Lecture 7 - Gain and stability of active networks Microwave Active Circuit Analysis and Design Clive Poole and Izzat Darwazeh Academic Press Inc. © Poole-Darwazeh 2015 Lecture 7 - Gain and stability of active networks Slide1 of 53 Intended Learning Outcomes I Knowledge I Understand the definitions of conditional and unconditional stability and be able to determine the stability of a microwave active device using defined stability criteria. I Understand and be able to calculate the various different definitions of power gain in small signal amplifiers. I Understand that an active two-port network has the possibility of becoming unstable, and therefore requires different treatment than a passive two-port. I Understand the various definitions of power at the input and output ports, and the corresponding definitions of ’power gain’ using either immittance parameters or S-parameters. I Skills I Be able to calculate the transducer gain, available gain, operating power gain and maximum available power gain of any two-port network with given source and load terminations based on immittance parameters or S-parameters. I Be able to calculate the optimum terminations to achieve simultaneous conjugate matching, and therefore maximum available gain, from an unconditionally stable transistor. I Be able to determine whether any two-port network is unconditionally stable or potentially unstable given a set of two-port immittance parameters or S-parameters. © Poole-Darwazeh 2015 Lecture 7 - Gain and stability of active networks Slide2 of 53 Table of Contents Power gain in terms of immittance parameters Power gain in terms of immittance parameters Stability in terms of immittance parameters Stability in terms of S-parameters Power gain in terms of S-parameters © Poole-Darwazeh 2015 Lecture 7 - Gain and stability of active networks Slide3 of 53 Voltage gain of an active two-port i1 i2 YS Two-port vS v1 v2 YL Network Figure 1: Two-port power gain definitions The voltage gain of the two-port in figure 1 is given by : i2 = Y21v1 + v2Y22 (3) v2 Av = (1) Solving (2) and (3) as simultaneous v 1 equations and applying (1) gives the small From figure 1 we can write : signal voltage gain for the two-port as follows: i2 = −v2YL (2) v2 −Y21 i v v Av = = (4) But 2 is also related to 1 and 2 via the v Y + Y Y-parameters of the two-port, so we can 1 L 22 also write: © Poole-Darwazeh 2015 Lecture 7 - Gain and stability of active networks Slide4 of 53 Table of Contents Power gain in terms of immittance parameters Power gain in terms of immittance parameters Stability in terms of immittance parameters Stability in terms of S-parameters Power gain in terms of S-parameters © Poole-Darwazeh 2015 Lecture 7 - Gain and stability of active networks Slide5 of 53 Power absorbed by the amplifier input (Pin) i1 i2 i1 i2 ZS vS v1 Zin Zout v2 ZL iS YS v1 Yin Yout v2 YL (a) Z-parameters (b) Y-parameters Figure 2: Two-port power gain definitions Pin is the power actually delivered to the input port of the amplifier, irrespective of whether the source is conjugately matched to the input port. Z-parameters Y-parameters 2 2 Pin = ji1j Re(Zin) (5) Pin = jv1j Re(Yin) (6) Re(Zin) Re(Yin) P = jv j2 (7) P = ji j2 · (8) in S 2 in S 2 jZS + Zinj jYS + Yinj © Poole-Darwazeh 2015 Lecture 7 - Gain and stability of active networks Slide6 of 53 Power available from the generator (PAVS) i1 i2 i1 i2 ZS vS v1 Zin Zout v2 ZL iS YS v1 Yin Yout v2 YL (a) Z-parameters (b) Y-parameters Figure 3: Two-port power gain definitions PAVS is the maximum power available from the source when it is conjugately matched ∗ ∗ to the input impedance of the amplifier. By setting ZS = Zin in (7) or YS = Yin in (8) we get: Z-parameters Y-parameters 2 2 jvSj jiSj PAVS = (9) PAVS = (10) 4Re(ZS) 4Re(YS) © Poole-Darwazeh 2015 Lecture 7 - Gain and stability of active networks Slide7 of 53 Power absorbed by the load (PL) i1 i2 i1 i2 ZS vS v1 Zin Zout v2 ZL iS YS v1 Yin Yout v2 YL (a) Z-parameters (b) Y-parameters Figure 4: Two-port power gain definitions PL is the power actually delivered to the load, irrespective of whether the load is conjugately matched to the output of the two-port. This is defined as: Z-parameters Y-parameters 2 2 PL = ji2j · Re(ZL) (11) PL = jv2j · Re(YL) (12) Re(Z ) Re(Y ) P = jv j2 · L (13) P = ji