IEEE Microwave Magazine - June 2015 - 32

The problem is the frequency
response of the device used for
coupling the control signal to the
variable-gain or variable-attenuation
element in the sweep generator.

C eff = S 22 - S 21

the forward transmission of a two-port device fed by
a signal with constant power, as shown in Figure 5 [7].
Applying the rule for reducing a loop in a signal
flow graph (see [1, pp. 189-192]) by dividing the coefficient of the forward path by (1 - t loop) yields
L = G const.

TX
,
1 - CL Ceff

(6)

where TX denotes the transmission coefficient and Ceff
denotes the reflection coefficient. This Ceff of the twoport device plays the same role as the effective source
match of the leveled signal source and, hence, uses the
same notation. Comparing (5) with (6) yields

Port 3
Z0
Z0/3
Port 1

Z0/3

Z0

Port 2
Z0

Figure 6. A three-resistor power divider is matched at all
three ports and transmits a quarter of the input power to
each of the outputs. This divider should not be used in a
leveling loop since the effective source match Ceff is -0.5.

We now have the tools for calculating Ceff for both
the divider and splitter (see "Power Leveling") [8]. If
a directional coupler with an ideal isolation of S 32 = 0
were used, the reflected wave from the DUT would
not influence the leveling loop and the effective source
match would reduce to S 22 .

Effective Source Match of Divider and Splitter
Three-Resistor Power Divider
A three-resistor divider (each resistor has the value
Z 0 /3; see Figure 6) exhibits matched ports ^S ii = 0 h and
transmits signals according to S ij = 0.5 for i ! j (see [1,
pp. 317-318]). Here, Z 0 denotes the wave impedance of
the devices and transmission lines and is usually 50 X.
With these values, (7) gives
Ceff, divider = 0 - 0.5 0.5 =-0.5 (!) .
0.5

Leveled

The forward transmission of a two-resistor splitter is the
same as before, that is, S 21 = S 31 = 0.5. Both resistors have
a value of Z 0 . Figure 7 shows the circuit of this device.
An incident wave at port 2 encounters an impedance of
Z 0 + Z 0 (Z 0 + Z 0) = ^5/3 h Z 0, which results in a reflection coefficient of S 22 = ^5/3 - 1 h /^5/3 + 1 h = 0.25.
The transmission coefficient S 32 from the output to
the detector, the root cause of the problem (the leakage
of a reflected wave to the leveling detector), can be calculated as follows. The voltage at port 3 is
V3 = V2

Z0
Port 2 (Output)
Z0

V2, incident

Figure 7. A two-resistor power splitter is matched only
at port 1 and transmits a quarter of the input power to
each of the outputs. This splitter is the device of choice for
a leveling loop since the effective source match Ceff for an
incident wave at port 2 is zero.

32

R $ Z 0 = 0.2 V
2
Z0 + R Z0 + Z0

with
R = Z 0 (Z 0 + Z 0),

Port 1 (Input)
Z0

(8)

As previously mentioned, using this divider in a leveling loop is ill-advised. Although the divider itself may
have an excellent return loss at port 2, the transmission
of an incident wave at port 2 to the detector corrupts
the power leveling and results in poor effective source
matching. In this case, the input power to the DUT
would depend on its reflection coefficient.

Port 3 (Detector)
Z0

(7)

Two-Resistor Power Splitter

Z0/3

Z0

S 32
.
S 31

(9)

where R/ ^Z 0 + R h describes the voltage divider from
port 2 to the branching point and Z 0 / ^Z 0 + Z 0 h is the
voltage divider from the branching point to port 3. Due
to the nonzero reflection, the voltage at port 2 is not
equal to the incident wave V2, incident
V2 = V2, incident (1 + S 22) = 1.25 V2, incident .

(10)

The transmission coefficient from the output to the
detector is therefore

June 2015



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

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