Welcome to Tim’s Blackboard! This is the place to find discussions on interesting topics related to signal integrity and power integrity.

This week we are taking a break from Signal Integrity. In this post, I will “demystify ultra-low impedance measurement.”

# Introduction

To measure ultra-low **DC** resistance, instead of using a traditional 2-terminal sensing, one uses 4-terminal Kelvin sensing to avoid contact resistance. Similarly, instead of using the 1-port method to measure low impedance of the Power Distribution Network (PDN), we use the 2-port shunt technique, shown in Fig. 1.

*Fig. 1: 2-port technique for ultra-low impedance measurement.*

In the following paragraphs, I will show you that not only does the 2-port technique give us more measurable signal levels, it also helps us reduce the effect of contact resistance.

# Why Use the 2-port Technique?

In the March 2010 issue of the PCB Design Magazine, Mr. Istvan Novak pointed out “S_{11} VNA Measurements Don’t Work for PDN Measurements.”

It is true. If I assume the device under test, the PDN, has 10 mOhm impedance and port impedance is 50 Ohm, the S_{11} calculation,

gives me -0.003 dB, which is easily masked by noise or bad calibration.

Even if I have an ideal VNA with no noise and with perfect calibration, the contact resistance from the test fixture to the device under test, typically in mOhm range, is large enough to influence the result of the measurement significantly.

*Fig. 2: Illustration of contact resistance in series with impedance under test.*

To tell how contact resistance impacts the impedance calculation, I need to derive the extracted impedance in terms of measured S_{11}. Using the schematic shown in Fig. 2, I would write

According to the impedance extraction equation, the contact resistance is directly influencing the extracted impedance. Worse yet, since the impedance and the contact resistance are of the same order of magnitude, I know the impedance extraction result is highly sensitive to the contact resistance.

# 2-port Ultra-low Impedance Measurement Technique

Shown in Fig. 3, the 2-port ultra-low impedance technique connects the device under test in shunt with the ports and uses S_{21} to extract the impedance under test.

*Fig. 3: Ultra-low impedance measurement uses S _{21 }to measure and extract the impedance under test.*

Note that because S_{21} is the response of port 2 by the excitation from port 1, it’s analogous to using port 1 as a current source and port 2 as a voltage probe in DC 4-terminal sensing.

Applying S-parameter analysis to the circuit in Fig. 3, the S_{21} of the device under test is:

Putting in the numbers (Z_{port} = 50 Ohm, Z_{DUT }= 10 mOhm),

Given a good VNA, I should be able to measure down to -68 dB.

As shown, the 2-port technique is more suitable for ultra-low impedance measurement. The measured S_{21} is in the -60 to -80 dB range, more approachable than the S_{11} in milli-dB range.

So far, I have shown S_{21 }produces more measurable signal levels than the S_{11} measurement. Next, I will demonstrate another great feature of 2-port measurement: insensitivity to contact resistance.

# 2-port Technique Reduces Impact of Contact Resistance

Using the previous result, I continue and solve for the ideal extracted impedance given measured S_{21},

As shown, if there were no contact resistance, above calculation with measured S_{21} gives exactly the impedance under test. Let’s see what happens when I put the contact resistance back in the setup.

*Fig. 4: Illustration of 2-port low impedance measurement setup with contact resistance.*

Found in Fig. 4 is the 2-port measurement setup with contact resistance included. With the contact resistance, the extracted impedance is no longer just the device under test. The result of the impedance extraction is

where

Now, knowing both the contact resistance and the impedance of the device under test are in the mOhm range, I know the resistance error constant, K_{r}, is dominated by the sum of the contact resistance:

In addition, if I am measuring a low impedance PDN, S_{21} is going to be a number much smaller than 1, that is,

Given the approximations, I can rewrite the 2-port extracted impedance,

Great news! Since both S_{21} and the contact resistance are small numbers, the product is going to be even smaller! Consequently, as long as I am measuring low impedance, where S_{21} is a small value, the 2-port measurement technique is **NOT** sensitive to the contact resistance.

# Ultra-low Impedance Measurement Demystified

Having derived the impedance extraction equations for both 1-port and 2-port measurements, I have demonstrated that the 2-port technique is a wonderful method to measure ultra-low impedance.

The 2-port low-impedance technique can examine more than just PDN. Because of the ability to measure ultra-low impedance, the technique is also useful to investigate the skin-depth of copper traces and a capacitor’s equivalent series resistance and equivalent inductance.

That's this week's Tim's Blackboard. See you in two weeks!

**To download ADS to create a virtual ultra-low impedance 2-port measurement test bench:**