Fundamentals of RF Circuit Design with Low Noise Oscillators Jeremy Everard Copyright © 2001 John Wiley & Sons Ltd ISBNs: 0-471-49793-2 (Hardback); 0-470-84175-3 (Electronic) Small Signal Amplifier Design and Measurement 3.1 Introduction So far device models and the parameter sets have been presented It is now important to develop the major building blocks of modern RF circuits and this chapter will cover amplifier design The amplifier is usually required to provide low noise gain with low distortion at both small and large signal levels It should also be stable, i.e not generate unwanted spurious signals, and the performance should remain constant with time A further requirement is that the amplifier should provide good reverse isolation to prevent, for example, LO breakthrough from re-radiating via the aerial The input and output match are also important when, for example, filters are used as these require accurate terminations to offer the correct performance If the amplifier is being connected directly to the aerial it may be minimum noise that is required and therefore the match may not be so critical It is usually the case that minimum noise and optimum match not occur at the same point and a circuit technique for achieving low noise and optimum match simultaneously will be described For an amplifier we therefore require: Maximum/specified gain through correct matching and feedback Low noise Low distortion Stable operation 98 Fundamentals of RF Circuit Design Filtering of unwanted signals Time independent operation through accurate and stable biasing which takes into account device to device variation and drift effects caused by variations in temperature and ageing It has been mentioned that parameter manipulation is a great aid to circuit design and in this chapter we will concentrate on the use of y and S parameters for amplifier design Both will therefore be described 3.2 Amplifier Design Using Admittance Parameters A y parameter representation of a two port network is shown in Figure 3.1 Using these parameters, the input and output impedances/admittances can be calculated in terms of the y parameters and arbitrary source and load admittances Stability, gain, matching and noise performance will then be discussed Figure 3.1 y parameter representation of an amplifier The basic y parameter equations for a two port network are: I1 = y11 V + y12 V (3.1) I = y 22 V + y 21V (3.2) From equation (3.1): Small Signal Amplifier Design and Measurement Yin = I1 y V = y11 + 12 V1 V1 99 (3.3) Calculate V1 from equation 3.2: V1 = I − y 22 V y 21 (3.4) Substituting (3.4) in (3.3): Yin = y V y I1 = y11 + 12 21 V1 I − y 22 V2 (3.5) Dividing top and bottom by V2: Yin = y12 y 21 I1 = y11 + V1 − y L − y 22 = y11 − y12 y 21 y L + y 22 (3.6) Similarly for Yout: Yout = y 22 − y 21 y12 YS + y11 (3.7) Yin can therefore be seen to be dependent on the load admittance YL Similarly Yout is dependent on the source admittance YS The effect is reduced if y12 (the reverse transfer admittance) is low If y12 is zero, Yin becomes equal to y11 and Yout becomes equal to y22 This is called the unilateral assumption 3.2.1 Stability When the real part Re(Yin) and/or Re(Yout) are negative the device is producing a negative resistance and is therefore likely to be unstable causing potential oscillation If equations (3.6) and (3.7) are examined it can be seen that any of the parameters could cause instability However, if y11 is large, this part of the input impedance is lower and the device is more likely to be stable In fact placing a resistor across (or sometimes in series with) the input or output or both is a 100 Fundamentals of RF Circuit Design common method to ensure stability This degrades the noise performance and it is often preferable to place a resistor only across the output Note that as y12 tends to zero this also helps as long as the real part of y11 is positive The device is unconditionally stable if for all positive gs and gL the real part of Yin is greater than zero and the real part of Yout is greater than zero The imaginary part can of course be positive or negative In other words the real input and output impedance is always positive for all source and loads which are not negative resistances Note that when an amplifier is designed the stability should be checked at all frequencies as the impedance of the matching network changes with frequency An example of a simple stability calculation showing the value of resistor required for stability is shown in the equivalent section on S parameters later in this chapter John Linvill [13] from Stanford developed the Linvill stability parameter: C= y12 y21 g11 g22 − Re( y12 y21) (3.8) where g11 is the real part of y11 The device is unconditionally stable if C is positive and less than one Stern [14] developed another parameter: k= 2( g11 + G S )(g 22 + G L ) y12 y 21 + Re( y12 y 21) (3.9) which is stable if k > This is different from the Linvill [13] factor in that the Stern [14] factor includes source and load admittances The Stern factor is less stringent as it only guarantees stability for the specified loads Care needs to be taken when using the stability factors in software packages as a large K is sometimes used to define the inverse of the Linvill or Stern criteria 3.2.1.1 Summary for Stability To maintain stability the Re(Yin) ≥ and the Re(Yout) ≥ for all the loads presented to the amplifier over the whole frequency range The device is unconditionally stable when the above applies for all Re(YL) ≥ and all Re(YS) ≥ Note that the imaginary part of the source and load can be any value Small Signal Amplifier Design and Measurement 3.2.2 101 Amplifier Gain Now examine the gain of the amplifier The