j2 · L (14) L 2 2 L 2 2 jZout + ZLj jYout + YLj © Poole-Darwazeh 2015 Lecture 7 - Gain and stability of active networks Slide8 of 53 Power available from the amplifier output (PAVN) i1 i2 i1 i2 ZS vS v1 Zin Zout v2 ZL iS YS v1 Yin Yout v2 YL (a) Z-parameters (b) Y-parameters PAVN is the maximum power available from the amplifier output when it is conjugately ∗ ∗ matched to the load. By setting ZS = Zin in (9) or YS = Yin in (10) we get : Z-parameters Y-parameters 2 2 jv2j ji2j PAVN = (15) PAVN = (16) 4Re(Zout ) 4Re(Yout ) © Poole-Darwazeh 2015 Lecture 7 - Gain and stability of active networks Slide9 of 53 Power gain definitions Using the power definitions (5) to (16), we can state the four most common definitions of two-port power gain as follows[9]: 1. Transducer Power Gain : PL GT = (17) PAVS 2. Available Power Gain : PAVN GA = (18) PAVS 3. Operating Power Gain : PL Go = (19) Pin © Poole-Darwazeh 2015 Lecture 7 - Gain and stability of active networks Slide10 of 53 Transducer power gain in terms of Y-parameters The transducer gain, GT , refers to the general case of any arbitrary source and load termination. The expression for transducer gain must therefore contain both source admittance, YS, and load admittance, YL. Applying (10) and (12) to (17) we can write the transducer gain as follows : P jv j2 G = L = 4 · 2 · Re(Y )Re(Y ) (20) T 2 S L PAVS jisj 2 2 We now need to replace the term jv2j =jisj in (20) by a term involving only the Y-parameters. With reference to figure 5(b) we can write the following : is v1 = (21) YS + Yin By replacing Yin in (21) with (??) we can write : is v1 = (22) Y12Y21 YS + Y11 − YL + Y22 is(Y + Y ) = L 22 (23) (YS + Y11)(YL + Y22) − Y12Y21 © Poole-Darwazeh 2015 Lecture 7 - Gain and stability of active networks Slide11 of 53 Transducer power gain in terms of Y-parameters We note that v2 is related to v1 and i2 by the definition of the Y-parameters in section ??, i.e. : i2 = Y22v2 + Y21v1 (24) Rearranging (24) gives: i2 − Y21v1 v2 = (25) Y22 From figure 5(b) we can see that i2 is also equal to −v2YL, so we can write : −v2YL − Y21v1 v2 = (26) Y22 −Y v = 21 1 (27) Y22 + YL Substituting (22) into (26) gives : v −Y 2 = 21 (28) is (YS + Y11)(YL + Y22) − Y12Y21 Finally, we substitute (28) back into (20) to get the required expression for transducer gain in terms of Y-parameters as follows : 4jY j2Re(Y )Re(Y ) G = 21 L S (29) T 2 2 jY11 + YSj jY22 + YLj © Poole-Darwazeh 2015 Lecture 7 - Gain and stability of active networks Slide12 of 53 Transducer power gain in terms of Y-parameters We can rearrange (29) as follows : 2Re(Y ) 2Re(Y ) G = S · jY j2 · L (30) T 2 21 2 jY11 + YSj jY22 + YLj We can thereby see that the transducer gain is comprised of three factors : 2 i) an intrinsic gain component : jY21j 2Re(Y ) ii) a source mismatch factor : S 2 jY11 + YSj 2Re(Y ) iii) a load mismatch factor : L 2 jY22 + YLj The first is the intrinsic gain of the Two-port when terminated with the reference impedance that is used to measure the Y-parameters (i.e. a short circuit). The second factor accounts for the degree of mismatch at the input ports. This can be demonstrated by considering what happens as YS ! 1, in which case this factor tends to unity. A similar argument applies to the load mismatch factor. © Poole-Darwazeh 2015 Lecture 7 - Gain and stability of active networks Slide13 of 53 Available power gain in terms of Y-parameters We derive available power gain, GA, in terms of Y-parameters by applying (10) and (16) to (18) to obtain : P ji j2 Re(Y ) G = AVN = 2 · S (31) A 2 PAVS jiSj Re(Yout ) We can see from figure 5(b) that iS = −v1(YS + Y11) (note that we use Y11, not Yin because i2 is defined here as the short circuit output current, i.e. YL = 1 and so Yin = Y22). We can therefore write : i i −Y 2 = 2 = 21 (32) iS −v1(YS + Y11) YS + Y11 If we now replace i2=iS in (31) with (32) we have : jY j2 Re(Y ) G = 21 · S (33) A 2 jYS + Y11j Re(Yout ) © Poole-Darwazeh 2015 Lecture 7 - Gain and stability of active networks Slide14 of 53 Operating power gain in terms of Y-parameters We now apply a similar reasoning to obtain the operating power gain, Go, in terms of Y-parameters. By applying (6) and (12) to (19) we have : 2 PL jv2j Re(YL) Go = = · (34) 2 Pin jv1j Re(Yin) From figure 5(b) we can see that i2 = −v2(YL + Y22).