RF Amplifier DesignRF Electronics Spring, 2018

Robert R. Krchnavek Rowan University

Objectives

• Be able to bias an RF amplifier

• Understand the meaning of various parameters used to describe RF amplifiers

• Understand the derivation of various amplifier gains and be able to use them to design an appropriate RF amplifier.

• Be able to determine stability regions for an RF amplifier.

Generic Transistor Amplifier

IC

VCE

IB1

Amplifier Classes

Amplifier Efficiency

See calculation on P. 460.

BJT Biasing Networks

BJT Biasing Networks

BJT Biasing Networks

FET Biasing Networks

A MESFET usually requires: VG < 0 and VD > 0.

FET Biasing Networks

Biasing the S-terminal can eliminate the need for a supply <0.

• Gain and gain flatness (in dB)

• Operating frequency and bandwidth (in Hz)

• Output power (in dBm)

• Power supply requirements (in V and A)

• Input and output reflection coefficients (VSWR)

• Noise figure (in dB)

RF Amplifiers

RF Amplifiers - Power RelationsCombining the matching networks into the source and load:

ZS

ZL

Z0

VS

Further simplifying:

RF Amplifiers - Power Relations

What is the input power to the amplifier?

Vin = V+ + V

− = V+ (1 + Γin) = VS

Zin

Zin + ZS

Γin =Zin − Z0

Zin + Z0

= Γ0 exp (−ȷ2βl)

Zin = Z0

1 + Γin

1 − Γin

ΓS =ZS − Z0

ZS + Z0 ZS = Z0

1 + ΓS

1 − ΓS

RF Amplifiers - Power Relations

V+ = VS

1 − ΓS

2 (1 − ΓinΓS)

Pin =1

2ℜ (VinI

∗

in)

Pin = P+

+ P−

=1

2ℜ

!

"

V+

+ V−

# "

I+

+ I−

#

∗

$

Pin = P+ + P

− =1

2ℜ

!

"

V+ (1 + Γin)

#

$

V +

Z0

−V −

Z0

%

∗&

Pin = P+

+ P−

=1

8

|VS |2

Z0

|1 − ΓS |2

|1 − ΓinΓS |2!

1 − |Γin|2"

RF Amplifiers - Power RelationsAn approach based on normalized power flow.

ZS

ZL

Z0

VS

a1aS

ΓSb1

a1 = aS + ΓSb1

b1 = a1Γin

a1 = aS + a1ΓinΓS

a1 =aS

1 − ΓinΓS

RF Amplifiers - Power Relations

Pin = Pinc

!

1 − |Γin|2"

=|a1|2

2

!

1 − |Γin|2"

aS =1

√

Z0

VSZ0

Z0 + ZS

=

√

Z0

Z0 + ZS

VS

Pin = Pinc

!

1 − |Γin|2"

=| aS

1−ΓinΓS|2

2

!

1 − |Γin|2"

Pin =1

2

|√

Z0

Z0+ZSVS |2

|1 − ΓinΓS |2!

1 − |Γin|2"

RF Amplifiers - Power RelationsDo the two approaches (voltage vs normalized power) yield the same result?

Pin =1

2

|√

Z0

Z0+ZSVS |2

|1 − ΓinΓS |2!

1 − |Γin|2"

Pin =1

8

|VS |2

Z0

|1 − ΓS |2

|1 − ΓinΓS |2!

1 − |Γin|2"

versus

RF Amplifiers - Power Relations Available Power - PA

The available power, PA, is the input power under conditions of maximum

transfer of power (Γin = ΓS*).

PA = Pin|Γin=Γ∗

S

=1

2

|√

Z0

Z0+ZSVS |2

|1 − |Γin|2|2!

1 − |Γin|2"

=1

2

|√

Z0

Z0+ZSVS |2

1 − |Γin|2=

1

2

|√

Z0

Z0+ZSVS |2

1 − |ΓS |2

PA = Pin|Γin=Γ∗

S

=1

8

|VS |2

Z0

|1 − ΓS |2

|1 − |Γin|2|2!

1 − |Γin|2"

=1

8

|VS |2

Z0

|1 − ΓS |2

1 − |Γin|2=

1

8

|VS |2

Z0

|1 − ΓS |2

1 − |ΓS |2

For the voltage analysis:

Similarly, for the normalized power analysis:

Note: The maximum available power is a function of ΓS.

RF Amplifiers - Power Relations Transducer Power Gain - GT

GT =power delivered to the load

available power from the source=

PL

PA

ZS

ZL

Z0

VS

RF Amplifiers - Power Relations Transducer Power Gain - GT

PL =1

2

|V +L|2

Z0

!

1 − |ΓL|2"

where VL+ is the incident voltage at the load. The derivation of this

expression is identical to the input power expression.

What is VL+?

V+S21 + V

+L

ΓLΓout = V+L

through the amp

reflected at the load and at the amp output

V+L

=V +S21

1 − ΓLΓout

=V +S21

1 − ΓLS22

RF Amplifiers - Power Relations Transducer Power Gain - GT

V+ = VS

1 − ΓS

2 (1 − ΓinΓS)and, from previously

V+L

=VS

1−ΓS

2(1−ΓinΓS)S21

1 − ΓLS22

and so on .....

