11 OSCILLATOR DESIGN Oscillator circuits are used for generating the periodic signals that are needed in various applications. These circuits convert a part of dc power into the periodic output and do not require a periodic signal as input. This chapter begins with the basic principle of sinusoidal oscillator circuits. Several transistor circuits are subsequently analyzed in order to establish their design procedures. Ceramic resonant circuits are frequently used to generate reference signals while the voltage-controlled oscillators are important in modern frequency synthesizer design using the phase-lock loop. Fundamentals of these circuits are discussed in this chapter. Diode-oscillators used at microwave frequencies are also summarized. The chapter ends with a description of the microwave transistor circuits using S- parameters. 11.1 FEEDBACK AND BASIC CONCEPTS Solid-state oscillators use a diode or a transistor in conjunction with the passive circuit to produce sinusoidal steady-state signals. Transients or electrical noise triggers oscillations initially. A properly designed circuit sustains these oscillations subsequently. This process requires a nonlinear active device. In addition, since the device is producing RF power, it must have a negative resistance. The basic principle of an oscillator circuit can be explained via a linear feedback system as illustrated in Figure 11.1. Assume that a part of output Y is fed back to the system along with input signal X . As indicated, the transfer function of the forward- 449 Radio-Frequency and Microwave Communication Circuits: Analysis and Design Devendra K. Misra Copyright # 2001 John Wiley & Sons, Inc. ISBNs: 0-471-41253-8 (Hardback); 0-471-22435-9 (Electronic) connected subsystem is A while the feedback path has a subsystem with its transfer function as b. Therefore, Y  AX  bY  Closed-loop gain T (generally called the transfer function) of this system is found from this equation as T  Y X  A 1 À Ab 11:1:1 Product Ab is known as the loop gain. It is a product of the transfer functions of individual units in the loop. Numerator A is called the forward pathgain because it represents the gain of a signal traveling from input to output. For the loop gain of unity, T becomes in®nite. Hence, the circuit has an output signal Y without an input signal X and the system oscillates. The condition, Ab  1, is known as the Barkhausen criterion. Note that if the signal Ab is subtracted from X before it is fed to A then the denominator of (11.1.1) changes to 1  Ab. In this case, the system oscillates for Ab À1. This is known as the Nyquist criterion. Since the output of an ampli®er is generally 180  out of phase with its input, it may be a more appropriate description for that case. A Generalized Oscillator Circuit Consider a transistor circuit as illustrated in Figure 11.2. Device T in this circuit may be a bipolar transistor or a FET. If it is a BJT then terminals 1, 2, and 4 represent the base, emitter, and collector, respectively. On the other hand, these may be the gate, source, and drain terminals if it is a FET. Its small-signal equivalent circuit is shown in Figure 11.3. The boxed part of this ®gure represents the transistor's equivalent, with g m being its transconductance, and Y i and Y o its input and output admittances, respectively. Figure 11.1 A simple feedback system. 