1

EKT 356

MICROWAVE COMMUNICATIONS

CHAPTER 6:

MICROWAVE AMPLIFIERS

2

INTRODUCTION

Most RF and microwave amplifiers today used transistor devices such as Si or SiGeBJTs, GaAs HBTs, GaAs or InP FETs, or GaAs HEMTs.

Microwave transistor amplifiers are rugged, low cost, reliable and can be easily integrated in both hybrid an monolithic integrated circuitry.

3

General Amplifier Block Diagram

( ) ( ) ( ) ( ) ...33

221 TOHtvatvatvatv iiio +++=

Input and output voltage relation of the amplifier

can be modeled simply as:

Vcc

PLPin

The active

component

vi(t)

vs(t)

vo(t)

DC supply

ZL

Vs

Zs AmplifierInput

Matching

Network

Output

Matching

Network

ii(t)

io(t)

4

Amplifier Classification

Amplifier can be categorized in 2 manners.

According to signal level:

Small-signal Amplifier.

Power/Large-signal Amplifier.

According to D.C. biasing scheme of the active component:

Class A.

Class B.

Class AB.

Class C.

There are also other classes, such as Class D (D stands for

digital), Class E and Class F. These all uses the transistor/FET as

a switch.

There are also other classes, such as Class D (D stands for

digital), Class E and Class F. These all uses the transistor/FET as

a switch.

Our approach in this chapter

5

Small-Signal Versus Large-Signal Operation

ZL

Vs

Zs

Sinusoidal waveform

Usually non-sinusoidal waveform

Large-signal: ( ) ( ) ( ) ( ) ...2

32

21 TOHtvatvatvatv iiio +++=

Nonlinear

Small-signal: ( ) ( )tvatv io 1≅Linear

vo(t)

vi(t)

6

Small-Signal Amplifier (SSA)

All amplifiers are inherently nonlinear.

However when the input signal is small, the input and output relationship of the amplifier is approximately linear.

This linear relationship applies also to current and power.

An amplifier that fulfills these conditions: (1) small-signal operation (2) linear, is called Small-Signal Amplifier (SSA). SSA will be our focus.

If a SSA amplifier contains BJT and FET, these components can bereplaced by their respective small-signal model, for instance the hybrid-Pi model for BJT.

( ) ( ) ( ) ( ) ( )tvaTOHtvatvatvatv iiiio 13

32

21 ... ≅+++=

When vi(t)→0 (< 2.6mV) ( ) ( )tvatv io 1≅ (1.1)

Linear relation

7

Example 1.1 - An RF Amplifier Schematic (1)

ZL

Vs

Zs AmplifierInput

Matching

Network

Output

Matching

Network

DC supply

L

L3

R=

L=100.0 nH

L

L2

R=

L=100.0 nH

R

RD1

R=100 Ohm

C

CD1

C=0.1 uF

C

CD2

C=100 pF

Port

VCC

Num=3

L

L4

R=

L=12.0 nH

C

C2

C=0.68 pF

C

Cc1

C=100.0 pF

R

RB2

R=1.5 kOhm

R

RB1

R=1 kOhm

L

LC

R=

L=100.0 nH

R

RC

R=470 Ohm

Port

Input

Num=1

Port

Output

Num=2L

L1

R=

L=4.7 nH

C

C1

C=3.3 pF

C

Cc2

C=100.0 pF

pb_phl_BFR92A_19921214

Q1

RF power flow

8

Typical RF Amplifier Characteristics

To determine the performance of an amplifier, the following characteristics are typically observed.

1. Power Gain.

2. Bandwidth (operating frequency range).

3. Noise Figure.

4. Phase response.

5. Gain compression.

6. Dynamic range.

7. Harmonic distortion.

8. Intermodulation distortion.

9. Third order intercept point (TOI).

Important to small-signal

amplifier

Important parameters of

large-signal amplifier

(Related to Linearity)

9

Power Gain

For amplifiers functioning at RF and microwave frequencies, usually of interest is the input and output power relation.

The ratio of output power over input power is called the Power Gain (G), usually expressed in dB.

There are a number of definition for power gain as we will see shortly.

