14. AMPLIFIERS 14.1 Introduction Microwave amplifiers combine active elements with passive transmission line circuits to provide functions critical to microwave systems and instruments. The history of microwave amplifiers begins with electron devices using resonant or slow-wave structures to match wave velocity to electron beam velocity. The design techniques used for BJT and FET amplifiers employ the full range of concepts we have developed in the study of microwave transmission lines, two-port networks and Smith chart presentation. In microwave systems, amplifiers play a significant role .We will start with the definitions of two port power gains and discuss the stability circles. Then we will design single-stage amplifiers for maximum gain, specified gain and low noise figure. The S parameters of a given microwave transistor can be derived from transistor equivalent circuit models based on device physics, or they can be measured directly. Generally, a manufacturer of a device intended for microwave applications will provide extensive S-parameter data to permit accurate design of microwave amplifiers. This can be verified by measurement, a step that has proven important on many occasions. For a bipolar junction transistor, in addition to intrinsic device parameters such as base resistance and collector-base capacitance, amplifier performance is strongly affected by the so-called parasitic elements associated with the device package, including base-lead and emitter-lead inductance internal to the package. Similar considerations apply to microwave field-effect transistors. The magnitude and phase angle of each of the S parameters typically vary with frequency, and characterization over the complete range of interest is necessary. Solid-state microwave amplifiers play an important role in communication where it has different applications, including low noise, high gain, and high power amplifiers. The high gain and low noise amplifiers are small signal low power amplifiers and are mostly used in the receiver side where the signal level is low. The small signal S parameter can be used in designing these low power amplifiers. The high power amplifier is used in the transmitter side where the signal should be at a high level to cross the desired distance. The intent of this chapter is to give an overview of some basic principles used in the analysis and design of the small signal low power amplifier and we will restrict ourselves to single-stage amplifier designs only. 14.2 Power Gains In this section, we will define various terms for power relating to the performance of the amplifier. Basically there are three types of power gains: s in out L Fig. 14.1 A general two-port network P • Operating power gain, G ≡ L , is defined as the ratio of the power delivered to Pin the load PL to the power input to the amplifier Pin. Pavn • Available power gain, GA ≡ , is defined as the ratio of the power available Pavs from the amplifier Pavs to the power available from the source Pavs. PL • Transducer power gain, GT ≡ , is defined as the ratio of the power delivered Pavs to the load PL to the power available from the source Pavs. The transducer power gain is equal to the operating power gain only when the amplifier is conjugate * matched to the source, i.e., Γin = ΓS . Therefore, GP>GT in general. Let us try to get the various expressions of power of an amplifier, viz., PL, Pin, Pavn and Pavs so that we can get their ratios to get the expressions of three power gains we have defined above. 1) By voltage division at the input port 1 (refer to Fig. 14.1), Zin + − + V1 = Vs = V1 + V1 = V1 (1+ Γin ) Zin + Zs Zin − Z0 Zs − Z0 ∴ Γin = , Γs = Zin + Z0 Zs + Z0 Γin Z in + Γin Z 0 = Z in − Z 0 Z in (Γin −1) = Z 0 (−1− Γin ) Z −1− Γ 1+ Γ in = in = in Z0 −1+ Γin 1− Γin + V1 1 Zin Hence, V1 = = ⋅ Vs ⋅ 1+ Γin 1+ Γin Zin + Zs 1+ Γin V 1− Γ = s . in 1+ Γ 1+ Γ 1+ Γ in in + s 1− Γin 1− Γs Vs ⋅(1− Γs ) = (1+ Γin )(1− Γs ) + (1+ Γs )(1− Γin ) V ⋅(1− Γ ) = s s 2(1− ΓsΓin ) Hence, the power delivered into the amplifier, i.e., the input power of the amplifier is + 2 1 V1 2 Pin = ⋅ ⋅(1− Γin ) 2 Z0 2 2 2 Vs (1− Γin )⋅ 1− Γs = ⋅ 2 8Z0 1− ΓsΓin 2 2 Vs 1− Γs Note that Pavs (Γs ) = Pin | ∗ . Therefore, Pavs = Pin | ∗ = ⋅ Γin =Γs Γin =Γs 2 8Z0 (1− Γs ) 2) From scattering matrix analysis at the output port 2 (refer to Fig. 14.1), − + + + − V2 = S21V1 + S22V2 and V2 = ΓLV2 − + − V2 =S21V1 + S22ΓLV2 + − S21V1 V2 = 1−S22ΓL S V (1− Γ ) = 21 s s (1−S22ΓL )(1− ΓsΓin ) − 2 2 2 2 1 V2 2 1 Vs S21 1− Γs 2 Therefore, PL = ⋅ (1− ΓL ) = ⋅ 2 2 (1− ΓL ) 2 Z0 8Z0 1−S22ΓL 1− ΓsΓin Note that Pavn (Γout ) = PL | ∗ ΓL =Γout 2 2 2 Vs S21 1− Γs 2 = ⋅ ⋅ (1− Γout ) 8Z ∗ 2 2 0 1− S22Γout 1− ΓsΓin S12S21ΓL S11 −S11S22ΓL + S12S21ΓL We also know that, Γin = S11 + = 1−S22ΓL 1−S22ΓL 2 putting this value in 1− ΓsΓin | ∗ ,we get ΓL =Γout 2 2 2 2 11−S Γ∗ − Γ Γ − Γ S S Γ∗ + Γ S S Γ∗ 1−S Γ (1− Γ )2 = ss out s 11 s 11 22 out s 12 21 out = 11 s out ∗ 2 ∗ 2 1−S22Γout 1−S22Γout 2 2 2 V S 1− Γ P = s 21 s avn 2 2 2 8Z0 1−S11Γs (1− Γout ) 3) Let us know get all the power gain ratios. 