RF Amplifiers - Power Relations Transducer Power Gain - GT

Using normalized power flow:

a1 = aS + b1ΓS

aS = a1 − b1ΓS

b′

1 = a1S11

b′′

1 = a2S12

b1 = b′

1 + b′′

1

aS = a1 − ΓS (a1S11 + a2S12)

ZS

ZLVS

[S]

aS

a1

a2b1

b2

b1'

b1"

ΓSΓin Γout ΓLPA =

1

2

|aS |2

1 − |ΓS |2

PL =1

2|b2|

2!

1 − |ΓL|2"

RF Amplifiers - Power Relations Transducer Power Gain - GT

a2 = b2ΓL

b2 = a1S21 + a2S22

b2 = a1S21 + b2ΓLS22

b2 =a1S21

1 − ΓLS22

a2 =a1S21ΓL

1 − ΓLS22

ZS

ZLVS

[S]

aS

a1

a2b1

b2

b1'

b1"

ΓSΓin Γout ΓL

RF Amplifiers - Power Relations Transducer Power Gain - GT

aS = a1 − ΓS (a1S11 + a2S12)

aS = a1 − ΓS

!

a1S11 +a1S21ΓL

1 − ΓLS22

S12

"

aS = a1

!

1 − ΓS

"

S11 +S21S12ΓL

1 − ΓLS22

#$

RF Amplifiers - Power Relations Transducer Power Gain - GT

GT =PL

PA

=|b2|2

|aS |2!

1 − |ΓL|2" !

1 − |ΓS |2"

GT =

!

!

!

a1S21

1−ΓLS22

!

!

!

2

!

!

!a1

"

1 − ΓS

#

S11 + S21S12ΓL

1−ΓLS22

$%!

!

!

2

&

1 − |ΓL|2' &

1 − |ΓS |2'

GT =

!

1 − |ΓL|2"

|S21|2!

1 − |ΓS |2"

|(1 − S11ΓS) (1 − S22ΓL) − S21S12ΓLΓS |2

RF Amplifiers - Power Relations Defining Reflection Coefficients - Γin , Γout

Γin = S11 +S21S12ΓL

1 − S22ΓL

Γout = S22 +S12S21ΓS

1 − S11ΓS

Γin =b1

a1

=b′1 + b′′1

a1

=a1S11 + S12

a1S21ΓL

1−ΓLS22

a1

Similarly,

RF Amplifiers - Power Relations Transducer Power Gain - GT

Two additional forms of GT using the previously defined reflection

coefficients:

GT =

!

1 − |ΓL|2"

|S21|2!

1 − |ΓS |2"

|1 − ΓSΓin|2 |1 − S22ΓL|

2

GT =

!

1 − |ΓL|2"

|S21|2!

1 − |ΓS |2"

|1 − ΓLΓout|2 |1 − S11ΓS |

2

RF Amplifiers - Power Relations Unilateral Power Gain - GTU

The unilateral power gain, GTU, assumes S12 = 0.

The unilateral power gain is often used to approximate the transducer power gain. It simplifies the design work.

GTU =

!

1 − |ΓL|2"

|S21|2!

1 − |ΓS |2"

|(1 − S11ΓS) (1 − S22ΓL)|2

RF Amplifiers - Power Relations Available Power Gain - GA

The available power gain, GA, is the transducer power gain under conditions of

load side matching (Γout = ΓL*).

GA = GT |Γout=Γ∗

L

=power available from the amplifier

power available from the source

GA =|S21|

2!

1 − |ΓS |2"

#

1 − |Γout|2$

|1 − S11ΓS |2

RF Amplifiers - Power Relations Operating Power Gain - G

The power gain, or operating power gain, G, is the ratio of the power delivered to the load to the power supplied to the amplifier.

G =power delivered to the load

power supplied to the amplifier

G =

PL

Pin

=

PL

PA

PA

Pin

= GT

PA

Pin

G =

!

1 − |ΓL|2"

|S21|2

#

1 − |Γin|2$

|1 − S22ΓL|2

StabilityTo maintain stability, we must eliminate positive feedback. We can guarantee stability by requiring the magnitude of the reflection coefficients be less than 1.

|ΓL| < 1

|ΓS | < 1

|Γin| =

!

!

!

!

S11 − ΓL∆

1 − S22ΓL

!

!

!

!

< 1

|Γout| =

!

!

!

!

S22 − ΓS∆

1 − S11ΓS

!

!

!

!

< 1

∆ = S11S22 − S12S21where

Stability Circles

|Γin| =

!

!

!

!

S11 − ΓL∆

1 − S22ΓL

!

!

!

!

< 1

S11 = SR11 + ȷSI

11

S22 = SR22 + ȷSI

22

∆ = ∆R + ȷ∆I

ΓL = ΓRL + ȷΓ

IL

Note: At a given frequency, the S parameters are fixed. Therefore, to maintain stability in a design, only the reflection coefficients, and can be varied.�L �S

The reflection coefficient on the output, , affects .

�L�in

Stability Circles

Stability Circles

Unconditional Stability

Design for Constant Gain Unilateral Design (S12 ≈ 0)

GTU =1 − |ΓS |2

|1 − S11ΓS |2× |S21|

2 ×1 − |ΓL|2

|1 − S22ΓL|2

= GS × G0 × GL

Design for Low Noise

Design for Constant VSWR

Broadband, High-Power, and Multistage Amplifiers