10 TRANSISTOR AMPLIFIER DESIGN Ampli®ers are among the basic building blocks of an electronic system. While vacuum tube devices are still used in high-power microwave circuits, transistorsÐ silicon bipolar junction devices, GaAs MESFET, heterojunction bipolar transistors (HBT), and high-electron mobility transistors (HEMT)Ðare common in many RF and microwave designs. This chapter begins with the stability considerations for a two-port network and the formulation of relevant conditions in terms of its scattering parameters. Expressions for input and output stability circles are presented next to facilitate the design of ampli®er circuits. Design procedures for various small-signal single-stage ampli®ers are discussed for unilateral as well as bilateral transistors. Noise ®gure considerations in ampli®er design are discussed in the following section. An overview of broadband ampli®ers is included. Small-signal equivalent circuits and biasing mechanisms for various transistors are also summarized in subsequent sections. 10.1 STABILITY CONSIDERATIONS Consider a two-port network that is terminated by load Z L as shown in Figure 10.1. A voltage source V S with internal impedance Z S is connected at its input port. Re¯ection coef®cients at its input and output ports are G in and G out , respectively. The source re¯ection coef®cient is G S while the load re¯ection coef®cient is G L . Expressions for input and output re¯ection coef®cients were formulated in the preceding chapter (Examples 9.6 and 9.7). 385 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) For this two-port to be unconditionally stable at a given frequency, the following inequalities must hold: jG S j < 1 10:1:1 jG L j < 1 10:1:2 jG in j S 11  S 21 S 12 G L 1 À S 22 G L         < 1 10:1:3 and, jG out j S 22  S 21 S 12 G S 1 À S 11 G S         < 1 10:1:4 Condition (10.1.3) can be rearranged as follows: S 11  S 22 S 21 S 12 G L S 12 S 21 À S 12 S 21  S 22 1 À S 22 G L          < 1 or, 1 S 22 D  S 21 S 12 1 À S 22 G L          < 1 10:1:5 where, D  S 11 S 22 À S 12 S 21 10:1:6 Since 1 À S 22 G L 3 G L 1 1 À S 22  1 ÀjS 22 j expjyA 10:1:7 Figure 10.1 A two-port network with voltage source at its input and a load terminating the output port. 386 TRANSISTOR AMPLIFIER DESIGN This traces a circle on the complex plane as y varies from zero to 2p. It is illustrated in Figure 10.2. Further, 1=1 ÀjS 22 j exp jy represents a circle of radius r with its center located at d, where r  1 2 1 1 ÀjS 22 j À 1 1 jS 22 j   jS 22 j 1 ÀjS 22 j 2 10:1:8 and, d  1 2 1 1 jS 22 j  1 1 ÀjS 22 j   1 1 ÀjS 22 j 2 10:1:9 Hence, for jG L j 1and G L  y, condition (10.1.5) may be written as follows: 1 jS 22 j D  S 21 S 12 1 jS 22 j exp jy 1 ÀjS 22 j 2          < 1 or, 1 jS 22 j D  S 12 S 21 1 ÀjS 22 j 2  S 12 S 21 jS 22 j exp jy 1 ÀjS 22 j 2         < 1 10:1:10 Now, using the Minkowski inequality, P n k1 ja k  b k j p  1=p P n k1 ja k j p  1=p  P n k1 jb k j p  1=p 10:1:11 we ®nd that (10.1.10) is satis®ed if 1 jS 22 j D  S 12 S 21 1 ÀjS 22 j 2          jS 12 S 21 j 1 ÀjS 22 j 2 < 1 or, 1 jS 22 j D  S 12 S 21 1 ÀjS 22 j 2         < 1 À jS 12 S 21 j 1 ÀjS 22 j 2 10:1:12 Since the left-hand side of (10.1.12) is always a positive number, this inequality will be satis®ed if the following is true 1 ÀjS 22 j 2 ÀjS 12 S 21 j > 0 10:1:13 STABILITY CONSIDERATIONS 387 Similarly, stability condition (10.1.4) will be satis®ed if 1 ÀjS 11 j 2 ÀjS 12 S 21 j > 0 10:1:14 Adding (10.1.13) and (10.1.14), we get 2 ÀjS 11 j 2 ÀjS 22 j 2 À 2jS 12 S 21 j > 0 or, 1 À 1 2 jS 11 j 2 jS 22 j 2  > jS 12 S 21 j10:1:15 From (10.1.6) and (10.1.15), we have jDj < jS 11 S 22 jjS 12 S 21 j < jS 11 S 22 j1 À 1 2 jS 11 j 2 jS 22 j 2  or, jDj < 1 À 1 2 jS 11 jÀjS 22 j 2 AjDj < 1 10:1:16 Multiplying (10.1.13) and (10.1.14), we get 1 ÀjS 22 j 2 ÀjS 12 S 21 j1 ÀjS 11 j 2 ÀjS 12 S 21 j > 0 or, 1 ÀjS 11 j 2 ÀjS 22 j 2 À 2jS 12 S 21 jz > 0 10:1:17 Figure 10.2 A graphical representation of (10.1.7). 388 TRANSISTOR AMPLIFIER DESIGN where, z jS 11 j 2 jS 22 j 2 jS 12 S 21 j 2 jS 12 S 21 jjS 11 j 2 jS 22 j 2  From the self-evident identity, jS 12 S 21 jjS 11 jÀjS 22 j 2 ! 