450 OSCILLATOR DESIGN Application of Kirchhoff's current law at nodes 1, 2, 3, and 4 gives Y 3 V 1 À V 3 Y 1 V 1 À V 2 Y i V 1 À V 2 0 11:1:2 ÀY 1 V 1 À V 2 ÀY 2 V 3 À V 2 ÀY i V 1 À V 2 Àg m V 1 À V 2 ÀY o V 4 À V 2 0 11:1:3 ÀY 3 V 1 À V 3 ÀY 2 V 2 À V 3 ÀY L V 4 À V 3 0 11:1:4 and g m V 1 À V 2 Y o V 4 À V 2 Y L V 4 À V 3 0 11:1:5 Figure 11.2 A schematic oscillator circuit. Figure 11.3 An electrical equivalent of the schematic oscillator circuit. FEEDBACK AND BASIC CONCEPTS 451 Simplifying (11.1.2)±(11.1.5), we have Y 1  Y 3  Y i V 1 ÀY 1  Y i V 2 À Y 3 V 3  0 11:1:6 ÀY 1  Y i  g m V 1 Y 1  Y 2  Y i  g m  Y o V 2 À Y 2 V 3 À Y o V 4  0 11:1:7 ÀY 3 V 1 À Y 2 V 2 Y 2  Y 3  Y L V 3 À Y L V 4  0 11:1:8 and g m V 1 Àg m  Y o V 2 À Y L V 3 Y o  Y L V 4  0 11:1:9 These equations can be written in matrix form as follows: Y 1  Y 3  Y i ÀY 1  Y i ÀY 3 0 ÀY 1  Y i  g m Y 1  Y 2  Y i  g m  Y o ÀY 2 ÀY o ÀY 3 ÀY 2 Y 2  Y 3  Y L ÀY L g m Àg m  Y o ÀY L Y o  Y L  2 6 6 6 6 6 4 3 7 7 7 7 7 5  V 1 V 2 V 3 V 4 2 6 6 6 6 6 4 3 7 7 7 7 7 5  0 11:1:10 For a nontrivial solution to this system of equations, the determinant of the coef®cient matrix must be zero. It sets constraints on the nature of circuit components that will be explained later. Equation (11.1.10) represents the most general formulation. It can be simpli®ed for speci®c circuits as follows: 1. If a node is connected to ground then that column and row are removed from (11.1.10). For example, if node 1 is grounded then the ®rst row as well as the ®rst column will be removed from (11.1.10). 2. If two nodes are connected together then the corresponding columns and rows of the coef®cient matrix are added together. For example, if nodes 3 and 4 are connected together then rows 3 and 4 as well as columns 3 and 4 are replaced by their sums as follows: Y 1  Y 3  Y i ÀY 1  Y i ÀY 3 ÀY 1  Y i  g m Y 1  Y 2  Y i  g m  Y o ÀY 2  Y o  ÀY 3  g m Àg m  Y 2  Y o Y 2  Y 3  Y o  2 4 3 5 V 1 V 2 V 3 2 4 3 5  0 11:1:11 452 OSCILLATOR DESIGN If output impedance of the device is very high then Y o is approximately zero. In this case, (11.1.11) can be simpli®ed further. For a nontrivial solution, the determinant of its coef®cient matrix must be zero. Hence, Y 1  Y 3  Y i ÀY 1  Y i ÀY 3 ÀY 1  Y i  g m Y 1  Y 2  Y i  g m ÀY 2 ÀY 3  g m Àg m  Y 2 Y 2  Y 3               0 11:1:12 For a common-emitter BJT (or a common-source FET) circuit, V 2  0, and therefore, row 2 and column 2 are removed from (11.1.12). Hence, it simpli®es further as follows: Y 1  Y 3  Y i ÀY 3 ÀY 3  g m Y 2  Y 3           0 11:1:13 Therefore, Y 2  Y 3 Y 1  Y 3  Y i Y 3 ÀY 3  g m 0 11:1:14 or, g m Y 3  Y 1 Y 2  Y 2 Y 3  Y 2 Y i  Y 1 Y 3  Y 3 Y i  0 11:1:15 If the input admittance Y i  G i (pure real) and the other three admittances (Y 1 , Y 2 , and Y 3 ) are purely susceptive then (11.1.15) produces g m jB 3 À B 1 B 2 À B 2 B 3  jB 2 G i À B 1 B 3  jB 3 G i  0 11:1:16 On separating its real and imaginary parts, we get B 1 B 2  B 2 B 3  B 1 B 3  0 11:1:17 and, g m B 3  B 2 G i  B 3 G i  0 11:1:18 Equation (11.1.17) is satis®ed only when at least one susceptance is different from the other two (i.e., if one is capacitive then other two must be inductive or vice versa). Similarly, (11.1.18) requires that B 2 and B 3 must be of different kinds. An exact relation between the two reactances can be established using (11.1.18) as follows: g m  G i B 3  G i B 2  0 11:1:19 FEEDBACK AND BASIC CONCEPTS 453 or g m G i  1  B 3  B 2  0  B 2 1  bB 3 11:1:20 or, X 3 À1  h fe X 2 11:1:21 Here, h fe represents the small signal current gain of common-emitter circuit. It is given by h fe  g m =G i 11:1:22 Equation (11.1.21) indicates that if X 2 is an inductor then X 3 is a capacitor or vice versa. Further, dividing (11.1.17) by B 1 B 2 B 3 , the corresponding