diode. For three-terminal devices, such as transistors, an appropriate feedback network must be added to achieve a negative resistance behavior When fixed frequency performances are required, the resonant structure is generally constituted by dielectric or ceramic resonator. When tunable oscillators are required, YIG spheres or ceramic resonators tuned with varactor diodes can be used The transition connects the source to the load, and can be used to improve some oscillator performance Oscillator scheme A R L CD C R RD C A’ e(t) Figure 2 In this circuit RD and CD model the active element (RD is a non linear resistance depending on the circuit current magnitude) The R, L, C network models the resonator around its resonance frequency and it also takes into account parasitic present in the circuit RC represents the load, and the generator e(t) models the noise sources present in the circuit 13 Oscillator theory This circuit can be studied in the complex frequency domain s = α + j (neglecting the noise generator). In this case, we consider the complex current I(s), and we place RT = R + RC + RDL (where RDL is a linear approximation of RD ). Applying the kirchhoff law to the network we obtain: [RT + sL + 1/(sCT) ] I(s) = 0 This equation admits non-zero solutions only if the part between square brackets is null, and therefore: 2 s LCT + sCTRT + 1 = 0 which admits two solutions: 2 푅푇 1 푅푇 푠1,2 = ±푗 − = 훼 ± 푗휔 2퐿 퐿퐶푇 2퐿 where 휔 is called the natural oscillation pulsation (pulsation = angular frequency) The circuit current evolves over time as: R T t I(t)Im I(s)est ˆI e 2L senˆ t 14 RT > 0 RT < 0 RT = 0 RD 0 ^I RT= 0 -5 RD= -10 -10 RL= 5 RT< 0 Steady state condition in terms of impedance If RT = 0, there are constant amplitude oscillations and the oscillation pulsation (resonance pulsation) is given by 2 0 = 1 / LCT from which it follows 0L - 1/0CT = 0 or XL + XC = XT = 0 15 In conclusion we have: RT = 0 XT = 0 That is: ZT = 0 These are the conditions for steady-state oscillations in terms of impedance Steady state condition in terms of impedances at a section The steady-state conditions can also be applied by looking at the left and right of any circuit section Therefore, if we consider the section AA‘ of the circuit scheme, indicating with ZD the impedance (with a negative real part) of the active device and with ZL the impedance of the whole passive circuit (resonator and load) we have for the total impedance ZT : A C R L ZT = ZD + ZL = 0 D C R RDL C RT = RD + RL = 0 A’ XT = XD + XL = 0 16 Steady-state condition in terms of reflection coefficients at a section The steady state condition can also be written according to the reflection coefficients D and L that are seen looking towards the active device and towards the load. In particular, we have: 2 Z Z Z Z ZDZL ZDZ0 ZLZ0 Z D 0 L 0 0 1 D L Z Z Z Z 2 D 0 L 0 ZDZL ZDZ0 ZLZ0 Z0 That becomes: D L = 1 D + L = 2n Steady-state condition in terms of admittances at a section A R C D C DL L R GDLRD C A’ It should be noted that if instead of the previous series circuit we study a parallel configuration constituted by a parallel resonant circuit and an active device modeled by a GDL in parallel with CD, the parallel resonance condition will be reached: 17 YT = YD + YL = 0 GT = GD + GL = 0 BT = BD + BL = 0 For the oscillator design these conditions can be used, instead of the previous.
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