EE 40458Power Gain and Amplifier Design,

Stability Considerations

11/13/2018 – 11/15/2018

Motivation• Brief recap:– We’ve studied matching networks (several types, how

to design them, bandwidth, how they work, etc…)– Studied network analysis techniques (matrices with S,

ABCD, …; flow graphs, etc.)– But why? Ultimate goal: design a circuit that does

something useful• Amplifiers are a good starting point– Useful for many (all?) systems– Fundamental principles can be applied to other

circuits

Design/Analysis Approach

• Could approach design many ways– “Regular” circuit design: use equivalent circuit model

(hybrid-π, T model)– We’ll take an alternative: treat active device as

(known) 2-port S-matrix• This approach allows us to be more general

(results will apply more widely)– Example: we’ll be able to project what the highest

gain possible is, whether or not the circuit is stable (i.e., will it oscillate?); not always so obvious from equivalent circuits without a lot of work

– Side bonus: we’ll get to see what features of S-matrices are desirable for circuits; help us choose devices

Amplifiers – It’s (Mostly) About Gain

• From previous classes—of course gain is the key thing• But it isn’t quite so simple for high-frequency systems• Recall: from “regular” circuit design: gain was (mostly)

voltage gain Av; sometimes (more rarely) current gain– But you don’t really need an amplifier

for either of these!– A transformer can produce voltage

gain (and a lot of it…)– More generally, a matching network

trades voltage gain for current gain (or visa-versa)

• What do amplifiers really do, then? – Power gain is the key feature: not just

voltage or current in isolation

Power Gain

• Power gain: not quite so simple as it might look:

• OK—but what is Pout and Pin, exactly? There are several (useful) possibilities, we need to be clear which we mean–With voltage, current gain these ambiguities are

less troublesome

Gain = PoutPin

Power Gain – Definitions• Since input, output power can be defined different ways,

multiple definitions for power gain (not just one gain; there are several different alternatives!)

GP =PLPin

=Power  delivered  to load

Power  delivered (absorbed  by) network

GT =PLPavs

=Power  delivered  to load

Power  available  from source 

GA =PavnPavs

=Power  available  from networkPower  available  from source

Power Gain - Comparisons• GP: “operating power gain,”

most obvious, but not all that useful

• GA: “available gain,” good baseline for comparisons, shows what performance is possible

• GT: “transducer gain,” most useful in system context, analysis• What’s the difference among these?

– Do you count what actually gets into the network (Pin) or what could have gotten in (Pavs)?

– Same idea at output (PL vs. Pavs)– Degree of mismatch is the

difference

GP =PLPin

=Power  delivered  to load

Power  delivered (absorbed  by) network

GT =PLPavs

=Power  delivered  to load

Power  available  from source 

GA =PavnPavs

=Power  available  from networkPower  available  from source

Distinctions Among Power Gain Definitions

• How much power is transferred depends on the impedances/reflection coefficients:

• Pin: power carried by a1, minus that reflected off (b1)

• Pavs: what’s the most power you could get from the source (if it were conjugately matched)?

• Pavn: what’s the most power available from S?

• PL: power dissipated in ZL

Numerical Comparisons?• How do these three definitions compare in terms of

magnitude?

• GA > GT (because PL < Pavn from mismatch)• GP > GT (because Pin < Pavs…)• But not clear if GP, GA larger…depends on relative degree of

source, load mismatch• Can derive each gain from flow graph:

GT =PLPavs

;  GP =PLPin;  GA =

PavnPavs

Power Gain Expressions• Power gains:

• Where:

GT =1− ΓS

2

1−ΓinΓSS21

2 1− ΓL2

1− S22ΓL2 =

1− ΓS2

1− S11ΓSS21

2 1− ΓL2

1−ΓoutΓL2

GP =1

1− Γin2 S21

2 1− ΓL2

1− S22ΓL2

GA =1− ΓS

2

1− S11ΓS2 S21

2 11− Γout

2

Γin = S11 +S12S21ΓL1− S22ΓL

Γout = S22 +S12S21ΓS1− S11ΓS

Amplifier Design• With these definitions, we can start designing

for real• Several useful special cases for designs:–Maximum gain– Specified gain (i.e. trade lower gain for more

bandwidth)– Low noise

• Let’s do an example of each one to see how they work

Design for Maximum Gain

• For maximum gain: want to conjugately match both the input and output—simultaneously

• Mathematically:

– Note: can work with reflection coefficients since Z G as a 1-to-1 map

– Also, in this case, GT = GA; maximizing power in and out

Γin = ΓS*

Γout = ΓL*

Maximum Gain Design, Cont.

