## 1.7 High-Frequency Measurements

High-frequency measurements represent a very important part of evaluating the effectiveness of decoupling capacitors, planes, and PDNs with surface-mount components. The magnitude and slope of change in resistance and inductance above the self-resonant frequency (SRF) depend on the relative dimensions of the capacitor body, pads, vias, and closest planes. Since *ESR*(*f*) and *ESL*(*f*) depend on the geometry, it is important to measure the parts in a fixture containing pads, vias, and planes with geometries similar to actual usage to ensure that the measured complex impedance reflects both the device under test (DUT) and, if necessary, the fixture. The low impedance values associated with today's bypass capacitors can be conveniently measured using vector network analyzers (VNA) in two-port connections [28].

The VNA consists of a tunable sinusoidal source and tracking receiver. The receiver can be connected to measure the incident wave (a_{l}) or reflected wave (b_{l}) to and from the DUT, or the transmitted wave (b_{2}) through the DUT. These waves are measured by measuring voltages. The VNA measures and displays the various combinations of incident and reflected waves, commonly known as *S*-parameters [29]. In a VNA measurement, the equivalent circuit can be represented as shown in Figure 1-41. In the figure, *Z*_{0} is the connecting impedance of the instruments, usually 50 W. A state-of-the-art VNA may have around one-tenth of a decibel repeatability and a fraction of a decibel absolute error in *S*_{11} measurements, which allows the accurate measurements of impedances not much smaller or greater than 50 W. The impedance of a one-port DUT can be calculated from its voltage reflection coefficient (*S*_{11}) by using this formula:

**Equation 1.37**

Figure 1-41 Schematic of a vector network analyzer measurement. By permission from M. Swaminathan, et al. [28], © 2004 IEEE.

The impedances of PDNs are typically low, much smaller than 50 W. With very low impedances, the nominator in equation (1.37) approaches zero, and therefore the accuracy gradually breaks down. As shown in [30], the two-port impedance measurement setup can extend the capabilities of one-port vector network analyzer measurements by more than an order of magnitude, enabling the measurement of PDN impedances below a milliohm. Hence, a two-port measurement is always preferable while characterizing PDNs.

### 1.7.1 Measurement of Impedance

The equivalent circuit of a PDN impedance measurement is shown in Figure 1-42. In the figure, *L _{p}*

_{1}and

*L*

_{p}_{2}represent the inductive discontinuities of the cable–DUT interface, including the loop inductance due to the current loop from the signal tip to the ground tip of a probe [31]. The PDN is represented as a two-port network with impedances

*Z*

_{11},

*Z*

_{12},

*Z*

_{21}, and

*Z*

_{22}associated with it, as defined earlier. The reference planes are set as shown in Figure 1-42, assuming that both the connectors (or probes) are calibrated up to the DUT interface. The DUT includes the entire PDN; hence the goal is to extract the PDN impedances between the ports of interest. Assuming a two-port measurement, the measured

*S*-parameters can be related to the

*Z*-parameters of the PDN through the transformation

**Equation 1.38**

Figure 1-42 Equivalent circuit for transfer impedance measurement. M. Swaminathan, W. Kim, and I. Novak, “Measurement problems in high speed networks,” *IEEE Instrumentation and Measurement Technology Conference*, pp. 1339–1346, Budapest, Hungary, 2001.

where *Z*_{0} = 50 W. Setting *Z*_{1} = 50 + *jwL _{p}*

_{1}and

*Z*

_{2}= 50 +

*jwL*

_{p}_{2}, the above equation can be rewritten in the form

**Equation 1.39**

Since *S*
_{12} measurement is preferable for low impedance measurements, from equation (1.39), the measured *S*_{12} parameter can be related to the impedance as