gain is dependent on the internal gain of the device and the closeness of the match that the device presents to the source and load As long as the device is stable maximum gain is obtained for best match It is therefore important to define the gain There are a number of gain definitions which include the ‘available power gain’ and ‘transducer gain’ The most commonly used gain is the transducer gain and this is defined here: GT = PL Power delivered to the load = PAVS Power available from the source (3.10) To calculate this, the output voltage is required in terms of the input current Using the block diagram in Figure 3.1 V1 = IS = YS + Yin IS V1 = I S (YL + y 22 ) (YS + y11 )(YL + y 22 ) − y12 y 21 y y YS + y11 − 12 21 YL + y 22 (3.11) (3.12) To calculate V2 remember that: I = y 22 V + y 21V (3.2) Taking (3.2) therefore: V2 = I − y 21V1 y 22 (3.13) As V2 is also equal to –I2/YL then I2 = -V2YL and: V2 = − V2YL − y 21V1 y 22 (3.14) 102 Fundamentals of RF Circuit Design  Y  − y 21V1 V2  + L  = y 22  y 22    y V  V2 = −  21    y 22   + YL  y 22  V2 = (3.15)       (3.16) − y 21V2 y 22 + YL (3.17) Substituting equation (3.12) in equation (3.17): V2 = − I S y 21 (YS + y11)(YL + y 22 ) − y12 y 21 (3.18) As PL = |V2| GL where GL is the real part of YL: 2 PL = I S G L y 21 (y s )( + y11 Y L + y 22 (3.19) )−y 12 y 21 The power available from the source is the power available when matched so: I  PAVS =  S    GS (3.20) Therefore the transducer gain is: GT = 4GS G L y 21 PL = PAVS YS + y11 YL + y 22 ( )( )− y 12 y 21 (3.21) Small Signal Amplifier Design and Measurement 103 For maximum gain we require a match at the input and the output; therefore YS = Yin* and YL = Yout*, where * is the complex conjugate Remember, however, that as the load is changed so is the input impedance With considerable manipulation it is possible to demonstrate full conjugate matching on both the input and output as long as the device is stable The source and load admittances for perfect match are therefore as given in Gonzalez [1]: GS = 1 [(2 g11 g 22 − Re( y12 y 21 )) − y12 y21 ]2 g 22 BS = − b11 + G L = GS Im (y12 y 21) g 22 (3.23) g22 g11 BL = − b 22 + (3.22) (3.24) Im (y12 y 21) g11 YS = G S + jB S (3.25) YL = G L + jB L (3.26) The actual transducer gain for full match requires substitution of equations (3.22) to (3.26) in the GT equation (3.21) 3.2.3 Unilateral Assumption A common assumption to ease analysis is to assume that y12 = 0, i.e assume that the device has zero feedback This is the unilateral assumption where YS = y11 * and YL = y 22 * As: GT = 4GS G L y 21 PL = PAVS YS + y11 YL + y 22 ( )( )− y 12 y 21 (3.27) 104 Fundamentals of RF Circuit Design GT = y 21 g 11 g 22 (3.28) This is the maximum unilateral gain often defined as GUM or MUG and is another figure of merit of use in amplifier design This enables fairly easy calculation of the gain achievable from an amplifier as long as y12 is small and this approximation is regularly used during amplifier design 3.3 Tapped LC Matching Circuits Using the information obtained so far it is now possible to design the matching circuits to obtain maximum gain from an amplifier A number of matching circuits using tapped parallel resonant circuits are shown in Figure 3.2 The aim of these matching circuits is to transform the source and load impedances to the input and output impedances and all of the circuits presented here use reactive components to achieve this The circuits presented here use inductors and capacitors Figure 3.2 Tapped parallel resonant RF matching circuits Small Signal Amplifier Design and Measurement 105 A tuned amplifier matching network using a tapped C matching circuits will be presented This is effectively a capacitively tapped parallel resonant circuit Both tapped C and tapped L can be used and operate in similar ways These circuits have the capability to transform the impedance up to the maximum loss resistance of the parallel tuned circuit The effect of losses will be discussed later Two component reactive matching circuits, in the form of an L network, will be described in the section on amplifier design using S parameters and the Smith Chart A tapped C matching circuit is shown in Figure 3.2a The aim is to design the component values to produce the required input impedance, e.g 50Ω for the input impedance of the device which can be any impedance above 50Ω To analyse the tapped C circuit it is easier to look at the circuit from the high impedance point as shown in Figure 3.3 C2 Yin C1 R Figure 3.3 Tapped C circuit for analysis The imaginary part is then cancelled using the inductor Often a tunable capacitor is placed in parallel with the inductor to aid tuning Yin is therefore required: Yin = Real + Imaginary parts = G + jB (3.29) Initially we calculate Zin: Z in = R / sC1 + sC R+ sC1 and with algebra: (3.30) 106 Yin = Fundamentals of RF Circuit Design s C1C R + sC sC R + sC1 R + (3.31) The real part of Yin is therefore: ω 2C2 R Yin = + ω R (C1 + C ) (3.32) The shunt resistive part of Yin is therefore Rin: + ω R (C1 + C ) Rin = (3.33) ω 2C2 R If we assume (or ensure) that ω R (C1 + C2) > 1, which occurs for loaded Qs greater than 10, then:  C  Rin = R 1 +   C2    2 (3.34) The imaginary part of Yin is: Yin (imag ) = sω C1C R (C1 + C )+ sC + ω R (C1 + C ) (3.35) Making the same assumption as above and assuming that C2 is smaller than 2 ω C1C2R (C1+C2) then: CT = C1C C1 + C This is equivalent to the two capacitors being added in series (3.36)     *% + /  é      : '           Figure 3.27 Example 1b: Match ρ = 0.4 angle +136° to 50Ω              C  $%        *#  $  ,      ±    , , é   , ,            %    *#     %   $   , ,    ,  $$    #(6,67$1&( &20321(17 5=R  25 &21'8&7$1&( &20321(17 *