• So: how to design?– Basic idea:

– Our job:• Make input matching network present Gs*=Gin• Make output matching network present GL*=Gout• (So it is really just about making matching networks—

once the correct “targets” for GS, GL are found)

Maximum Gain Design, Cont.• Design approach: know input, output must be conjugately

matched. So:

• But we also know

• So: GS depends on GL (and visa-versa)– Must solve for both GS and GL simultaneously (a single solution that

satisfies all the constraints)– Book goes through the algebra in some detail…

Γin = ΓS*

Γout = ΓL*

Γin = S11 +S12S21ΓL1− S22ΓL

Γout = S22 +S12S21ΓS1− S11ΓS

Maximum Gain Design, Cont.• Bottom line: there is a unique solution for GS and GL

Where:

ΓS =B1 ± B1

2 − 4 C12

2C1

ΓL =B2 ± B2

2 − 4 C22

2C2

B1 =1+ S112− S22

2− Δ

2

B2 =1+ S222− S11

2− Δ

2

C1 = S11 −ΔS22*

C2 = S22 −ΔS11*

Δ = S11S22 − S12S21     (det(S))

Maximum Gain Design, Cont.• So: what to do with this?• Find [S] of

amplifying element(transistor, usuallyplus bias networks)

• Compute GS, GL from [S]• Design input matching

network to go from source to GS

• Design output matching network to go from load to GL

Input & Output Matching• So: maximum gain amplifier design is really designing two

matching networks:

Input & Output Matching• So: maximum gain amplifier design is really designing two

matching networks:

Specified Gain Design

• Key point: already know how to design an amplifier with maximum gain. Why would we ever go for less?– Need specific signal level– More likely: need more bandwidth than max. gain design

can deliver; to get bandwidth, need to give up something else

• Approach: look at GT expression:

• Note: 3 basic terms (source, S21, and load)

GT =1− ΓS

2

1−ΓinΓSS21

2 1− ΓL2

1− S22ΓL2 =

1− ΓS2

1− S11ΓSS21

2 1− ΓL2

1−ΓoutΓL2

GT =GS ⋅GO ⋅GL

Specified Gain Design, Cont.• But a little complicated: GS depends on load, GL

depends on source• Yuck. So let’s make a simplifying assumption—that

[S] is unilateral (e.g. S12 = 0). Then Gin=S11 and Gout=S22

• Net result:

• So how can we set/change the gain? Introduce (intentionally) mismatch at source and/or load– If the match is perfect, the gain will be maximized. We’re

intentionally mismatching to control the gain

GS =1− ΓS

2

1− S11ΓS2 ;  GL =

1− ΓL2

1− S22ΓL2

Specified Gain Design, Cont.

• For maximum gain, conjugate match at both input and output: GS=S11*, GL=S22* (remember, we’re assuming unilateral)

• So the maximum GS, GL become:

• It turns out (read: after much algebra, you can show if you are lucky and factor things just exactly right) that:

constant |GS| and |GL| plot as circles in the GS and GL planes

GS,max =1

1− S112 ;  GL,max =

11− S22

2

Specified Gain Design, Cont.• These are the “constant gain circles”• Mathematically: gs =

GSGS,max

;  gL =GLGL,max

CS =gsS11

*

1− (1− gs ) S112

RS =1− gs (1− S11

2 )1− (1− gs ) S11

2

CL =gLS22

*

1− (1− gL ) S222

RL =1− gL (1− S22

2 )1− (1− gL ) S22

2

Source side: Load side:

C: coordinates (on imaginary plane) of center of circleR: radius

Specified Gain Design, Cont.• So: idea is to figure out how much mismatch to

introduce at input, output to get desired gain• Total gain:– Designer chooses GS, GL so that GT comes out at value

desired (but remember, GS≤ GS,max; GL≤GL,max)– In principle, any combination is ok (results in the correct

gain)– But usually want to get something in exchange for the

trade (e.g. bandwidth)– That additional trade is where design insight comes in

GT =GS ⋅GO ⋅GL

Example: Specified Gain

• Example 12.4 in Pozar is a good example• Summary: design an amplifier at 4 GHz with 11 dB

gain; try to maximize the bandwidth– Given: transistor S-parameters:

• Can compute GS,max; GL,max

Numerically: GS,max = 2.29 = 3.6 dB; GL,max = 1.56 = 1.9 dBGo = |S21|2 = 6.25 = 8 dB– Max gain: 13.5 dB. We can give up 2.5 dB in combination of GS, GL

S = 0.75∠−120 02.5∠80 0.6∠− 70

#

$%

&

'(

GS,max =1

1− S112 ;  GL,max =

11− S22

2

Example: Specified Gain (cont.)• Pick a few choices for GS, GL• Plot the gain circles:

• OK…• Remember:

GS,max = 3.6 dBGL,max=1.9 dB

• Options:– GS=3 dB,

GL=0 dB – GS=2 dB

GL=1 dB – Both of these sets are 2.5 dB less than the maximum, give

us 11 dB total

Example: Specified Gain (cont.)