reactance relation is found as X 1  X 2  X 3  0 11:1:23 Hence, at least one of the reactance is different from the other two. That is, if X 3 is an inductor then the other two must be capacitors or vice versa. From (11.1.21) and (11.1.23), X 1  X 2 À1  h fe X 2  0 11:1:24 or, X 1  h fe X 2 11:1:25 Since h fe is a positive number, X 1 and X 2 must be of the same kind. If B 1 and B 2 are inductive then B 3 must be a capacitive susceptance. This kind of oscillator circuit is called the Hartley oscillator. On the other hand, B 3 is an inductor if capacitors are used for B 1 and B 2 . This circuit is called the Colpitts oscillator. Figure 11.4 illustrates the RF sections of these two circuits (excluding the transistor's Figure 11.4 Simpli®ed circuits of (a) Hartley and (b) Colpitts oscillators. 454 OSCILLATOR DESIGN biasing network). A BJT Hartley oscillator with its bias arrangement is shown in Figure 11.5. Resonant frequency of the Hartley oscillator is obtained from (11.1.23) as follows: oL 1  oL 2 À 1 oC 3  0 or, o 2  1 C 3 L 1  L 2  11:1:26 Similarly, the resonant frequency of a Colpitts oscillator is found to be À 1 oC 1 À 1 oC 2  oL 3  0 or, o 2  C 1  C 2 C 1 C 2 L 3 11:1:27 Resistors R B1 , R B2 and R E in Figure 11.5 are determined according to the bias point selected for a transistor. Capacitors C B and C E must bypass the RF, and therefore, these should have relatively high values. C E is selected such that its reactance at the design frequency is negligible in comparison with R E . Similarly, the parallel combination of R B1 and R B2 must be in®nitely large in comparison with the reactance of C B . The RF choke (RFC) offers an in®nitely large reactance at the RF Figure 11.5 A biased BJT Hartely oscillator circuit. FEEDBACK AND BASIC CONCEPTS 455 while it passes dc with almost zero resistance. Thus, it blocks the ac signal from reaching the dc supply. Since capacitors C B and C E have almost zero reactance at RF, the node that connects L 1 and C 3 is electrically connected to the base of BJT. Also, the grounded junction of L 1 and L 2 is effectively connected to the emitter. Hence, the circuit depicted in Figure 11.5 is essentially the same for the RF as that shown in Figure 11.4 (a). Capacitor C 3 and total inductance L 1  L 2 are determined such that (11.1.26) is satis®ed at the desired frequency of oscillations. L 1 and L 2 satisfy (11.1.25) as well when the oscillator circuit operates. A BJT-based Colpitts oscillator is shown in Figure 11.6. Resistors R B1 , R B2 , and R E are determined from the usual procedure of biasing a transistor. Reactance of the capacitor C B1 must be negligible in comparison with parallel resistances R B1 and R B2 . Similarly, the reactance of C B2 must be negligible in comparison with that of the inductor L 3 . The purpose of capacitor C B2 is to protect the dc supply from short- circuiting via L 3 and RFC. Since capacitors C B1 and C B2 have negligible reactance, the ac equivalent of this circuit is same as that shown in Figure 11.4 (b). C 1 , C 2 , and L 3 are determined from the resonance condition (11.1.27). Also, (11.1.25) holds at the resonance. As described in the preceding paragraphs, capacitor C B2 provides almost a short circuit in the desired frequency range and the inductor L 3 is selected such that Figure 11.6 A biased BJT Colpitts oscillator circuit. Figure 11.7 A FET-based Clapp oscillator circuit. 