2 2 PL (1− ΓL ) S21 Operating power gain, G ≡ = 2 2 Pin 1−S22ΓL (1− Γin ) 2 2 2 2 2 P S (1− Γ )(1− Γ ) S (1− Γ ) Available power gain, G ≡ avn = 21 out s or 21 s A ∗ 2 2 2 Pavs 1−S22Γout 1− ΓsΓin 1−S11Γs (1− Γout ) 2 2 2 P S (1− Γ )(1− Γ ) Transducer power gain, G = L = 21 L s T 2 2 Pavs |1−S22ΓL | 1− ΓsΓin A special case for the transducer power gain occurs when S12 is zero or negligible. Then the device is non-reciprocal and hence in = S11 and S12=0, therefore the unilateral transducer gain is 2 2 2 P S (1− Γ )(1− Γ ) G = L = 21 L s TU 2 2 Pavs |1− S22ΓL | 1− ΓsS11 14.3 Stability s in out L Fig. 14.2 A general amplifier two-port network The stability of an amplifier is a very important consideration in a microwave circuit design. Stability or resistance to oscillation in a microwave circuit can be determined by the S-parameters. Oscillations are possible in a two-port network if either or both the input and the output port have negative resistance. In the above circuit, oscillation is possible if either input or output port impedances has a negative real part which would implies that Γin >1or Γout >1. There are two types of amplifier stability, unconditionally stable and conditionally stable. In the former, the real part of the input and output impedances of the amplifier is greater than zero for all passive load and source impedances. However, the amplifier is said to be conditionally stable or potentially unstable if the real part of the input or output impedances of the amplifier is less than zero for at least a passive load or source impedances. The stability test should be done for every frequency in the desired range. • Unconditional stability: network is unconditionally stable if Γin < 1 and Γout < 1 for all passive source and load impedances. For unconditionally stable: S12S21ΓL Γin = S11 + < 1 1−S22ΓL S12S21Γs Γout = S22 + <1 1−S11Γs If the device is unilateral,S12 = 0 , these condition reduce to S11 <1and S22 <1. The necessary and sufficient conditions for a two-port network to be unconditional stable are: 2 2 2 1− S − S + ∆ K = 11 22 >1; ∆ <1 also called as Rollet’s condition 2 S12S21 2 1− S Or, µ = 11 > 1 also known as -test. * S 22 − S11∆ + S 21S12 In practice, most of the microwave transistor amplifiers are potentially unstable because of the internal feedback. There are two ways to overcome the stability problem of the transistor amplifier. The first is to use some form of feedback to stabilize the amplifier. The second is to use a graphical analysis to determine the regions where the values of ΓS and ΓL (source and load reflection coefficients) are less than one, which means the real parts of ZIN and ZOUT are positive. • Conditional stability: Network is conditionally stable if Γin < 1and Γout < 1only for certain range of passive source and load impedances. This case is referred to as potentially unstable. Note that the stability condition of network is frequency dependent as it is possible for an amplifier to be stable at its design frequency and unstable at other frequencies. For conditionally stable: Stability circles are defined as the loci of ΓS (ΓL ) plane for which Γin =1 or ( Γout =1). Then the stability circles defines the boundary between stable and potentially unstable of ΓS and ΓL . * * (S22 − ∆S11 ) 1) CL = 2 2 S22 − ∆ S12S21 R L = 2 2 , where ∆ = S11S22 −S12S21 S22 − ∆ Output stability circles * * (S11 − ∆S22 ) 2) CS = 2 2 S22 − ∆ S12S21 RS = 2 2 S22 − ∆ Input stability circles |S11|>1 |S11|<1 CL CL RL RL (a) in < 1;stable |S22|<1 |S22|>1 CS CS RS RS < 1;stable (b) out Fig. 14.3 (a) Output stability circles and (b) input stability circles for conditionally stable device Given the S parameters of the device, we can plot the input and output stability circle to define where Γin =1 and Γout =1.