Furthermore G is a function of frequency and the input signal level.

dB log10PowerInput

PowerOutput 10

=GPower Gain (1.2)

10

Why Power Gain for RF and Microwave Circuits? (1)

Power gain is preferred for high frequency amplifiers as the impedance encountered is usually low (due to presence of parasitic capacitance).

For instance if the amplifier is required to drive 50Ω load the voltage across the load may be small, although the corresponding currentmay be large (there is current gain).

For amplifiers functioning at lower frequency (such as IF frequency), it is the voltage gain that is of interest, since impedance encountered is usually higher (less parasitic).

For instance if the output of IF amplifier drives the demodulator circuits, which are usually digital systems, the impedance looking into the digital system is high and large voltage can developed across it. Thus working with voltage gain is more convenient.

Power = Voltage x Current

11

Why Power Gain for RF and Microwave Circuits? (2)

Instead on focusing on voltage or current gain, RF engineers focus on power gain.

By working with power gain, the RF designer is free from the constraint of system impedance. For instance in the simple receiver block diagram below, each block contribute some power gain. A large voltage signal can be obtained from the output of the final block by attaching a high impedance load to it’s output.

LO

IF Amp.BPF

LNABPF

RF Portion

(900 MHz)

IF Portion

(45 MHz)

RF signal

power

1 µW

15 µW

IF signalpower

75 µW

7.5 mW

400Ω

t

v(t) 4.90 V

R

VaverageP

2

2

=

12

Harmonic Distortion (1)

ZL

Vs

Zs

When the input driving signal issmall, the amplifier is linear. Harmonic components are almost non-existent.

Small-signal

operation

region

Pout

Pin

13

Harmonic Distortion (2)

ZL

Vs

Zs

When the input driving signal istoo large, the amplifier becomesnonlinear. Harmonics areintroduced at the output.

Harmonics generation reduces the gain

of the amplifier, as some of the output

power at the fundamental frequency is

shifted to higher harmonics. This result in

gain compression seen earlier!

Harmonics generation reduces the gain

of the amplifier, as some of the output

power at the fundamental frequency is

shifted to higher harmonics. This result in

gain compression seen earlier!

ff1

f

harmonics

f1 2f1 3f1 4f10

Pout

Pin

14

Power Gain, Dynamic Range and Gain Compression

Dynamic range (DR)

1dB compression

Point (Pin_1dB)

Saturation

Device

Burn

out

Ideal amplifier

1dBGain compression

occurs here

Noise Floor

-70 -60 -50 -40 -30 -20 -10 0 10

Pin

(dBm)20

Pout

(dBm)

-20

-10

0

10

20

30

-30

-40

-50

-60

Power gain Gp =

Pout(dBm) - Pin(dBm)

= -30-(-43) = 13dB

Linear RegionLinear Region

Nonlinear Region

Nonlinear Region

Pin Pout

Input and output at same frequency

15

Bandwidth

Power gain G versus frequency for small-signal amplifier.

f / Hz0

G/dB

3 dB

Bandwidth

Po dBm

Pi dBmPo dBm

Pi dBm

16

Noise Figure (F)

Vs

• The amplifier also introduces noise into the output inaddition to the noise from the environment.

• Assuming small-signal operation.

Noise Figure (F)= SNRin/SNRout

• Since SNRin is always larger

than SNRout, F > 1 for an

amplifier which contribute noise.

SNR:

Signal to Noise

Ratio

SNR:

Signal to Noise

Ratio

Smaller SNRin

Larger SNRout

Zs

ZL

17

Power Gain Definition

From the power components, 3 types of power gain can be defined.

GP, GA and GT can be expressed as the S-parameters of the amplifier and the reflection coefficients of the source and load networks. Refer to Appendix 1 for the derivation.

As

LT

As

AoA

in

Lp

P

PG

P

PG

P

PG

==

==

==

powerInput Available

load todeliveredPower Gain Transducer

powerInput Available

Power load AvailableGain Power Available

Amp. power toInput

load todeliveredPower Gain Power

The effective power gain

(2.1a)

(2.1b)

(2.1c)

18

Naming Convention

ZLVs

Zs Amplifier

2 - port Network

Γin Γout

Source

NetworkLoad

Network

ΓsΓL

2221

1211

ss

ss

In the spirit of high-

frequency circuit design,

where frequency response

of amplifier is characterized

by S-parameters and

reflection coefficient is

used extensively

instead of impedance,

power gain can be expressed

in terms of these parameters.