• How to choose?• Trade for something

else—like

bandwidth

• Recall: for largest

bandwidth,

matching

network should not “work hard”

• Pick GS, GL nearest center of Smith Chart—”laziest” matching network

• If pick roughly equal input, output matching networks, overall BW maximized (filters in cascade)

Example: Specified Gain (cont.)

Low Noise Amplifier Design• Last case: low noise amplifier• Noise limits many systems (radio, wireline, fiber

receivers, etc.) • Performance dominated (usually) by first amplifier in

cascade• Unfortunate fact: best noise performance not

obtained at maximum gain. Have to match for either noise or gain; a tradeoff is inevitable

Noise in Two-Port Networks• Noise in amplifiers (as represented by 2-port

networks) can be quantified by noise factor:

– Noise factor is a measure of how much worse the SNR (power) is at the output compared to the input, due to noise added by the network

– Noise factor depends on input termination (GS):

F = SNRinSNRout

F = Fmin +4RNZO

ΓS −Γopt2

(1− ΓS2 ) 1+Γopt

2

Zs

vsomgi.rsJIaina Igz'idtcannot tell SinfronNia

Signal amplifiesnoisetooSoma Also adds its ownnoise

Noise in Two-Port Networks• Noise factor depends on input termination (GS):

– Gopt, RN, Fmin are “noise parameters” of the device• Can show (if you are patient) that constant F

contours are (gasp) circlesin the GS plane:

• CF is center (complex plane), RF is radius

F = Fmin +4RNZO

ΓS −Γopt2

(1− ΓS2 ) 1+Γopt

2

!" =Γ%&'( + 1

+" =( ( + 1 − Γ%&'

-

( + 1

( = Γ. − Γ%&'-

1 − Γ. -= / − /0124+4/67

1 + Γ%&'-

Low Noise Amplifier Design

• Example 12.5 in Pozar:– Design an amplifier with 2 dB noise figure and maximum

gain (consistent with this noise level)– Plot noise figure circles– Note that noise figure is related to GS; but gain is too

(specified gain: GS term)– So we look at both constant gain and noise figure circles on

the same plot:

Low Noise Amplifier Design (cont.)• Noise figure: NF=2 dB circle• Several GS circles

1.0, 1.5, 1.7 dB• Need to satisfy

both: findintersectionbetweenNF, highestgain GScircle

• What about output? Conjugately match once input is finalized

Another Important Consideration: Stability

• In the design techniques so far, we’ve neglected an important issue, stability

• In microwave design, “stability” has a very specific meaning– Stability immunity to oscillation– A stable circuit won’t oscillate, and unstable one

might (or will, depending on the situation)• In “regular old” circuit design, stability usually

considered using feedback theory• Using s-parameters gives us an alternative

approach that can be helpful

S-parameter Based Stability Considerations

• If we treat the circuit (e.g. amplifier) as a black box with s-parameters, what would it mean if the circuit were oscillating?– An oscillator is a signal generator: produces signal power at

some frequency even when no input is present– Looking into the ports, “reflection coefficient” > 1: more power

“reflected” than put in– Not necessarily “reflected,” but “reflection coefficient” is just

the ratio of the outgoing wave (b) to incoming wave (a)– If a wave is zero, but b wave isn’t--circuit is oscillating

––

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S-parameter Based Stability Considerations

• What does this mean from a circuit perspective?– In terms of impedances, real part of input (Zin) and/or

output (Zout) impedances < 0– “negative resistances” can generate power—and

that’s what oscillators do

• S-parameter-based approach is sometimes easier– In terms of reflection coefficients (instead of

impedances): |Gin| and/or |Gout| > 1 (more reflected than incident) indicates instability

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Reflection Coefficient View• Let’s look at this with some analysis• Basic circuit:

• To check to see if the circuit can oscillate, choose GS and GL so that |GS|, |GL|

