The scattering matrix for a two-port is a very important quantity for measurements at high frequencies. It relates the voltages of a two-port. The voltages are decomposed into waves traveling to the ports and the waves traveling from the ports. When the characteristic impedances of the transmission lines at the two-ports are the same, the S-matrix is defined
by
V1− V2−
=
S11 S12 S21 S22
V1+ V2+
= [S]
V1+ V2+
(3.37) where superindex + indicates that the wave travels towards the two-port and − that it travels from the two-port. Then S11 and S22 are the reflection coefficients at port 1 and 2, respectively, and the element S21 is the transmission coefficients from port 1 to port 2, and vice versa for S12.
The transmitted power from an incident wave at port one to port two and the reflected power at port one are given by
Pt=|S21|2Pi
Pr=|S11|2Pi (3.38)
3.7.1 S-matrix when the characteristic impedance is not the same
Now assume a two-port where the transmission line connected to port 1 has characteristic impedance Z1 and the transmission line that is connected to port 2 has characteristic impedance Z2. We like to keep the expressions for the transmitted and reflected powers in (3.38) and for this reason we need to alter the definition of S21 and S12
S21= V2+ V1+
rZ1 Z2
S12= V1+ V2+
rZ2 Z1
(3.39)
The scattering matrix S 53
The definitions of S11and S22are the same as in (3.37) and the expressions for the powers in (3.38) still hold.
3.7.2 Relation between S and Z
There is a relation between the scattering matrix and the impedance matrix. We let the two transmission lines that are connected to the two ports have the same characteristic impedance Z0. We let [U ] denote the 2 by 2 unit matrix and utilize that
V = V++ V−= ([U ] + [S])V+ I = I++ I−= 1
Z0(V+− V−) = 1
Z0([U ]− [S])V+ Since V = [Z]I we get
1 Z0
[Z]([U ]− [S]) = [U] + [S]
and
[Z] = Z0([U ] + [S])([U ]− [S])−1 [S] = (Z0[U ] + [Z])−1([Z]− Z0[U ])
The impedance matrix for a reciprocal two-port is symmetric and that means that also the corresponding scattering matrix is symmetric. A lossless two-port satisfies
Re{VtI∗} = 1 Z0
Re{(V++ V−)t(V+− V−)∗}
= 1
Z0Re{V+tV+∗+ V−tV+∗− V+tV−∗− V−tV−∗} = 0
Since V−tV+∗− V+tV−∗ is purely imaginary and V+tV+∗− V−tV−∗ purely real we get V+tV+∗− V−tV−∗= V+t([U ]− [S]t[S]∗)V+∗ = 0
This is valid for all input signals V+ and then [S] satisfies [S]t[S]∗ = [U ]
We see that the scattering matrix of a lossless two-port is a unitary matrix and hence its inverse is equal to its Hermite conjugate
[S]−1= [S]∗t
If we use the tables in subsection 3.2.5 we can relate the S-matrix to the hybrid and cascade matrices. The S-matrix can be obtained from measurements.
3.7.3 Matching of load impedances
When a load is impedance matched to a transmission line, the transmission of the signal is optimized in some sense, often w.r.t. the transmitted power. If the signal is a single frequency then there are many ways to obtain an exact match. If the signal contains a band of frequencies and if the load is frequency dependent it is mostly not possible to match the load exactly in the whole frequency band. Instead we try to find an optimal matching.
jB jB
jX jX
ZL ZL
Figure 3.20: Two-ports for matching of ZL. Conjugate matching
In circuit theory we know that we maximize the power from a source with inner impedance Ziby choosing the load impedance equal to the complex conjugate of the inner impedance, i.e., ZL= Zi∗. In many cases we have a given load impedance ZL. In order to get maximum power to the load we have to use a lossless matching circuit.
Matching a load to a transmission line
A load impedance ZL is impedance matched to a transmission line with characteristic impedance Z0 only if ZL= Z0. Otherwise power is reflected at the load. We can match a load to a line by adding reactive components. It is quite easy to match a load at a single frequency. We have already seen that the resistive load impedance can be matched to a transmission line with a characteristic impedance Z0 by using quarter wave transmission line with characteristic impedance, c.f., (3.22)
Z1 = Z02 ZL
If the load has a reactance X we can get rid of the reactance by first adding a reactance
−X in series, or a susceptance B = −X/√
R2+ X2 in parallel with ZL, before we add the quarter wave transformer. Below we present two other methods that can be used.
All three methods described here manage to match the impedance at one frequency. It is more difficult to get an impedance match for a band of frequencies and we do not discuss that in detail here.
Two-port matching
We can match the impedance ZLto a characteristic impedance Z0by adding a capacitance and an inductance. The two possible matching two-ports are depicted in figure 3.20
For the two-port to the left we get
Z0= jX + 1
jB + (GL+ jBL) (3.40)
and for the two-port to the right we get 1
Z0 = jB + 1
jX + RL+ jXL (3.41)
The scattering matrix S 55
where ZL= RL+jXLand YL= GL+jBL. We assume that the transmission line is lossless, i.e., Z0 is real. By identifying the real and imaginary parts in the two expressions we can solve for X and B. The upper equation gives
X=±Z0
r1− Z0GL Z0GL B=±
s
GL(1− Z0GL) Z0 − BL
We see that we must have Z0GL ≤ 1, i.e., RL+ XL2/RL ≥ Z0 to satisfy the relations.
This is satisfied if RL≥ Z0. Equation (3.41) gives X=±p
RL(Z0− RL)− XL
B=±
p(Z0− RL)/RL Z0
This requires that RL ≤ Z0. Evidently we should use the two-port to the left when RL≥ Z0 and the two-port to the right when RL≤ Z0.
Example 3.11
Apparently we have two choices when RL = Z0. If we use the left circuit we see that X = 0 and B =−BLand if we choose the right circuit we get B = 0 and X =−XL.
3.7.4 Matching with stub
A short transmission line is sometimes called a stub. We can match the impedance ZL to a characteristic impedance Z0 by adding a shortened stub with length ` and characteristic impedance Z0 at a distance d from the load. We can either connect the stub in series or in parallel. We have the two parameters d and ` to play with and that is enough to achieve impedance match.
The input impedance of the stub is purely reactive and is given by Zst= jZ0tan β`
The input impedance of the transmission line at the distance d from the load is Zin= Z0ZLcos(βd) + jZ0sin(βd)
Z0cos(βd) + jZLsin(βd)
We first connect the stub in parallel with the load. We get impedance match when 1
Z0 = 1
Zstub + 1 Zin This gives
1 =− j
tan β` +1 + jzLtan βd
zL+ j tan βd (3.42)
The real part of the equation gives the expression for d
tan βd =
xL±q
rL((rL− 1)2+ x2L) rL− 1
We pick the smallest d that satisfies this equation. We then get ` from the imaginary part of (3.42)
tan β` = rL2 + (xL+ tan βd)2
rL2tan βd− (1 − xLtan βd)(xL+ tan βd)
This equation has infinitely many solutions and we pick the smallest positive solution. If ZL is purely resistive the expression is simplified.
The other alternative is to connect the stub in series which gives Z0 = Zstub+ Zin
from which ` and d are solved.