In the spirit of high-

frequency circuit design,

where frequency response

of amplifier is characterized

by S-parameters and

reflection coefficient is

used extensively

instead of impedance,

power gain can be expressed

in terms of these parameters.

19

TWO-PORT POWER GAIN

Figure 6.1: A two port network with general source and load impedance.

20

TWO-PORT POWER GAINPower Gain = G = PL / Pin is the ratio of power dissipated in the load ZL to

the power delivered to the input of the two-port network. This gain is

independent of Zs although some active circuits are strongly dependent on

ZS.

Available Gain = GA = Pavn / Pavs is the ratio of the power available from

the two-port network to the power available from the source. This assumes

conjugate matching in both the source and the load, and depends on ZS but

not ZL.

Transducer Power Gain = GT = PL / Pavs is the ratio of the power delivered

to the load to the power available from the source. This depends on both ZS

and ZL.

If the input and output are both conjugately matched to the two-port, then

the gain is maximized and G = GA = GT

21

TWO-PORT POWER GAIN

−+++−

−+++−

Γ+=+=

Γ+=+=

2221212221212

2121112121111

VSVSVSVSV

VSVSVSVSV

L

L

0

0

22

211211

1

1

1 ZZ

ZZ

S

SSS

V

V

in

in

L

Lin

+

−=

Γ−

Γ+==Γ

+

−

From the definition of S parameters:

[6.1a]

[6.1b]

Eliminating V2- from [6.1a]:

[6.2]

0

0

11

211222

2

2

1 ZZ

ZZ

S

SSS

V

V

out

out

S

Sout

+

−=

Γ−

Γ+==Γ

+

−

[6.3]

22

TWO-PORT POWER GAIN

in

inin ZZ

Γ−

Γ+=

1

10

( )( )inS

SSVV

ΓΓ−

Γ−=+

1

1

21

By voltage division:

( )in

inS

inS VVV

ZZ

ZVV Γ+=+=

+= +−+ 11111

Using:

Solving for V1+:

[6.4]

[6.5]

[6.6]

23

TWO-PORT POWER GAIN

( ) ( )2

2

2

0

2

22

1

0

11

1

81

2

1in

inS

SS

ininZ

VV

ZP Γ−

ΓΓ−

Γ−=Γ−= +

( )

( )22

22

222

21

0

2

2

22

22

21

0

2

1

11

11

8

1

1

2

inSL

SLS

L

L

L

S

S

Z

V

S

S

Z

VP

ΓΓ−Γ−

Γ−Γ−=

Γ−

Γ−=

+

( )2

0

2

21

2LL

Z

VP Γ−=

−

The average power delivered to the network:

The power delivered to the load is:

[6.7]

[6.8]

[6.9]

24

TWO-PORT POWER GAIN

( )( ) 2

22

2

22

21

11

1

Lin

L

in

LP

S

S

P

PGG

Γ−Γ−

Γ−===

( )2

2

0

2

1

1

8S

Ss

inavsZ

VPP

Sin Γ−

Γ−== ∗Γ=Γ

( )∗

∗

Γ=Γ

∗Γ=ΓΓΓ−Γ−

Γ−Γ−==

outL

outL

inSout

Souts

Lavn

S

S

Z

VPP

22

22

222

21

0

2

11

11

8

The power gain can be expressed as:

The available power from the source:

The available power from the network:

[6.10]

[6.11]

[6.12]

25

TWO-PORT POWER GAIN

( )22

11

22

21

0

2

11

1

8outS

Ss

avn

S

S

Z

VP

Γ−Γ−

Γ−=

( )( ) 2

11

2

22

21

11

1

Sout

S

avs

avnA

S

S

P

PG

Γ−Γ−

Γ−==

( )( )22

22

222

21

11

11

inSL

LS

avs

LT

S

S

P

PG

ΓΓ−Γ−

Γ−Γ−==

The power available from the network:

The available power gain:

The transducer power gain:

[6.13]

[6.14]

[6.15]

26

Summary of Important Power Gain Expressions

and the Gain Dependency Diagram

( )( )22

22

22

21

11

1

inL

L

P

s

sG

Γ−Γ−

Γ−=

( )( )22

11

2

21

2

11

1

outs

s

A

s

sG

Γ−Γ−

Γ−=

( ) ( )22

22

22

21

2

11

11

sinL

sL

T

s

sG

ΓΓ−Γ−

Γ−Γ−=

Note:

All GT, GP, GA, Γ1 and Γ2

depends on the S-

parameters.

