Transcript 6 - web page for staff


Power and power-gains are two main considerations in the
design of a microwave transistor amplifier. To derive power
and power-gains using traveling waves concept, we need to
determine the reflection coefficients in the form of traveling
waves and S parameters.
GS
ZS
-
ZL
Transistor
b1
GL
a2
a1
+
ES
GOUT
GIN
b2

the source and the load reflection coefficient in a
Z0 system are
Z S  Z0
GS 
Z S  Z0
Z L  Z0
GL 
Z L  Z0

For the transistor, the input and output traveling
waves measured in a Z0 system are related by
b1  S11a1  S12 a2
b2  S21a1  S22 a2 .


The concepts of a reflection coefficient and
traveling waves can be used even if there are
no transmission lines at port 1 and port 2.
We can show that the input reflection
coefficient
G IN
b1
S12 S21G L

 S11 
a1
1  S22 G L

For the output reflection coefficient:
S12 S21G S
b2
GOUT 
 S22 
a2
1  S11G S
PIN 
1 2 1 2 1 2
2
a1  b1  a1 (1  G IN )
2
2
2
and can be shown in terms of S parameters as
Ig
1
PIN  bS
2
+
ZS
+
ES
-
Vg
-
Input
port
1  G IN
2
2
1  G S G IN
2
where
a1 
Vg
Z0
,b1 
Vg
Z0
, and bS 
ES Z0
Z S  Z0
.

The power available from the source is equal to the
input power when GIN = GS* and can be expressed as
1 2
bS
PAVS  PIN G G  2
2
IN
S
1  GS

We can also express PIN in the form
2
PIN  PAVS
or
2
(1  G S )(1  G IN )
1  G S G IN
PIN = PAVSMS
2
where MS is the source mismatch factor which is equal to
2
2
(1  G S )(1  G IN )
1  G S G IN
2
.

The power delivered to the load
ZL is
1 2 1
1 2
2
2
PL  b2  a2  b2 (1  G L )
2
2
2
and can be shown in terms of S parameters as
1
PIN  bTH
2
IL
+
ZOUT
ETH
+
-
VL
-
ZL
2
1  GL
2
1  GOUT G L
2
where
ETH Z0
VL
VL
and
a2 
, b2 
b

.
,
TH
ZOUT  Z0
Z0
Z0

The power available from the network PAVN is equal
to the power delivered to the load when GL = GOUT*
and can be expressed as
1
2
bTH
PAVN  PL G G  2
2
L
OUT
1  GOUT

We can also express PL in the form
2
2
PL  PAVN
or
(1  G L )(1  GOUT )
1  GOUT G L
PL = PAVNML
2
where ML is the load mismatch factor which is equal to
2
2
(1  G L )(1  GOUT )
1  GOUT G L
2
.

The power gain GP is given by
2
PL
1
2 1  GL
GP 

S21
2
PIN (1  G IN )
1  S22 G L

2
The transducer power gain GT is given by
PIN
PL
PL PIN
GT 

 GP
 GP M S
PAVS PIN PAVS
PAVS

1  GS
2
1  G S G IN
2
S21
2
1  GL
2
1  S22 G L
2
Manipulating the denominator, GT can be also written in the form
GT 
1  GS
2
1  S11G S
S21
2
1  GL
2
2
1  GOUT G L
2
The unilateral power gain GTU is an often employed
approximation for the transducer power gain. GTU which neglects
the feedback effect of the amplifier (S12 = 0) can be expressed as

GTU 
1  GS
2
1  S11G S
2
S21
2
1  GL
2
1  S22 G L
2
The available power gain GA can be expressed in the
form
P
P P
G
GA  AVN  L AVN  T
PAVS PAVS PL
ML


1  GS
2
1  S11G S
2
S21
2
1
1  GOUT
2
.
S11  0.6 160
S21  2.5 30
S12  0.045 16
S22  0.5 90
,the input side of the amplifier is connected to a voltage source
with E1 = 10 V and source impedance ZS = 50 . The output is
connected to a load which also has an impedance ZL = 50 ,
given GS  0.5120
GL  0.4 90
find the following quantities:
a) transducer power gain GT, unilateral transducer gain GTU,
available power gain GA, and operating power gain GP.
b) power delivered to the load PL, power available from the
source PAVS, and input power PIN.


