IEEE Microwave Magazine - June 2015 - 30

The calculation of Ceff is a good
example of how to use a signal flow
graph and apply Mason's rule.
return loss, corrupts the power measurement used for
leveling. A directional coupler with good isolation
would block this signal path, but a power divider directs a substantial part of the reflected signal to the
detector or power sensor, thus adjusting the level to
an incorrect value. For this reason, a power divider
with perfectly matched ports results in a poor effective
source match.
A two-resistor power splitter, by contrast, exhibits a
substantial mismatch at its output ports (see the determination of S 22 of the two-resistor power splitter in the
"Effective Source Match of Divider and Splitter" section), but the degraded leveling signal compensates for
this reflection of the backward wave. Figure 1(c) illustrates this setup.

Effective Source Match C eff
The basic equipment for generating a swept frequency signal of constant power is a generator with
a leveling loop. A coupling device directs part of the
forward wave to a detector, which, in turn, controls
the power of the generator via a closed-loop control.
Any leakage of the backward wave to this leveling
detector interferes with the power control. To the circuit driven by this generator, this imperfect signal
generator leveling appears like a nonzero reflection
coefficient of the source. In other words, the generator exhibits poor source matching. Consequently, this
degradation is usually simulated as an additional
reflection coefficient at the output port of the source.
The effective source match C eff accounts for both the
reflection coefficient of the coupling device itself and
for the leveling error.

S31

S12

(1)

The transmission coefficient between two nodes is
the sum of all possible paths t k between these nodes,
corrected by the influences of loops (parts of the signal
playing ping-pong between various reflection points).
These corrections are introduced via the variables D k
and D:
D =1
- sum of all loops
+ sum of all products of two nontouching loops
- sum of all products of three nontouching loops
...
and

CD
Port 3
S32

S13

/ tk Dk

S23

Port 2
L

S22 CL

Figure 2. The signal flow graph of a three-port device.
G, L, and D are the signals from the generator, at the load,
and at the detector, respectively. The detector output is the
control voltage for leveling the power of the generator and is
thus kept constant.

S21
S11

The aim of this article is to demystify the concept
of effective source matching by investigating the signal
flow graph of the coupling structure used for obtaining
the control signal of the leveling loop. Figure 2 shows
the signal flow graph for the Y-branch circuit that can
represent any of the three three-port devices shown in
Figure 1. This Y-branch structure has three ports: port 1
is the input port driven by the sweep frequency generator via an adjustable attenuator, port 2 is the output port
connected to the load, and port 3 is the measurement
port connected to a detector diode or a power sensor.
The signal emerging from port 3 is kept constant by
means of a leveling loop. The challenge is to keep the
signal emerging from port 2 constant, irrespective of
any backward wave from the load. The reflection coefficients of the generator, the load, and the detector are C G,
CL, and CD, respectively. The input signal coming from
the generator is denoted by G, the signal at the load by
L, and the signal at the detector by D.
To analyze the signal flow chart, we have to apply
Mason's nontouching-loop rule [4], [6], which reads for
a path t between the source and target nodes

S33

Port 3
Port 1
G

Signal Flow Graph

Port 1 S31
G

S11

S33

S13

S32

S23

Port 2
L

S21

S12

S22

Figure 3. The signal flow graph for determining t LG . The
direct path S 21 from generator to load is shown in green,
the loop S 33 C D not touching S 21 is shown in blue, and the
path S 31 C D S 23 via the detector diode is shown in red.

June 2015

Table of Contents for the Digital Edition of IEEE Microwave Magazine - June 2015