456 OSCILLATOR DESIGN (11.1.27) is satis®ed. An alternative design procedure that provides better stability of the frequency is as follows. L 3 is selected larger than needed to satisfy (11.1.27), and then C B2 is determined to bring it down to the desired value at resonance. This kind of circuit is called the Clapp oscillator. A FET-based Clapp oscillator circuit is shown in Figure 11.7. It is very similar to the Colpitts design and operation except for the selection of C B2 that is connected in series with the inductor. At the design frequency, the series inductor-capacitor combination provides the same inductive reactance as that of the Colpitts circuit. However, if there is a drift in frequency then the reactance of this combination changes rapidly. This can be explained further with the help of Figure 11.8. Figure 11.8 illustrates the resonant circuits of Colpitts and Clapp oscillators. An obvious difference between the two circuits is the capacitor C 3 that is connected in series with L 3 . Note that unlike C 3 , the blocking capacitor C B2 shown in Figure 11.6 does not affect the RF operation. Reactance X 1 of the series branch in the Colpitts circuit is oL 3 whereas it is X 2  oL 3 À 1 oC 3 in the case of the Clapp oscillator. If inductor L 3 in the former case is selected as 1.59 mH and the circuit is resonating at 10 MHz, then the change in its reactance around resonance is as shown in Figure 11.9. The series branch of the Clapp circuit has the same inductive reactance at the resonance if L 3  3:18 mH and C 3  159 pF. However, the rate of change of reactance with frequency is now higher in comparison with X 1 . This characteristic helps in reducing the drift in oscillation frequency. Another Interpretation of the Oscillator Circuit Ideal inductors and capacitors store electrical energy in the form of magnetic and electric ®elds, respectively. If such a capacitor with initial charge is connected across an ideal inductor, it discharges through that. Since there is no loss in this system, the inductor recharges the capacitor back and the process repeats. However, real inductors and capacitors are far from being ideal. Energy losses in the inductor and the capacitor can be represented by a resistance r 1 in this loop. Oscillations die out because of these losses. As shown in Figure 11.10, if a negative resistance Àr 1 can be introduced in the loop then the effective resistance becomes zero. In other words, if a circuit can be devised to compensate for the losses then oscillations can be sustained. This can be done using an active circuit, as illustrated in Figure 11.11. Figure 11.8 Resonant circuits for Colpitts (a) and Clapp (b) oscillators. FEEDBACK AND BASIC CONCEPTS 457 Figure 11.10 An ideal oscillator circuit. Figure 11.9 Reactance of inductive branch versus frequency for Colpitts (X 1 ) and Clapp (X 2 ) circuits. Figure 11.11 A BJT circuit to obtain negative resistance. 458 OSCILLATOR DESIGN [...]... Capacitors CB and CE must bypass the RF, and therefore, these should have relatively high values CE is selected such that its reactance at the design frequency is negligible in comparison with RE Similarly, the parallel combination of RB1 and RB2 must be in®nitely large in comparison with the reactance of CB The RF choke (RFC) offers an in®nitely large reactance at the RF 456 OSCILLATOR DESIGN Figure... hyperabrupt junctions Abrupt junction diodes provide very high Q and also operate over a very wide tuning voltage range (typically, 0 to 60 V) These diodes provide an excellent phase noise performance because of their high Q Hyperabrupt-type diodes exhibit a quadratic characteristic of the capacitance with applied voltage Therefore, these varactors provide a much more linear tuning