**Equation 1.40**

### 1.7.2 Measurement of Self-Impedance

Assuming the self-impedance of a PDN is to be measured, the two connectors (or probes) can be placed next to each other such that *Z*_{11} = *Z*_{12} = *Z*_{21} = *Z*_{22}. Under such a measurement, the equivalent circuit simplifies to the circuit shown in Figure 1-43. Then equation (1.40) simplifies to

**Equation 1.41**

Figure 1-43 Equivalent circuit for self-impedance measurement. M. Swaminathan, W. Kim, and I. Novak, “Measurement problems in high speed networks,” *IEEE Instrumentation and Measurement Technology Conference*, pp. 1339–1346, Budapest, Hungary, 2001.

from which the self-impedance can be extracted from the measured *S*-parameter as

**Equation 1.42**

Assuming the inductive discontinuities are negligible, resulting in a nearly 0 dB through measurement, the self-impedance simplifies to

**Equation 1.43**

Although equation (1.42) is an exact expression, inductances L_{p1} and L_{p2} result in very small perturbation to the magnitude of *Z*_{11} in equation (1.43), which is illustrated in the following example. Equation (1.43) is especially useful for PDNs that have a very small impedance since *S*_{11} measurements are not very reliable.

When small impedance values are measured with a 50-W system, *S*_{12} is much less than 1 and therefore a good approximation for the self-impedance measurement is

**Equation 1.44**

The insertion loss *S*_{12} is typically measured in decibels and hence must be converted to the appropriate scale. Using equation (1.42), (1.43), or (1.44), accurate measurements are possible for small values of impedances typical of PDNs provided the measurement setup is well calibrated up to the probe tips. A simple calibration method is the short-open-load-through (SOLT) calibration.

### 1.7.3 Measurement of Transfer Impedance

The transfer impedance *Z*_{12} between two ports placed far apart can be extracted from equation (1.40) as

**Equation 1.46**

where *Z*_{1} and *Z*_{2} have the same definitions as in the self-impedance measurements, while *Z*_{11}, *Z*_{22}, and *Z*_{12} are the two-port impedance parameters of the structure. Assuming *L _{p}*

_{1}=

*L*

_{p}_{2}= 0, equation (1.46) can be simplified to

**Equation 1.47**

For low impedance measurements *S*_{12}*Z*_{21}/100 ≪ 1, *Z*_{11}/50 ≪ 1, and *Z*_{22}/50 ≪ 1. Therefore, the transfer impedance measurement can be simplified similar to equation (1.44). Several of the transfer impedance measurements in the book make use of the simplified equation in (1.44).

### 1.7.4 Measurement of Impedance by Completely Eliminating Probe Inductance

For PDNs with a moderate impedance and measurements involving a high probe inductance, a different methodology can be used. Let's once again assume that the 2 x 2 impedance matrix, including the probe parasitics, can be approximated as

**Equation 1.48**

This equation shows that the transfer impedance is not affected by the probe inductance. Hence, whatever the value of *L _{p}*

_{1}and

*L*

_{p}_{2}is,

*Z*

_{12}can be extracted very accurately, assuming the VNA is calibrated up to the probe tips. This property is used in Chapter 5, where the material properties are extracted from

*Z*

_{12}. The 2 x 2 impedance matrix in equation (1.48) can be obtained from measured

*S*-parameters.

For accurate measurement of an input impedance, say *Z*_{11}, the probe parasitic inductances must be known. However, there is a simple way of avoiding this problem as well. A separate measurement is taken, where the two probes are placed very close to this input port. Since the two ports are very close to each other, the input and transfer impedances of such a network can be assumed to be equal (*Z*_{11} = *Z*_{12} = *Z*_{21} = *Z*_{22}), as mentioned earlier. Hence, once again, the transfer impedance *Z*_{12} can be measured, which is not affected by the probe inductance. Since the input and transfer impedances of this DUT are equal, the input impedance *Z*_{11} can be found to be equal to the measured transfer impedance *Z*_{12}. This procedure provides accurate measurement results for PDNs in the presence of high probe inductance.