The figure shows a linear circuit, consisting of a perfect a.c. voltage generator (zero internal impedance) having r.m.s. voltage V, connected through a resistance r and a reactance jx, representing a series internal impedance z = r + jx, to a load impedance which for the purposes of this web page we call Z = R + jX.
The question we ask is, if we hold the voltage of the generator constant, then as R and X (of the load) are varied, at what values of R and X is the maximum power dissipated in the resistance R?
The answer to this problem is that jX = -jx and R = r. Such a situation is called a conjugate match and it may be expressed alternatively by the formula
Now, if we consider a transmission line of characteristic impedance Zo, then there will be reflection coefficients gamma and GAMMA respectively for reflection coefficients from the impedances z and Z attached to the transmission line.
It is easy to see from the relation between reflection coefficient and load impedance, that the condition
is entirely equivalent to the condition
We next interpose a 2-port network having scattering matrix S, between the source z and the load Z.
We would like the source impedance z to be a conjugate match to the input impedance of the combination of 2-port and load Z.
So, let us consider a two-port linear network having the output connected to a load impedance Z with reflection coefficient GAMMA. This is entirely equivalent to a one-port network at the input terminals, port one, and we may ask, what is the reflection coefficient at the input to port one?
The situation is shown in the next figure.
Some maths. Here, the terms s11 s12 s21 s22 are the s-parameters of the two-port, a1 is the input wave complex amplitude looking into port 1, a2 is the input wave complex amplitude looking into port 2, and b1 and b2 are the outgoing wave complex amplitudes from ports 1 and 2 respectively.
The output wave from port 2 (b2) is reflected by the load impedance Z (reflection coefficient GAMMA) to form the input wave to port 2 (a2).
The reflection coefficient looking in to port 1 is therefore just b1/a1, given the load impedance Z. In general (if s12 is not zero) this input reflection coefficient will change if we alter the value of Z.
b1 = s11 a1 + s12 a2
b2 = s21 a1 + s22 a2
a2 = GAMMA b2
so
b2 = s21 a1 + s22 GAMMA b2
rearranging
b2 = (s21 a1)/(1 - s22 GAMMA)
so
a2 = a1 (s21 GAMMA)/(1 - s22 GAMMA)
so using the first equation
b1 = a1 [s11 + (s12 s21 GAMMA)/(1 - s22 GAMMA)]
and so everything above inside the square brackets [] is the input
reflection coefficient of the one-port.
As in our first example, we can choose a source impedance
having reflection coefficient gamma = []* as a conjugate match.
We observe that the reflection coefficient [] involves all four
scattering parameters, and also the output load reflection
coefficient, GAMMA. For purposes of calculation it may
be rewritten as
[] = [s11 + (s12 s21)/(1/GAMMA - s22)]
= [(s11/GAMMA - DELTA)/(1/GAMMA - s22)]
= [(s11 AMMAG - DELTA)/(AMMAG - s22)]
where we have written AMMAG = (1/GAMMA)
and where
DELTA = (s11 s22 - s12 s21)
DELTA is the determinant of the S matrix of the 2-port.
Thus we require
gamma = ((s11 AMMAG - DELTA)/(AMMAG - s22))*
or
ammag* = 1/gamma* = (AMMAG - s22)/(s11 AMMAG - DELTA)
Similarly, we might ask, for a given source impedance having reflection coefficient gamma, feeding port 1, what is the reflection coefficient looking into the ouput port (port 2)?
More maths
b1 = s11 a1 + s12 a2
b2 = s12 a1 + s22 a2
a1 = gamma b1
so
b1 = s11 gamma b1 + s12 a2
or
b1 = [s12 a2]/[1 - s11 gamma]
and
a1 = gamma b1
= a2 [gamma s12]/[1 - s11 gamma]
so
b2 = a2 {s22 + (s12 s21 gamma)/(1 - s11 gamma)}
and thus the reflection coefficient looking into the output
port is just everything inside the {}.
Just as before, we can rewrite the function {} thus
{} = ((s22/gamma - DELTA)/(1/gamma - s11))
= {(s22 ammag - DELTA)/(ammag - s11)}
As in our first example, we can choose a load impedance having
reflection coefficient GAMMA to be a conjugate match at port 2
GAMMA = {}*
= ((s22 ammag - DELTA)/(ammag - s11))*
As before, the {} term involves all four s parameters and also the input source internal impedance's reflection coefficient, gamma.
We now ask for a simultaneous conjugate match. We wish to choose, for a given S matrix, source and load impedances such that the entire system is conjugately matched on both ports. The algebra is rather complicated, but it is done formally in the book by David M Pozar, called "Microwave and RF design of wireless systems", John Wiley, 0-471-32282-2, published 2001, on pages 205-207 (section 6.4).
GAMMA = ((s22 ammag - DELTA)/(ammag - s11))*
and also we had (above)
ammag* = (AMMAG - s22)/(S11 AMMAG - DELTA)
we substitute ammag*
into GAMMA = (s22* ammag* - DELTA*)/(ammag* - s11*)
or into AMMAG = (ammag* - s11*)/(s22 ammag* - DELTA*)
and solve the resulting quadratic for AMMAG and therefore
GAMMA (the reflection coefficient of the load attached to port 2),
from which we
can extract gamma (the reflection coefficient looking into the
source impedance of the generator on port 1).
For practical matches we require the modulus of GAMMA to be less than unity, and the modulus of gamma to be less than unity also.
The interested reader is referred to the citation above for details.
There is a Javascript conjugate match calculator which may be accessed via a page which is pointing to a local student project website. This still needs minor work but is returning, we think, correct values with the nomenclature developed above. Try it, feedback is welcome to David Jefferies dj@eryptick.net
Copyright © D.Jefferies 2001.