## Measuring Parasitic Capacitance and Inductance Using TDR

Time-domain refiectometry (TDR} is commonly used as a convenient method of determining the characteristic impedance of a transmission line or quantifying reflections caused by discontinuities along or at the termination of a transmission line. TDR can also be used to measure quantities such as the input capacitance of a voltage probe, the inductance of a jumper wire, the end-to-end capacitance of a resistor, or the effective loading of a PCI card. Element values can be calculated directly from the integral of the reflected or transmitted waveform.

by David J. Dasclier

W hy would anyone use TDR to measure an inductance or capacitance when there ;ire many inductance-capacitance-resistance (LCEt) meters available that have excellent resolution and are easy to use? First of all, TDR allows measurements 10 be made on devices or structures as they reside in the circuit. When measuring parasitic quantities, the physical surroundings of a device may have a dominant effect on Ihe quantity that is being measured. If the measurement cannot be made on the device as it resides in the circuit, then ihe measurement may be invalid. Also, when measuring the effects of devices or structures in systems containing transmission lines, TDR allows the user to separate the characteristics of the transmission lines from the characteristics of the device or structure being measured without physically separating anything in the circuit To illustrate a case where TDR can directly measure a quantity that is very difficult to measure with an LCR meter, consider the following example.

A printed circuit board has a long, narrow trace oyer a ground plane, which formsa microstrip transmission line. At some point, the trace goes from the top of the printed circuit board, through a via, to the bottom of the printed circuit board and continues on. The ground plane has a small opening w here the via passes through il. Assuming that the via adds capacitance to ground, a model of this structure would be a discrete capacitance to ground between the top and bottom transmission lines. For now, assume that the characteristics of the transmission lines are known and all (hat needs to be measured is the value of capacitance to ground between the two transmission lines.

I sing an LCR meter, the total capacitance between Ihe Irace-via-trace structure and ground can be measured but the capacitance of the via cannot be separated from the capacitance of the traces. To isolate the via from the traces, the traces are removed from the board. Now ihe capacitance between just the via and ground can be measured. I Infort 11 nately, the measured value is not the correct value of capacitance for the model.

I sing TDR instead of an LCR meter, a step-shaped wave is sent down the trace on Ihe printed circuit board and the wave that gets reflected from the discontinuity caused by the via is observed. The amount of "excess" capacitance caused by I he via can be calculated by integrating and scaling the reflected waveform. Csing this method, the measured value of capacitance is the correct value of capacitance to be used in the model.

Tile discrepancy between t he two measurements exists because the LCR meter was used to measure the lota! capacitance of the via while TDR was used to measure the excess capacitance of the via. If Ihe series inductance of the via were zero, Ihen its total capacitance would be the same as its excess capacitance. Since the series inductance of the via is not zero, a complet e model of the via must include both its series inductance and its Shunt capacitance. Assuming thai the via is eapacilive, the complete model can be simplified accurately by removing the series inductance and including only the excess capacitance rather than the total capacitance.

Il should be no suqirise thai the value of excess capacitance measured using TDR is the correct value for the model. The reason to model the tracc-via-trace structure in the first place is to predict what effect the via will have on signals propagating along the traces, TDR propagates a signal along the trace lo make the measurement In this sense, TDR provides a direct measurement 0! the unknown quantity.

To derive expressions that relate TDR waveforms to excess capacitance and inductance, an understanding of fundamental transmission line parameters is required. This article presents it cursory review of transmission lines and the use of TDR to characterize them, and then derives expressions for excess capacitance and inductance. If you are already familiar with transmission lines and TDR, you may wish to skip the review sections.

### Fundamental Transmission Line Parameters

First a few words about "ground." Twin-lead {or twisted-pair J wire forms a two-conductor transmission line structure that can be modeled as shown in Fig. I. The model includes the

Al CH

Coaxial Cable

A2 O

A2 O

Coax/Microslrip/Stfipline Lumped LC Model US t/4 L/4 1/4 L/B

Coax/Microslrip/Stfipline Lumped LC Model US t/4 L/4 1/4 L/B

Fig. 1. TVi.i conductor transmission lines and lumped LC models.

The other component is a result of the current through the center conductor and the mutual inductance bet ween I he center and outer conductors (Vg - Lmcii/d1). Current that enters (he center conductor returns through the outer conductor. so the two currents are equal but in opposite directions. The unique property of coax is that the self-inductance of the outer conductor is exactly equal to the mutual inductance between the center and the outer conductors. Hence the two components that contribute to the voltage generated across the inductance of the outer conductor exactly cancel each other and the resulting voltage is zero, When current is injected into a coax transmission line, no voltage is generated along the outer conductor if the current in the center conductor is returned in the outer conductor.

The point here is thai the generalized model for a two-conductor transmission line can be simplified for microstrip and coax construct ions. The simplified model has zero inductance in series With one of the conductors since, in both cases, no voltage appears across the conductor. This conductor is commonly referred to as the ground plane in microstrip transmission lines and as the shield in coax transmission lines. The other conductor, the one that develops a voltage across it. is referred to as the transmission line, even though it is really only half of the structure,

There are two ways to model a lossless transmission line. One method defines the transmission line in terms of characteristic impedance (Zo) and time delay (td) and the other

Fig. 1. TVi.i conductor transmission lines and lumped LC models.

self-inductance of each conductor, mut ual inductance between the self-inductances, and capacitance between the two conductors. Skin effect and dielectric loss are assumed to be negligible in this model. Injecting a current transient i into one side of the transmission line, from node AI to node CI, causes a voltage v = iZn to appear between nodes Al and C1 and also causes a voltage v = Ldi/dt to appear across the series inductance of both the A-B and C-D conductors. Referring bark to the physical structure, I bis means that there is a voltage difference between the left side and the right side of each conductor. Even if you name the C-D conductor "ground," it still develops a voltage between its left side and its right side, across its series inductance.

Microstrip traces and coaxial cables ("coax ) are two special cases of two-conduct or transmission line structures. Injecting a current transient into one side of a microstrip or coax transmission line causes a voltage to appear across only one of the two conductors. In the case of an ideal microstrip, where one of the conductors is infinitely wide, the wide conductor can be thought of as a conductor with zero inductance. Hence the voltage generated across the infinitely wide conductor is zero. With coax, the inductance of the outer conductor is not zero. However, the voltage generated across the inductance of the outer conductor has two c omponents. One component is a result of the current through the self-inductance of the outer conductor (vj = L&di/dt),

One-Segment Model

UZ Ul

Five-Segment Model