In a two-port network, oscillations are possible when
either input or output port represents a negative
resistance. This occurs when G IN  1 or GOUT  1.
For a unilateral device (S12 = 0), the oscillations
occur when S11  1 or S22  1 .
GS
ZS
-
ZL
Transistor
b1
GL
a2
a1
+
ES
GOUT
GIN
b2
The two-port network is unconditionally stable if the
real parts of ZIN and ZOUT are greater than zero for all
passive load and source impedances.
(1)
G 1

S
(2)
(3)
(4)
GL  1
S12 S21G L
G IN  S11 
1
1  S22G L
S12 S21G S
GOUT  S22 
1
1  S11G S
Note: all coefficients are normalized to the same characteristic
impedance Z0.

This happens when some passive source and load
terminations (some but not all values of GS and GL)
produce input and output impedances having a
negative real part.

The graphical analysis is useful in the analysis of
potentially unstable transistors. First, the regions
where values of GS and GL produce G IN  1 and GOUT  1
are determined, respectively. The solutions for GS and
GL lie on circles (called stability circles) whose
equations are given by
GL


 
S22  S11
GS


 
S11  S22
and

2
S22  
2

2
S11  
2


S12 S21
2
S22  
2
S12 S21
2
S11  
2
.

The radii and centers of the circles where G IN  1
and GOUT  1 in the GL plane and GS plane,
respectively, are obtained, namely
GL values for G IN  1 (Output Stability Circle):
radius
rL 
center
CL
S12 S21
2
S22  


2

S22  S11

2
2
S22  


GS values for G OUT  1 (Input Stability Circle):
radius
center
rS 
CS
S12 S21


where  = S11S22-S12S21.
2
S11  
2

S11  S22

2
2
S11  

GOUT  1
G IN  1
rL
CL
CL
(a) GL plane
rS
CS
CS
(b) GS plane

The region where values of GL (where G L  1 )
produce G IN  1 are the stable regions.
G IN  1
G IN  1
G IN  1
G IN  1
rL
rL
CL
CL
CL
CL
G IN  1
G IN  1
S11  1
S11  1

The region where values of GS (where GS  1 )
produce GOUT  1 are the stable regions.
GOUT  1
GOUT  1
rS
GOUT  1
rS
GOUT  1
CS
CS
CS
CS
GOUT  1
GOUT  1
S22  1
S22  1
K>1
where
2
and
or
and
2
1  S11  S22  
K
2 S12 S21
2
.
 1
K>1
2
2
2
B1  1  S11  S22    0

From practical point of view, most microwave
transistors produced by manufacturers are either
unconditionally stable or potentially unstable with
0< K < 1 and   1.
rL
rS
CL
CL
CS
CS
S11  1
S22  1
Conditions for unconditionally stability for GL plane and Gs plane.
f (GHz)
S11
0.5
0.761 151
1
0.770 166
2
4
S12
S21
S22
11.84 102
0.429 35
0.029 35
6.11 89
0.365 34
0.760 174
0.040 44
3.06 74
0.364 43
0.756 179
0.064 48
1.53 53
0.423 66
0.025 31
Determine the stability. If the transistor is potentially
unstable at a given frequency, draw the input and
output stability circles.

Even when the selection of GS and GL produces G IN  1
or GOUT  1 , the circuit can be made stable if
and

Re(ZS +ZIN) > 0
Re(ZL+ZOUT) > 0
A potentially unstable transistor can be made
unconditionally stable by either resistively loading
the transistor or by adding negative feedback.
These techniques are not recommended in
narrowband amplifiers.