### Introduction

While the output voltage of a Voltage Regulator Module (VRM) might be very stable this is not an indication of the stability of the voltage that is seen at the pads of a chip. The variable current draw of the chip across the impedance of the PDN will cause an associated voltage drop at the pads. By keeping the impedance of the PDN below a target threshold we can be reasonably certain that the voltage excursions at the pads of the chip can be kept in an acceptable range.

### Method

Impedance is generally measured with a vector network analyzer (VNA) using a 1-port measurement. This technique works quite well for most interconnects designed to carry signals on a PCB where the impedance is on the order of the 50 Ohm port impedance of the analyzer. The impedance of a PDN might be under 100 millohms which makes accurate 1-port measurements impractical as virtually all of the incident energy is reflected. With such a large amount of reflected energy, the measurement will be close to 0 dB and dominated by system noise and probing or fixturing effects. By making a 2-port measurement we can accurately measure impedances below 100 milliohms without these artifacts.

The 2-port measurement technique presented in this application note is similar to the 4-terminal sensing measurement method to eliminate contact resistance effects from the measurement system and measure only the impedance of the device under test. The concept is to use one port to generate a current through the device and the other port to measure the voltage drop across the device. Figure 1 below illustrates how the impedance is calculated from 2-port measurement. In practice the two ports will not be connected to the exact same points on the DUT but to nearby connection points.

The goal is to measure ZDUT but the SPARQ will only directly measure S21 so we’ll start with the following relationship where V_{2} and V_{1} are the output voltages of each port.

$$S_{21} = V_2 \over V_1$$

We know that V_{1} = 1/2*V_{source} so

$$S_{21} = {V_2 \over {{1\over2}V_{source}}}$$

V_{2} is easily calculated as

$$V_2 = {V_{source} { { {Z_{DUT}R_2} \over {Z_{DUT}+R_2} } \over {R_1 + {{Z_{DUT}R_2}\over{Z_{DUT}+R_2}} } }}$$

Plugging this in and solving for Z_{DUT} we get

$$Z_{DUT} = {25\Omega{S_{21}\over{1-S_{21}}}}$$

### About the SPARQ Network Analyzer

The SPARQ series analyzers take S-parameter measurements using TDR and TDT to characterize the behavior of a device under test. Although it uses TDR/T, it is operated like a VNA, with one important difference: users do not need to perform a calibration manually; the SPARQ performs this step automatically using an internally connected OSLT calibration kit. This automatic, internal calibration capability allows the SPARQ to be used quickly and easily. SPARQ models with 2 through 12 ports are available, and up to 40 GHz for 2 and 4 port units.

When making measurements, user can select to run the SPARQ in either “normal” or “long” DUT length mode. This setting controls the length of the TDR and TDR acquisitions. In long mode, the acquisition window is 250ns long (220ns “post-edge”), providing an intrinsic frequency resolution for S-parameter measurements of about 4 MHz, with 5 MHz being a standard value.

### Execution

This technique can be accomplished in the SPARQ software by selecting S1 and S2 to display the real and imaginary parts of S21 (figure 3) and using the built in MATLAB math function to perform the calculation (figure 4.)

**The impedance is calculated and plotted with a short MATLAB script as shown below**

`S21 = WformIn1 + WformIn2 * i;`

Z21 = 25 * (S21 ./ (1 - S21));

WformOut = abs(Z21);

x=linspace(0,25E9,5E3);

x=x(:);

figure('Name','Impedance','NumberTitle','off');

The maximum size of the data array that can be created using this built in MATLAB math function is 5000 points which will give us an end frequency of 25 GHz. This script plots the data on a log-log scale which is typically the most convenient way to view the impedance measurement. A linear plot is displayed in the SPARQ software.

### Tips

The SPARQ is a time domain based instrument and so the frequency spacing and minimum measured frequency point in the impedance profile is a function of the total capture time for the TDR and TDT waveforms. Using the maximum number of points and setting the DUT Length Mode to Long will ensure that you achieve the minimum delta frequency of 5 MHz.

At high frequency the effects of fixturing or probing can start to dominate the measurement. Use the Gating feature in the SPARQ to de-embed the effects of your fixture or probes. For an explanation of how to use this feature please see the application note __De-embedding Gigaprobes Using Time Domain Gating with the LeCroy SPARQ__ on the Teledyne LeCroy website.