Two-Port Circuits and Power Gain

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EECS 142

Lecture 4: Two-Port Circuits and Power Gain

Prof. Ali M. Niknejad

University of California, Berkeley Copyright c 2005 by Ali M. Niknejad

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 1/23 – p. 1/23 A Generic Ampliﬁer

YS + y11 y12 vs Y · y21 y22 ¸ L −

Consider the generic two-port ampliﬁer shown above. Note that any two-port linear and time-invariant circuit can be described in this way. We can use any two-port parameter set, including admittance parameters Y , impedance parameters Z, or hybrid or inverse-hybrid parameters H or G.

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 2/23 – p. 2/23 Choosing Two-Port Parameters

Feedback

YS Yout +

vs Amp YL −

Yin

The choice of parameter set is usually determined by convenience. For instance, if shunt feedback is applied, Y parameters are most convenient, whereas series feedback favors Z parameters. Other combinations of shunt/series can be easily described by H or G. ABCD parameters are useful for cascading two-ports.

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 3/23 – p. 3/23 Y Parameters

We’ll primarily use the Y parameters

i y y v 1 = 11 12 1 i2! y21 y22! v2!

But in fact the choice depends largely on convenience. Often the form of feedback determines the best choice. All 2-port parameters are equivalent. Many of the results that we derive carry in terms of Y-parameters can be applied to other two-port parameters (input impedance, output impedance, gain, etc).

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 4/23 – p. 4/23 Admittance Parameters

Notice that y11 is the short circuit input admittance i y = 1 11 v 1 v2=0

The same can be said of y22. The forward transconductance is described by y21 i y = 2 21 v 1 v2=0

whereas the reverse transconductance is described by y12.

If a two-port ampliﬁer is unilateral, then y12 =0

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 5/23 – p. 5/23 Hybrid-Π Admittance Parameters

Cµ

Yπ Yo + Yµ gmvin r Rin Cin vin o Co −

Let’s compute the Y parameters for the common hybrid-Π model

i2 Yµ y11 = yπ + yµ + + v1 Yπ Yo gmvin v2 =0 − − y21 = gm yµ

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 6/23 – p. 6/23 Admittance Parameters (cont)

i1 Yµ y22 = yo + yµ + + v1 =0 Yπ Yo gmvin v2 − − y12 = yµ

Note that the hybrid-π model is unilateral if yµ = sCµ =0. Therefore it’s unilateral at DC. A good ampliﬁer has a high ratio y21 because we expect y12 the forward transconductance to dominate the behavior

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 7/23 – p. 7/23 Why Use Two-Port Parameters?

The parameters are generic and independent of the details of the amplifier → can be a single transistor or a multi-stage amplifier High frequency transistors are more easily described by two-port parameters (due to distributed input gate resistance and induced channel resistance) Feedback amplifiers can often be decomposed into an equivalent two-port unilateral amplifier and a two-port feedback section We can make some very general conclusions about the “optimal” power gain of a two-port, allowing us to define some useful metrics

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 8/23 – p. 8/23 Voltage Gain and Input Admittance

Since i2 = v2YL, we can write

(y22 + YL)v2 = y21v1 Which leads to the “internal” two-port gain

v2 y21 Av = = v1 y22 + YL

Check low freq limit for a hybrid-Π: Av = gmZo||ZL X The input admittance is easily calculated from the voltage gain i1 v2 Yin = = y11 + y12 v1 v1 y12y21 Yin = y11 y22 + YL

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 9/23 – p. 9/23 Output Admittance

By symmetry we can write down the output admittance by inspection

y12y21 Yout = y22 y11 + YS

Note that for a unilateral ampliﬁer y12 =0 implies that

Yin = y11

Yout = y22 The input and output impedance are de-coupled!

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 10/23 – p. 10/23 External Voltage Gain

The gain from the voltage source to the output can be derived by a simple voltage divider equation

′ v2 v2 v1 YS YSy21 Av = = = Av = vs v1 vs Yin + YS (y22 + YL)(YS + Yin) If we substitute and simplify the above equation we have ′ YSy21 Av = (YS + y11)(YL + y22) y12y21 Verify that this makes sense at low frequency for hybrid-Π:

′ YSy21 Zin Av(DC) = = × gmRL||ro (YS + y11)(YL + y22) Zin + ZS

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 11/23 – p. 11/23 Feedback Ampliﬁers and Y -Params

Note that in an ideal feedback system, the ampliﬁer is y A unilateral and the closed loop gain is given by x = 1+Af We found last lecture that the voltage gain of a general two-port driven with source admittance YS is given by

′ YSy21 Av = (YS + y11)(YL + y22) y12y21 If we unilaterize the two-port by arbitrarily setting y12 =0, we have an “open” loop forward gain of Y y A = A′ = S 21 vu v y12=0 (YS + y11)(YL + y22)

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 12/23 – p. 12/23 Identiﬁcation of Loop Gain

′ Re-writing the gain Av by dividing numerator and denominator by the factor (YS + y11)(YL + y22) we have