0 it can be proved that z jDj 2 Therefore, (10.1.17) can be written as follows: 1 ÀjS 11 j 2 ÀjS 22 j 2 À 2jS 12 S 21 jjDj 2 > 0 or, 1 ÀjS 11 j 2 ÀjS 22 j 2 jDj 2 > 2jS 12 S 21 j or, 1 ÀjS 11 j 2 ÀjS 22 j 2 jDj 2 2jS 12 S 21 j > 1 Therefore, k  1 ÀjS 11 j 2 ÀjS 22 j 2 jDj 2 2jS 12 S 21 j > 1 10:1:18 If S-parameters of a transistor satisfy conditions (10.1.16) and (10.1.18) then it is stable for any passive load and generator impedance. In other words, this transistor is unconditionally stable. On the other hand, it may be conditionally stable (stable for limited values of load or source impedance) if one or both of these conditions are violated. It means that the transistor can provide stable operation for a restricted range of G S and G L . A simple procedure to ®nd these stable regions is to test inequalities (10.1.3) and (10.1.4) for particular load and source impedances. An alternative graphical approach is to ®nd the circles of instability for load and generator re¯ection coef®cients on a Smith chart. This latter approach is presented below. STABILITY CONSIDERATIONS 389 From the expression of input re¯ection coef®cient (9.4.7), we ®nd that G in  S 11  S 21 S 12 G L 1 À S 22 G L A G in 1 À S 22 G L S 11 1 À S 22 G L S 21 S 12 G L or, G in  S 11 À G L S 11 S 22 À S 12 S 21 À G in S 22 AG L  S 11 À G in D À G in S 22 or, G L  S 11 À G in D À G in S 22 S 22 S 22  1 S 22 S 11 S 22 À G in S 22 À S 12 S 21  S 12 S 21 D À G in S 22  or, G L  1 S 22 1  S 12 S 21 D À G in S 22   1 DS 22 D  S 12 S 21 1 À G in D À1 S 22  10:1:19 As before, 1 À G in S 22 D À1 represents a circle on the complex plane. It is centered at 1 with radius jG in S 22 D À1 j; the reciprocal of this expression is another circle with center at 1 2 1 1 jD À1 S 22 j  1 1 ÀjD À1 S 22 j   1 1 ÀjD À1 S 22 j 2 and radius 1 2 1 1 ÀjD À1 S 22 j  1 1 jD À1 S 22 j   jD À1 S 22 j 1 ÀjD À1 S 22 j 2 Since jG in j < 1, the region of stability will include all points on the Smith chart outside this circle. From (10.1.19), the center of the load impedance circle, C L ,is C L  1 DS 22 D  S 12 S 21 1 ÀjD À1 S 22 j 2   1 DS 22 D  S 12 S 21 jDj 2 jDj 2 ÀjS 22 j 2  or, C L  1 S 22 1  S 12 S 21 D* jDj 2 ÀjS 22 j 2   1 S 22 jDj 2 ÀjS 22 j 2  S 12 S 21 D* jDj 2 ÀjS 22 j 2  390 TRANSISTOR AMPLIFIER DESIGN or, C L  1 S 22 D*D  S 12 S 21 ÀjS 22 j 2 jDj 2 ÀjS 22 j 2   1 S 22 D*S 11 S 22 ÀjS 22 j 2 jDj 2 ÀjS 22 j 2  Therefore, C L  D*S 11 À S* 22 jDj 2 ÀjS 22 j 2  S 22 À DS** 11 * jS 22 j 2 ÀjDj 2 10:1:20 Its radius, r L , is given by r L  1 jDS 22 j S 12 S 21 jD À1 S 22 j 1 ÀjD À1 S 22 j 2            S 12 S 21 jDj 2 ÀjS 22 j 2         10:1:21 As explained following (10.1.19), this circle represents the locus of points over which the input re¯ection coef®cient G in is equal to unity. On one side of this circle, the input re¯ection coef®cient is less than unity (stable region) while on its other side it exceeds 1 (unstable region). When load re¯ection coef®cient G L is zero (i.e., a matched termination is used), G in is equal to S 11 . Hence, the center of the Smith chart (re¯ection coef®cient equal to zero) represents a stable point if jS 11 j is less than unity. On the other hand, it represents unstable impedance for jS 11 j greater than unity. If G L  0 is located outside the stability circle and is found stable then all outside points are stable. Similarly, if G L  0 is inside the stability circle and is found stable then all enclosed points are stable. If G L  0 is unstable then all points on that side of the stability circle are unstable. Similarly, the locus of G S can be derived from (10.1.4), with its center C S and its radius r S given as follows: C S  D*S 22 À S** 11 jDj 2 ÀjS 11 j 2  S 11 À DS** 22 * jS 11 j 2 ÀjDj 2 10:1:22 and, r S  1 jDS 11 j S 12 S 21 jD À1 S 11 j 1 ÀjD À1 S 11 j 2            S 12 S 21 jDj 2 ÀjS 11 j 2         10:1:23 This circle represents the locus of points over which output re¯ection coef®cient G out is equal to unity. On one side of this circle, output re¯ection coef®cient is less than unity (stable region) while on its other side it exceeds 1 (unstable region). When the source re¯ection coef®cient G S is zero then G out is equal to S 22 . Hence, the center of the Smith chart (re¯ection coef®cient equal to zero) represents a stable point if jS 22 j is less than