characteristic than... protect the dc supply from shortcircuiting via L3 and RFC Since capacitors CB1 and CB2 have negligible reactance, the ac equivalent of this circuit is same as that shown in Figure 11.4 (b) C1 , C2 , and L3 are determined from the resonance condition (11.1.27) Also, (11.1.25) holds at the resonance As described in the preceding paragraphs, capacitor CB2 provides almost a short circuit in the desired... voltage law, we can write Vi ˆ Ii …X1 ‡ X2 † À Ib …X1 À bX2 † …11:1:28† 0 ˆ Ii …X1 † À Ib …X1 ‡ rp † …11:1:29† and, Equation (11.1.29) can be rearranged as follows: Ib ˆ X1 I X1 ‡ rp i …11:1:30† Substituting (11.1.30) into (11.1.28), we ®nd that  X À bX2 Vi ˆ Ii X1 ‡ X2 À 1 X X1 ‡ rp 1  …11:1:31† Impedance Zi across its input terminal can now be determined as follows:   Vi X1 À bX2 …1 ‡ b†X1 X2... (11.1.21) indicates that if X2 is an inductor then X3 is a capacitor or vice versa Further, dividing (11.1.17) by B1 B2 B3, the corresponding reactance relation is found as X1 ‡ X2 ‡ X3 ˆ 0 …11:1:23† Hence, at least one of the reactance is different from the other two That is, if X3 is an inductor then the other two must be capacitors or vice versa From (11.1.21) and (11.1.23), X1 ‡ X2 À …1 ‡ hfe †X2 ˆ 0... from reaching the dc supply Since capacitors CB and CE have almost zero reactance at RF, the node that connects L1 and C3 is electrically connected to the base of BJT Also, the grounded junction of L1 and L2 is effectively connected to the emitter Hence, the circuit depicted in Figure 11.5 is essentially the same for the RF as that shown in Figure 11.4 (a) Capacitor C3 and total inductance L1 ‡ L2 are... using the crystal is shown in Figure 11.15 It is known as the Pierce oscillator A comparison of its RF equivalent circuit with that shown in Figure 11.4 (b) indicates that the Pierce circuit is similar to the Colpitts oscillator with inductor L3 replaced by the crystal As mentioned earlier, the crystal provides very stable frequency of oscillation over a wide range of temperature The main drawback of... frequency, the series inductor-capacitor combination provides the same inductive reactance as that of the Colpitts circuit However, if there is a drift in frequency then the reactance of this combination changes rapidly This can be explained further with the help of Figure 11.8 Figure 11.8 illustrates the resonant circuits of Colpitts and Clapp oscillators An obvious difference between the two circuits is the... the corresponding values of V and C Then o2 ˆ o 1 1  ˆ  A L…Cf ‡ Co † L Cf ‡ n Vo …11:3:4† Further, the carrier frequency deviates from oo by do for a voltage change of dV Therefore, …oo ‡ do†2 ˆ  L Cf ‡ 1 A …Vo ‡ dV †n  A …oo ‡ do†À2 ˆ L…Cf ‡ A…Vo ‡ dV †Àn † …11:3:5† Dividing (11.3.5) by (11.3.4), we have  oo ‡ do oo 2 ˆ Cf ‡ Co ˆ Cf ‡ A…Vo ‡ dV †Àn Cf ‡ Co  Àn dV Àn Cf ‡ AVo 1 ‡ Vo or, ... Oscillations die out because of these losses As shown in Figure 11.10, if a negative resistance Àr1 can be introduced in the loop then the effective resistance becomes zero In other words, if a circuit can be devised to compensate for the losses then oscillations can be sustained This can be done using an active circuit, as illustrated in Figure 11.11 458 OSCILLATOR DESIGN Figure 11.9 Reactance of inductive . subsequently. This process requires a nonlinear active device. In addition, since the device is producing RF power, it must have a negative resistance. The basic. in comparison with the reactance of C B . The RF choke (RFC) offers an in®nitely large reactance at the RF Figure 11.5 A biased BJT Hartely oscillator