(2.2a)

(2.2c)

(2.2d)

(2.2e)

Γs ΓL

Γin Γout

GA GP

GT

2221

1211

ss

ss

The Gain Dependency Diagram

0

0

22

211211

1

1

1 ZZ

ZZ

S

SSS

V

V

in

in

L

Lin

+

−=

Γ−

Γ+==Γ

+

−

0

0

11

211222

2

2

1 ZZ

ZZ

S

SSS

V

V

out

out

S

Sout

+

−=

Γ−

Γ+==Γ

+

−

(2.2b)

27

TWO-PORT POWER GAIN

2

21SGT =

A special case of the transducer power gain occurs when both input and

output are matched for zero reflection (in contrast to conjugate

matching).

Another special case is the unilateral transducer power gain, GTU where

S12=0 (or is negligibly small). This nonreciprocal characteristic is

common to many practical amplifier circuits. Γin = S11 when S12 = 0, so

the unilateral transducer gain is:

( )( )2

22

2

11

222

21

11

11

LS

LS

TU

SS

SG

Γ−Γ−

Γ−Γ−=

[6.16]

[6.17]

28

TWO-PORT POWER GAIN

Figure 6.2: The general transistor amplifier circuit.

29

TWO-PORT POWER GAIN

2

22

2

2

210

2

2

1

1

1

1

L

L

L

Sin

S

S

SG

SG

G

Γ−

Γ−=

=

ΓΓ−

Γ−=

The separate effective gain factors:

[6.18a]

[6.18b]

[6.18c]

30

TWO-PORT POWER GAINIf the transistor is unilateral, the unilateral transducer gain reduces

to GTU = GSG0GL , where:

2

22

2

2

210

2

11

2

1

1

1

1

L

L

L

S

S

S

SG

SG

SG

Γ−

Γ−=

=

Γ−

Γ−=

[6.19c]

[6.19b]

[6.19a]

31

Example 1 – Familiarization with the Gain Expressions

An RF amplifier has the following S-parameters at fo: s11=0.3<-70o, s21=3.5<85o, s12=0.2<-10o, s22=0.4<-45o. The system is shown below. Assuming reference impedance (used for measuring the S-

parameters) Zo=50Ω, find:

(a) GT, GA, GP.

(b) PL, PA, Pinc.

Amplifier

2221

1211

ss

ssZL=73Ω

40Ω

5<0o

32

STABILITY

In the circuit of Figure 6.2, oscillation is possible if either the input or

output port impedance has the negative real part; this would imply that

|Γin|>1 or |Γout|>1.

Γin and Γout depends on the source and load matching networks, the

stability of the amplifier depends on ΓS and ΓL as presented by matching

networks.

Unconditionally stable: The network is unconditionally stable if |Γin| < 1

and |Γout| < 1 for all passive source and load impedance (ex; |ΓS| < 1 and

|Γ| < 1).

Conditionally stable: The network is conditionally stable if |Γin| < 1 and

|Γout| < 1 only for a certain range of passive source and load impedance.

This case also referred as potentially unstable.

The stability condition of an amplifier circuit is usually frequency

dependent.

33

STABILITY CIRCLES

11 22

211211 <

Γ−

Γ+=Γ

L

Lin

S

SSS

11

11

2112

22<

Γ−

Γ+=Γ

S

S

out

S

SSS

The condition that must be satisfied by ΓS and ΓL if the amplifier

is to be unconditionally stable:

[6.20a]

[6.20b]

The determinant of the scattering matrix:

21122211 SSSS −=∆ [6.21]

34

STABILITY CIRCLES

( )

22

22

2112

22

22

1122

∆−=

∆−

∆−=

∗∗

S

SSR

S

SSC

L

L

( )

22

11

2112

22

11

2211

∆−=

∆−

∆−=

∗∗

S

SSR

S

SSC

S

S

The output stability circles:

The input stability circles:

[6.22a]

[6.22b]

[6.23a]

[6.23b]

35

STABILITY CIRCLES

Figure 6.3: Output stability circles for conditionally stable device.