−YS y21 (YS +y11)(YL+y22) A′ = v 1 y12y21 (YS +y11)(YL+y22)

We can now see that the “closed” loop gain with y12 =06 is given by A A′ = vu v 1 + T where T is identiﬁed as the loop gain

y12y21 T = Avuf = (YS + y11)(YL + y22)

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 13/23 – p. 13/23 The Feedback Factor and Loop Gain

Using the last equation also allows us to identify the feedback factor Y f = 12 YS

If we include the loading by the source YS, the input admittance of the ampliﬁer is given by

y12y21 Yin = YS + y11 YL + y22 Note that this can be re-written as y y Y = (Y + y ) 1 12 21 in S 11 (Y + y )(Y + y ) S 11 L 22

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 14/23 – p. 14/23 Feedback and Input/Output Admittance

The last equation can be re-written as

Yin = (YS + y11)(1 + T )

Since YS + y11 is the input admittance of a unilateral ampliﬁer, we can interpret the action of the feedback as raising the input admittance by a factor of 1 + T . Likewise, the same analysis yields

Yout = (YL + y22)(1 + T ) It’s interesting to note that the same equations are valid for series feedback using Z parameters, in which case the action of the feedback is to boost the input and output impedance.

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 15/23 – p. 15/23 Power Gain

Pin PL

YS + y11 y12 vs Y · y21 y22 ¸ L −

Pav,s Pav,l

We can deﬁne power gain in many different ways. The power gain Gp is deﬁned as follows

PL Gp = = f(YL, Yij) =6 f(YS) Pin We note that this power gain is a function of the load admittance YL and the two-port parameters Yij.

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 16/23 – p. 16/23 Power Gain (cont)

The available power gain is deﬁned as follows

Pav,L Ga = = f(YS, Yij) =6 f(YL) Pav,S

The available power from the two-port is denoted Pav,L whereas the power available from the source is Pav,S. Finally, the transducer gain is deﬁned by

PL GT = = f(YL, YS, Yij) Pav,S This is a measure of the efﬁcacy of the two-port as it compares the power at the load to a simple conjugate match.

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 17/23 – p. 17/23 Derivation of Power Gain

The power gain is readily calculated from the input admittance and voltage gain

|V |2 P = 1 ℜ(Y ) in 2 in

|V |2 P = 2 ℜ(Y ) L 2 L 2 V2 ℜ(YL) Gp = V ℜ(Y ) 1 in

|Y |2 ℜ(Y ) G 21 L p = 2 |YL + Y22| ℜ(Yin)

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 18/23 – p. 18/23 Derivation of Available Gain

Y11 Y12 IS YS Ieq Yeq · Y21 Y22 ¸

To derive the available power gain, consider a Norton equivalent for the two-port where

Y21 Ieq = I2 = Y21V1 = IS Y11 + YS The Norton equivalent admittance is simply the output admittance of the two-port

Y21Y12 Yeq = Y22 Y11 + YS

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 19/23 – p. 19/23 Available Gain (cont)

The available power at the source and load are given by 2 2 |IS| |Ieq| Pav,S = Pav,L = 8ℜ(YS) 8ℜ(Yeq)

|Y |2 ℜ(Y ) G 21 S a = 2 |Y11 + YS| ℜ(Yeq)

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 20/23 – p. 20/23 Transducer Gain Derivation The transducer gain is given by

1 2 2 PL 2 ℜ(YL)|V2| V2 GT = = 2 =4ℜ(YL)ℜ(YS) |IS | Pav,S IS YS 8ℜ( )

We need to ﬁnd the output voltage in terms of the source current. Using the voltage gain we have and input admittance we have

V Y 2 = 21 V Y + Y 1 L 22

IS = V (YS + Yin) V Y 1 2 = 21 I Y + Y |Y + Y | S L 22 S in

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 21/23 – p. 21/23

Transducer Gain (cont)

Y Y |Y + Y | = Y + Y 12 21 S in S 11 Y + Y L 22

We can now express the output voltage as a function of source current as

V 2 |Y |2 2 = 21 I 2 S |(YS + Y11)(YL + Y22) Y12Y21|

And thus the transducer gain

2 4ℜ(YL)ℜ(YS)|Y21| GT = 2 |(YS + Y11)(YL + Y22) Y12Y21|

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 22/23 – p. 22/23 Comparison of Power Gains

It’s interesting to note that all of the gain expression we have derived are in the exact same form for the impedance, hybrid, and inverse hybrid matrices.

In general, PL ≤ Pav,L, with equality for a matched load. Thus we can say that

GT ≤ Ga The maximum transducer gain as a function of the load impedance thus occurs when the load is conjugately matched to the two-port output impedance

∗ PL(YL = Yout) GT,max,L = = Ga Pav,S

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 23/23 – p. 23/23 Comparison of Power Gains (cont)

Likewise, since Pin ≤ Pav,S, again with equality when the the two-port is conjugately matched to the source, we have GT ≤ Gp The transducer gain is maximized with respect to the source when

∗ GT,max,S = GT (Yin = YS ) = Gp

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 4 p. 24/23 – p. 24/23