Reflection Coefficient View of Stability

• So for this circuit, what does this imply?• Mathematically:– Require: Γ" < 1, Γ& < 1– For stability (no oscillations):Γ'( = *++ + "-.".-/0+1"../0

Alternative Approach: Look for Edge of Stability

• Stability requires |Gin|, |Gout| < 1; conversely if > 1, then unstable

• “Edge of stability”: |Gin|, |Gout| = 1• Recall: Γ"# = %&& + ()*(*)+,&-(**+, (depends on GL), andΓ/01 = %22 + ()*(*)+3&-())+3 (depends on GS)

• Idea: look for set of GS, GL that make the corresponding |Gin|, |Gout| = 1– This will define the “border” between stability and

instability, and then we can just figure out which regions are which

Alternative Approach: Look for Edge of Stability

• Basic approach (math): solving

• Γ"# = %&& + ()*(*)+,&-(**+, = 1 à yields set of GL’s that satisfy the equation

• Math is messy, tedious. Bottom line: set of GL’s that solve the equation is a circle in the GL plane

• Can show that:

./ = (**-0())∗ ∗

(** *- 0 *and 2/ = ()*(*)(** *- 0 *

where CL is the center of the circle and rL is the radius

What does this mean?• GL’s that solve the equation is a circle in the GL plane• !" = $%%&'$((

∗ ∗

$%% %& ' %

• *" = $(%$%($%% %& ' %

• If GL is chosen on this circle, then |Gin|=1 – on edge of stability

• Circle divides GL plane into two regions; inside & outside– One region is stable, the other isn’t

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What does this mean?• Which region is stable, unstable?

– It could be either way– the circle is just the boundary• Quick check: check any point (choose some GL) not on the circle,

and see if Gin < or > 1– Easiest one? GL=0 (center of Smith chart)– For GL=0, Gin=S11 (Γ"# = %&& + ()*(*)+,&-(**+,)– If |S11|

How about source side?• So far, have focused entirely on effect of GL on Gin

– But Gout depends on GS (symmetry of the overall architecture)– Nothing special about the port numbers

• Can follow a very similar derivation for effect of GS• In the GS plane, GS’s that cause |Gout|=1 is a circle• !" = "$$%&"''

∗ ∗

"$$ '% & '

• )" = "$'"'$"$$ '% & '• To check region of stability or instability, look at |S22|

(just like |S11| check for the GL plane)• Bottom line: need to check both source and load sides;

circuit might be unstable at only one port, etc.

Careful: GL, GS are in Different Planes

• Warning: it is common to show both source & load circles on the same Smith chart

• Careful! These circles are in different planes; there isno significance to overlaps, etc.

• Remember: source stabilitycircle tells us what GS might cause problems (input); loadcircle tells us what GL might causeproblems (load)

• Treat them separately

Unconditional Stability• Special case: unconditional stability• In English:– A circuit is unconditionally stable if there is no combination

of passive source & load terminations that will cause oscillation

• In math:– |Gin|

Unconditional Stability• That’s a little tedious…there is an easier way.

Conditions on stability can be expressed mathematically

• Two main approaches• Classic approach: Rowlett stability test

! = 1 − %&&' − %'' ' + Δ '2 %&' %'&

> 1 ,-.Δ = %&&%'' − %'&%&' < 1

– Both must be satisfied simultaneously for unconditional stability

• Alternative: “µ test”. Circuit is unconditionally stable if

0 = 1 − %&&'

%'' − Δ%&&∗ + %&'%'&> 1

Stability Interpretation• A key point is that these expressions are frequency dependent

(since S is); circuits are often unconditionally stable at some frequencies and potentially unstable at others

• In general, stability must be checked/controlled for all frequencies

• Remember: the circuit doesn’t know (or care) what you want it to do; it does what it wants– The “frequency of interest” or the “design frequency” doesn’t mean

anything. A circuit that is “unconditionally stable” at a 1 GHz design frequency might well make an excellent oscillator at 10 MHz

– A circuit that is oscillating by accident will (usually) not work as intended; the large oscillation signals will corrupt the bias point, etc.

Stability Interpretation• The inequalities (K-D or µ) are the easiest way to quickly check

all frequencies for “yes/no” unconditional stability • Often the case that unconditional stability cannot be

achieved, have to “work around” unstable regions (i.e. operate within the stable region of a potentially unstable device)

• The stability circles give a lot more information about what regions are stable and unstable, and can help you avoid undesired areas (e.g. source, load impedances that would cause oscillations)

• Typically, check stability circles in regions that the µ or K-Dtests say are not unconditionally stable to see where your termination conditions are; often not at design frequency