(a) |S11| < 1 (b) |S11| > 1

36

STABILITY CIRCLESIf the device is unconditionally stable, the stability circles must be

completely outside (or totally enclose) the Smith chart.

11

11

22

11

<→→>−

<→→>−

SRC

SRC

SS

LL[6.24a]

[6.24b]

37

STABILITY TEST

12

1

2112

22

22

2

11>

∆+−−=

SS

SSK

121122211 <−=∆ SSSS

Rollet’s condition:

the auxiliary condition:

11

21121122

2

11>

+∆−

−=

∗SSSS

Sµ

the μ test:

[6.25]

[6.26]

[6.27]

38

Example 2

The S parameters for the HP HFET-102 GaAs FET at 2 GHz with a

bias voltage of Vgs = 0 are given as follow (Z0 = 50 Ohm):

S11 = 0.894 < -60.6

S21 = 3.122 < 123.6

S12 = 0.020 < 62.4

S22 = 0.781 < -27.6

Determine the stability of this transistor using the K-∆ test and the μ

test, and plot the stability circles on the Smith Chart

39

SINGLE STAGE TRANSISTOR AMPLIFIER

DESIGN

∗

∗

Γ=Γ

Γ=Γ

Lout

Sin

2

22

2

2

2121

1

1

1max

L

L

S

T

SSG

Γ−

Γ−

Γ−=

Maximum power transfer from the input matching network to the

transistor and the maximum power transfer from the transistor to the

output matching network will occur when:

Then, assuming lossless matching sections, these conditions will

maximize the overall transducer gain:

[6.28a]

[6.28b]

[6.29]

40

SINGLE STAGE TRANSISTOR

AMPLIFIER DESIGN

In the general case with a bilateral transistor, Γin is affected by

Γout, and vice versa, so that the input and output sections must be

matched simultaneously.

S

SL

L

LS

S

SSS

S

SSS

Γ−

Γ+=Γ

Γ−

Γ+=Γ

∗

∗

11

211222

22

211211

1

1

[6.30a]

[6.30b]

41

SINGLE STAGE TRANSISTOR

AMPLIFIER DESIGN

2

2

2

2

22

1

2

1

2

11

2

4

2

4

C

CBB

C

CBB

L

S

−±=Γ

−±=Γ

The solution is:

[6.31a]

[6.31b]

42

SINGLE STAGE TRANSISTOR

AMPLIFIER DESIGN

∗

∗

∆−=

∆−=

∆−−+=

∆−−+=

11222

22111

22

11

2

222

22

22

2

111

1

1

SSC

SSC

SSB

SSB

The variables are defined as:

[6.32a]

[6.32b]

[6.32c]

[6.32d]

43

SINGLE STAGE TRANSISTOR

AMPLIFIER DESIGN

2

22

2

212

11 1

1

1

1max

SS

SGTU

−−=

( )12

12

21

max−−= KK

S

SGT

12

21

S

SGmsg =

When S12 = 0, it shows that ΓS = S11* and ΓL = S22*, and the maximum

transducer gain for unilateral case:

The maximum stable gain with K = 1:

[6.33]

[6.34]

[6.35]

When the transistor is unconditionally stable, K > 1, and the max

transducer power gain can be simply re-written as:

44

Example 3

Design an amplifier for a maximum gain at 4.0 GHz. Calculate the

overall transducer gain, GT, and the maximum overall transducer gain

GTmax. The S parameters for the GaAs FET at 4 GHz given as follow

(Z0 = 50 Ohm):

S11 = 0.72 < -116o

S21 = 2.60 < 76o

S12 = 0.03 < 57o

S22 = 0.73 < -68o

45

UNILATERAL FORM

22 )1(

1

)1(

1

UG

G

UTU

T

−<<

+

• In many practical cases |S12| is small enough to be ignored, the device

then can be assumed to be unilateral, which greatly simplifies design

procedure

• Error in the transducer gain caused by approximating |S12| as zero is

given by the ratio GT/GTU, and be bounded by:

Where U is defined as the unilateral figure of merit

)1)(1(2

22

2

11

22112112

SS

SSSSU

−−=

46

Example 4

An FET is biased for minimum noise figure, and has the following S

parameters at 4 GHz:

S11 = 0.60 < -60o

S21 = 1.90 < 81o

S12 = 0.05 < 26o

S22 = 0.50 < -60o

For design purposes, assume the device is unilateral and calculate the

max error in GT resulting from this assumption.

