Part 1: S₁₁ One-Port Shunt, Two-Port Shunt and Two-Port Series Methods

This is the first of two articles devoted to the topic of capacitor impedance evaluation from the s parameter measurements using a network analyzer. Part 1 describes the impedance measurements and calculations from the s₁₁ parameter using the one-port shunt method, two-port shunt, and two-port series methods. Part 2 will discuss impedance measurements and calculations using the s₂₁ parameter with two-port shunt and two-port series methods.

Configurations, Circuit Models, S₁₁ – Impedance Relationships

One-Port Shunt Method

Note: The one-port shunt method is also called a (one-port) reflection method [1]. One-port configuration for a two-terminal DUT is shown in Figure 1.

Figure 2 shows the transmission line circuit model at Port 1.

The network analyzer sends the incident waves (at different frequencies) to Port 1, terminated with Z_x (interconnects are taken out of the measurements through the calibration process).

Upon the arrival at load Z_x, the incident waves get reflected (unless the load impedance Z_x equals Z₀. The reflected voltage waves, v_r, are related to incident voltage waves, v_i, by the load reflection coefficient, Γ, defined as

(1)

This load reflection coefficient equals the s₁₁ parameter and can be computed from [2],

 (2)

Eq. (1) is used to obtain the DUT impedance in terms of s₁₁ parameter, as follows

(3)

(4)

(5)

(6)

Resulting in the DUT impedance in terms of the s₁₁ parameter as [1],

(7)

Two-Port Shunt Method

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

The simplified circuit model of this shunt configuration is shown in Figure 4.

Note that the DUT impedance Z_x is in parallel with the Port 2 impedance Z₀, resulting in an equivalent load impedance of

(8)

The s₁₁ parameter equals the load reflection coefficient and can be computed from

(9)

Utilizing Eq. (8) in Eq. (9) we get

(10)

or

(11)

(12)

which simplifies to

(13)

Eq. (13) is now solved for Z_x in terms of s₁₁.

(14)

or

(14)

(15)

(16)

resulting in

(17)

Two-Port Series Method

The two-port series configuration for a two-terminal DUT is shown in Figure 5.

The simplified circuit model of this series configuration is shown in Figure 6.

Note that the DUT impedance Z_x is in series with the Port 2 impedance Z₀, resulting in an equivalent load impedance of

(18)

The s₁₁ parameter equals the load reflection coefficient and can be computed from

(19)

Utilizing Eq. (18) in Eq. (19) we get

(20)

or

(21)

Eq. (21) is now solved for Z_x in terms of s₁₁.

(22)

or

(23)

(24)

(25)

resulting in

(26)

Impedance Measurement Setup and Results

The impedance measurement setup and the PCB boards are shown in Figure 7. 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 for a 47 nF capacitor are shown in Figures 8 and 9. Figure 8 compares the results between the one-port shunt and two-port shunt configurations, while Figure 9 compares the two-port series and two-port shunt configurations.

Clearly, the two-port series measurement is not reliable. Figure 10 shows the capacitor impedance curve obtained from the Murata Design Support Software “SimSurfing” [3].

The one-port shunt, two-port shunt, and Murata measurements at 0 dB and at self-resonant frequencies are shown in Table 1.

Table 1: Impedances at 0 dB and self-resonant frequencies

Clearly, the one-port shunt and two-port shunt measurements do not agree with the Murata values. The measurement results for the other two capacitors (4.7 nF, 470 nF), not presented here, showed similar trends.

The overall conclusion is that the capacitor impedance evaluation from s₁₁ parameter measurements is not accurate. The next article will discuss the capacitor impedance estimation from s₂₁ parameters and show its superiority over the s₁₁-based methods.

References

1. Microwaves & RF Application Note, Make Accurate Impedance Measurements Using a VNA. 
2. Keysight Application Note, Impedance Measurements of EMC Components with DC Bias Current. 
3. Murata Design Support Software “SimSurfing.”