1. Use s parameters-determining_inductance_capacitance
2. Relationship Between Common Circuits and the ABCD Parameters
3. Converts Z-parameters to S-parameters
4. Relationships Between Two-Port S and ABCD Parameters
5. Via and equivalent circuit
3. 3
Converts Z-parameters to S-parameters
• where U is the identity or unit matrix and Zn is the termination impedance of each port.
• The derivation assumes that each port is terminated in the same value.
• Equation above converts Z-parameters to S-parameters.
• conversion back from S-parameters to Z-parameters for an arbitrary-sized matrix:
• Derive the transformation of the ABCD to S-parameters.
• The final solution is summarized in
• where Zn is the termination impedance at the ports, which are all assumed to be equal.
• The final relationship between a two-port Z-matrix and the ABCD matrix is shown as:
[ ] [ ] [ ]A Z S→ →
5. 5
Via and equivalent circuit
Ex1: Extract an equivalent circuit for the via shown in Figure from the following S-parameter matrix
measured at 5 GHz assuming port impedance values (Zn) of 50 Ohm:
Step 1: Transform the S-matrix into ABCD parameters using the relations between 2-port S and ABCD
parameters.
Sol:
Step 2: Choose the form of the equivalent circuit. A signal propagating through the via will experience
the capacitance of the via pad, the inductance of the barrel, and then the capacitance of the second
pad. This configuration fits the pi model.
Pi model
6. 6
Step 3: Use the relations in Table to calculate the admittance of each segment in the pi model
Due to symmetry in the circuit, Y1 = Y2.
Step 4: Calculate the circuit values:
7. 7
The model can be extracted as either a Pi or a T network now we show T model.
The inductance values will include the L of the trace and the via barrel (it is assumed that the test
setup minimizes the trace length, and subsequently the trace capacitance is minimal).
The capacitance represents the via pads
Ex2: Assume the following s-matrix measured at 5 GHz
Convert to ABCD parameters
T model
11 12
21 22
0.110 0.153 0.798 0.572
0.798 0.572 0.110 0.153
S S j j
S S j j
− − −
= − − −
0.827 20.08
0.0157 0.827
A B j
C D j
=
Relating the ABCD parameters to the T circuit
topology, the capacitance and inductance is
extracted from C & A.
3
1
1 2
3
1 1
0.0157 0.5
1
2
2
1 0.827 1 0.35
1/ ( 2 )
VIA
VIA
VIA
C j C pF
Z
j fC
Z j fL
A L L nH
Z j fC
π
π
π
= = = ⇒ =
⋅
⋅
= + = = + ⇒ = =
⋅
1 1 2
1 2
3 3
2
3 3
1
0.827 20.08
0.0157 0.8271
1
Z Z Z
Z Z
Z Z j
Z j
Z Z
+ + + = +
8. 8
Ex3: Using the two independently measured values of S-parameters for a via and a loss-free
transmission line at 5 GHz, calculate the, equivalent S-parameters that would be measured if the two
circuits were cascaded as shown Figure. Assume that the termination impedance is 50 Ohm.
(a) configuration of the independently measured via;
(b) transmission line; (c) via cascaded with the transmission line.
9. 9
Sol:
Step 1: Convert to ABCD parameters
Step 2: Multiply the ABCD matrices
Step 3: Convert back to S-parameters using Table where Zn = 50 Ohm (the termination impedance)