47

CONSTANT GAIN CIRCLES

• In many cases it is desirable to design for less than the max obtainable gain, to improve bandwidth or to obtain a specific value for an amplifier gain.

• Mismatches are purposely introduced to reduce the overall gain• Procedure is facilitated by plotting constant gain circles on the Smith Chart

• Represents loci of ΓS and ΓL, that give fixed values of GS and GL.

• To simplify the discussion, we will only treat the case of a unilateral device

48

CONSTANT GAIN CIRCLES

2

11

2

1

1

S

S

S

SG

Γ−

Γ−=

2

22

2

1

1

L

L

L

SG

Γ−

Γ−=

2

111

1max

SG

S

−=

The expression for the GS and GL for the unilateral case is given by:

These gains are maximized when ΓS = S11* and ΓL = S22* :

2

221

1max

SG

L

−=

49

CONSTANT GAIN CIRCLES

)1(1

1 2

112

11

2

max

SSG

Gg

S

S

S

S

S−

Γ−

Γ−==

Now we define normalized gain factors gS and gL as;

Thus we have that: 0 ≤ gS ≤ 1, and 0 ≤ gL ≤ 1. A fixed value of gS and gL

represents circles in the ΓS and ΓL planes.

)1(1

1 2

222

22

2

max

SSG

Gg

L

L

L

L

L−

Γ−

Γ−==

50

CONSTANT GAIN CIRCLES

( )

( )( ) 2

11

2

11

2

11

11

11

11

11

Sg

SgR

Sg

SgC

S

S

S

S

SS

−−

−−=

−−=

∗

( )

( )( )

2

22

2

22

2

22

22

11

11

11

Sg

SgR

Sg

SgC

L

L

L

L

LL

−−

−−=

−−=

∗

Input constant gain circles:

Output constant gain circles:

[6.37a]

[6.37b]

[6.38a]

[6.38b]

51

Example 5Design an amplifier to have a gain of 11 dB at 4 GHz. Plot constant gain circles for GS = 2 dB and 3 dB; and GL = 0 dB and 1 dB. The FET has the following S parameters (Z0 = 50 Ω):

S11 = 0.75 < -120o

S21 = 2.50 < 80o

S12 = 0.00 < 0o

S22 = 0.60 < -85o

52

LOW NOISE AMPLIFIER DESIGNIn receiver applications especially, it is often required to have a preamplifier with as low a noise figure as possible since, the first stage of a receiver front end has the dominant effect on the noise performance of the overall system.

Generally it is not possible to obtain both minimum noise figure and maximum gain for an amplifier, so some sort of compromise must be made. This can be done by using constant gain circles and circles of constant noise figure to select a usable trade of between noise figure and gain.

2

min optS

S

N YYG

RFF −+= [6.39]

53

LOW NOISE AMPLIFIER DESIGN

2

0

min 14

opt

N ZR

FFN Γ−

−=

( )1

1

1

2

+

Γ−+=

+

Γ=

N

NNR

NC

opt

F

opt

F

For a fixed noise figure, F, the noise figure parameter, N, is given as:

The circles of constant noise figure:

[6.40]

[6.41a]

[6.41b]

54

Example 6An GaAs FET amplifier is biased for minimum noise figure and has

the following S-parameters (Z0 = 50 Ω):

S11 = 0.75 < -120

S21 = 2.50 < 80

S12 = 0.00 < 0

S22 = 0.60 < -85

Γopt = 0.62 < 100

Fmin = 1.6 dB

RN = 20 Ω

For design purposes, assume the unilateral. Then design an amplifier

having 2.0 dB noise figure with the max gain that is compatible with

this noise figure.