unity. On the other hand, it represents an unstable impedance point for jS 22 j greater than unity. If G S  0 is located outside the stability circle and is STABILITY CONSIDERATIONS 391 found stable then all outside source-impedance points are stable. Similarly, if G S  0 is inside the stability circle and is found stable then all enclosed points are stable. If G S  0 is unstable then all points on that side of the stability circle are unstable. Example 10.1: S-parameters of a properly biased transistor are found at 2 GHz as follows (50-O reference impedance): S 11  0:894À60:6  ; S 12  0:02 62:4  ; S 21  3:122 123:6  ; S 22  0:781À27:6  Determine its stability and plot the stability circles if the transistor is potentially unstable (see Figure 10.3). From (10.1.16) and (10.1.18), we get jDjjS 11 S 22 À S 12 S 21 j0:6964 and, k  1 jDj 2 ÀjS 11 j 2 ÀjS 22 j 2 2jS 12 S 21 j  0:6071 Figure 10.3 Input and output stability circles for Example 10.1. 392 TRANSISTOR AMPLIFIER DESIGN Since one of the conditions for stability failed above, this transistor is potentially unstable. Using (10.1.20)±(10.1.23), we can determine the stability circles as follows. For the output stability circle: C L  1:36 46:7  and, r L  0:5 Since jS 11 j is 0.894, G L  0 represents a stable load point. Further, this point is located outside the stability circle, and therefore, all points outside this circle are stable. For the input stability circle: C S  1:13 68:5  and, r S  0:2 Since jS 22 j is 0.781, G S  0 represents a stable source impedance point. Further, this point is located outside the stability circle, and therefore, all points outside this circle are stable. For the output stability circle, draw a radial line at 46.7  . With the radius of the Smith chart as unity, locate the center at 1.36 on this line. It can be done as follows. Measure the radius of the Smith chart using a ruler scale. Supposing that it is d mm. Location of the stability circle is then at 1.36 d mm away on this radial line. Similarly, the radius of the stability circle is 0.5 d mm. Load impedance must lie outside this circle for the circuit to be stable. Following a similar procedure, the input stability circle is drawn with its radius as 0.2 d mm and center at 1.13 d mm on the radial line at 68.5  . In order to have a stable design, the source impedance must lie outside this circle. 10.2 AMPLIFIER DESIGN FOR MAXIMUM GAIN In this section, ®rst we consider the design of an ampli®er that uses a unilateral transistor (S 12  0) and has maximum possible gain. A design procedure using a bilateral transistor (S 12 T 0) is developed next that requires simultaneous conjugate matching at its two ports. AMPLIFIER DESIGN FOR MAXIMUM GAIN 393 Unilateral Case When S 12 is zero, the input re¯ection coef®cient G in reduces to S 11 while the output re¯ection coef®cient G out simpli®es to S 22 . In order to obtain maximum gain, source and load re¯ection coef®cients must be equal to S* 11 and S* 22 ; respectively. Further, the stability conditions for a unilateral transistor simplify to jS 11 j < 1 and, jS 22 j < 1 Example 10.2: S-parameters of a properly biased BJT are found at 1 GHz as follows (with Z o  50 O): S 11  0:606À155  ; S 22  0:48À20  ; S 12  0; and S 21  6 180  Determine the maximum gain possible with this transistor and design an RF circuit that can provide this gain. (i) Stability check: k  1 ÀjS 11 j 2 ÀjS 22 j 2 jDj 2 2jS 12 S 21 j I; , S 12  0 and, jDjjS 11 S 22 À S 12 S 21 jjS 11 S 22 j0:2909 Since both of the conditions are satis®ed, the transistor is unconditionally stable. (ii) Maximum possible power gain of the transistor is found as G TU  1 ÀjG S j 2 j1 À G 11 G S j 2 ? jS 21 j 2 ? 1 ÀjG L j 2 j1 À S 22 G L j 2 and G TU max  1 ÀjS* 11 j 2 j1 ÀjS 11 j 2 j 2 ? jS 21 j 2 ? 1 ÀjS* 22 j 2 j1 ÀjS 22 j 2 j 2  1 1 À 0:606 2 ? 6 2 ? 1 1 À 0:48 2  73:9257 394 TRANSISTOR AMPLIFIER DESIGN . system. While vacuum tube devices are still used in high-power microwave circuits, transistorsÐ silicon bipolar junction devices, GaAs MESFET, heterojunction. source impedance) if one or both of these conditions are violated. It means that the transistor can provide stable operation for a restricted range of G S and