Part 2: S₂₁ Two-Port Shunt and Two-Port Series Methods

This is the second of two articles devoted to the topic of capacitance impedance evaluation from the S parameter measurements using a network analyzer. The previous article [1] described the impedance measurements and calculations from the S₁₁ parameters using the one-port shunt, two-port shunt, and two-port series methods. This article is devoted to the impedance measurements and calculations from the S₂₁ parameters using the two-port shunt and two-port series methods.

Two-Port Shunt Method

The two-port configuration for a two-terminal DUT is shown in Figure 1.

Figure 2 shows the transmission line circuit model of this configuration.

The network analyzer sends the incident waves, v_i, (at different frequencies) from Port 1 to Port 2. Between the ports, there is a shunt discontinuity, Z_x. Upon the arrival at the discontinuity, the incident waves get reflected and transmitted.

The reflection coefficient at the discontinuity was derived in [1, Eq. (13)] as

(1)

The transmission coefficient at the discontinuity, which is equal to s₂₁, is related to the reflection coefficient by

(2)

Thus

(3)

or

(4)

leading to, [2],

(5)

Eq. (5) is now solved for Z_x in terms of S₂₁.

(6)

or

(7)

(8)

(9)

resulting in

(10)

Two-Port Series Method

The two-port series configuration for a two-terminal DUT is shown in Figure 3.

For this two-port series configuration, we will use the circuit theory (not the transmission line theory) and the two circuit models shown in Figure 4.

Voltage at port 2, V_L1, (with Z_x = 0, is obtained from the voltage divider as

(11)

Voltage at port 2, V_L2, (with Zx ≠ 0, is obtained as

(12)

The s₂₁ parameter is determined from

(13)

Thus,

(14)

or

(15)

Eq. (15) is now solved for Z_x in terms of S₂₁.

(16)

or

(17)

(18)

(19)

resulting in

(20)

Impedance Measurement Setup and Results

The impedance measurement setup and the PCB boards are shown in Figure 5. The boards were populated with Murata X7R ceramic capacitors, GCM188R71H472KA37, GCM188R71H473KA55, GCM188R71C474KA55, of the values 4.7 nF, 47 nF, and 470 nF, respectively.

Impedance curves (obtained from the S₂₁ parameter measurements) for a 4.7 nF capacitor are shown in Figure 6.

Figure 7 shows the capacitor impedance curve obtained from the Murata Design Support Software “SimSurfing” [4].

The two-port series, two-port shunt, and Murata measurements at 0 dB and self-resonant frequencies for a 4.7 nF capacitor are shown in Table 1.

It is apparent that the two-port shunt measurements, at 0 dB and self-resonant frequencies, are significantly closer to the Murata results, than the two-port series measurements.

Impedance curves for a 47 nF capacitor are shown in Figure 8.

Figure 9 shows the Murata impedance curve.

The two-port series, two-port shunt, and Murata measurements at 0 dB and self-resonant frequencies for a 47 nF capacitor are shown in Table 2.

Again, the two-port shunt measurements, at 0 dB and self-resonant frequencies, are significantly closer to the Murata results than the two-port series measurements.

Impedance curves for a 470 nF capacitor are shown in Figure 10.

Figure 11 shows the Murata impedance curve.

The two-port series, two-port shunt, and Murata measurements at 0 dB and self-resonant frequencies for a 470 nF capacitor are shown in Table 3.

Once again, the two-port shunt measurements, at 0 dB and self-resonant frequencies, are significantly closer to the Murata results than the two-port series measurements.

The overall conclusion is that the two-port shunt method is the most accurate method for the capacitor impedance evaluation from S₂₁ parameter measurements.

References

1. Bogdan Adamczyk, Patrick Cribbins, and Khalil Chame, “Capacitor Impedance Evaluation from S Parameter Measurements – Part 1: S₁₁ One-Port Shunt, Two-Port Shunt, and Two-Port Series Methods,” In Compliance Magazine, February 2025.
2. Keysight Application Note, Impedance Measurements of EMC Components with DC Bias Current.
3. Microwaves & RF Application Note, Make Accurate Impedance Measurements Using a VNA.
4. Murata Design Support Software “SimSurfing.”