IMPEDANCE MATCHING
We can define two types of impedance matching.
Conjugate Matching: The matching of load impedance, to a generator for maximum transfer of power.
-
Maximum power is delivered to a load when Z L  Z *g  Rg  jX g . For this condition, the power
absorbed by the load is exactly the available power of the generator PA which is PA 
-
Vg2
4 Rg
, Vg
is in RMS.
If the load impedance is not adjustable, a matching network may be placed between the
generator and the fixed load. In this case, Zin  Rg  jX g and the power into the network is
equal to PA . If the matching network is lossless, all the available generator power is delivered to
the load Z L .
-


When Z g is real Z g  Rg , the conjugate match condition reduces to Z
in
 Zg .
Z 0 Matching: In this case, Z L  Z0 and  L  0 , SWR=1.
If Z L  Z0 , a matching network may be used to eliminate the standing waves on the line.
We require Zin  Z 0 while Zin  Z 0
For a well-designed source, Z g  Z 0 , with Z 0 real, matching the load to the line ( Z L  Z 0 )
results in a conjugate match between the generator and the load. Namely, Z g  Z0  Z L .
If Z g  Z 0 , matching network can be placed at the generator end to match the generator to the
line. This and the load matching network are shown below:
The equivalent circuit is:
For properly designed networks, Z L  Z 0 and Z g  Z 0 , where Z L is the impedance to the
right of plane B and Z g is the Thevenin impedance to the left of plane A. When the matching
network is lossless, the power available from the Thevenin equivalent must be the same as the
available generator power. Then Vg  Vg Z 0 Rg .
If the generator matching network is not built into the source, it should be placed as close as
possible to its output terminals. Similarly, it is good engineering practice to place the load
matching network as close as possible to the load.
GENERATOR AND LOAD MISMATCHES
POWER FLOW ALONG TERMINATED TL’s
-
The time-averaged power flow into an impedance Z is given by:
*
1
1  V  V
 1 
Pav  Re VI *   Re  V *  
Re  * 
2
2  Z  2
Z 
2
-
-
This equation also applies to the average power flow at any point along a TL, the direction of
flow being from the generator toward the load.
V
I
1   
Z0
V  V 1   

So, Pav 
  z
0

V V e

1
1 
V *
Re VI *   Re V  1    * 1  * 
2
2 
Z0

 2
1
1 V
2
*
Pav  Re VI   Re
1      *
*
2
2  Z0







If [   a  jb, *  a  jb,   *  a  jb  a  jb  j 2b ]
So,
Pav 
V
2
1    Re  1 
2
 Z* 
 0
2
For low-loss TL’s, Z 0
Pav 
V
2
2Z 0
1     P
2

 P  ....................*

V0
V0
e 2 z
2

1V

= Time-average power in the forward traveling wave.
P 
2 Z0
P    P  = Time-average power in the reflected wave.
2
For Z 0 real, the net power flow at any point on the line is simply the difference between the
power in the forward wave ( P  ) and that in the reflected wave P  .
Equation (*) can be applied to any point along the line. For example,




At the input  Pin  Pin 1   in and PL  PL 1   L  At the load, where
2
in   L e2   2
2
L 
Z L  Z0
Z L  Z0
If the line is lossless, Pin  PL  P  .
In order to calculate the net power flow at any point on the TL we must first determine P  at
that point. With real Z 0 , V  and I  are in phase and
2
2
V *
V0 2 z
1
1
 *

P  Re V I   Re(V
)
e
2
2
Z0
2Z 0

Determination of V0 in terms of input and load variables:
From the above figure:
Vg  Vin  I in Z g  (V0  inV0 ) 
where  g 
Z g  Z0
Z g  Z0
Z0
Z0
Vg Z 0
Z g  Z 0  in  Z g  Z 0 

Z
g
 Z 0 1   g in 
. Since, in   Le2
 Z  Z  1    e 
Given V , Z , Z  and V ,  ,  , V
2
g
g
0
g
g
L
in
0

0 may
V0
Z 0Vg
Vg Z 0
V0 
(V0  inV0 )
Z 0 1  in   Z g 1  in 
Vg 
V0 
Zg
be determined.