A sense resistor is not only a resistor. A better model includes series inductance. The terminal leads (or terminal traces for surface-mount resistors) contribute an inductive element. In most resistor applications, this inductance is of no consequence, for it forms a time constant that is very small. But when resistance is also small in value, the time constant, t  = L/R, becomes large – large enough so that the frequency 1/t lies too close to the loop bandwidth of the power circuit.

It is not difficult to encounter parasitic inductances in the 50 to 200 nH range. This is too small in value to measure accurately on common RLC meters (or "bridges"), but can be measured conveniently on the lab bench by different methods (covered below). A 100 mW sense resistor with 100 nH series inductance has a time constant of 1 μs. And a 10 mW resistor will have a time constant of 10 μs, or a break frequency of about 16 kHz, within the bandwidth of many power-circuit feedback loops.

The series RL combination has an impedance of:

where s = j× w for frequency-response analysis (and w  = 2× p × f , where f is frequency in Hertz). The additional zero at radian frequency 1/(L_s/R_s) introduces an additional pole in a current-amplifier feedback loop if it is in the feedback path. Consequently, the zero cannot be ignored and some estimate of the parasitic inductance becomes worthwhile.

We will examine two possible methods for series-inductance measurement, based on the frequency and time domains, in that order.

The test circuit show below can be easily built on a lab bench and used to measure the parasitic series L. Typical function generators have 50 W outputs and can be used for this measurement.

Let the generator source resistance and its output terminating resistance (both 50 W ) be combined (in parallel) to form the equivalent series source resistance of R_g = 25 W . The transfer function of this circuit is:

For R_g >> R_s, this is approximated by

By frequency-sweeping the generator, the frequency, f_a , at which the amplitude increases by a times, is then substituted into

to produce the value of L_s. (Use a value of a >> 1 to avoid the knee of the frequency-response curve around the break frequency.) For a = 5, sweep the generator upward in frequency until the measured amplitude is 5 times that of its unchanging, low-frequency value. Then substitute a and this frequency, f_a, into the above equation for L_s.

Several factors which limit the usefulness of this method are:

1. Resistance (distinct from reactance) increases with frequency due to the skin effect, the self-induced eddy-current effect within a conductor. Above a frequency at which the effective depth of current penetration into the conductive material is reduced, resistance increases. Thin-film resistors suffer this effect at relatively higher frequencies than bulk resistors because skin depth exceeds their thin conductive dimension at lower frequencies, causing no change in their effective resistance.



2. The source cable, if not terminated into 50 W , will set up standing waves which degrade amplitude measurement accuracy. If possible, set up the test at the generator output terminals, as shown below, but preferably not with a ´ 10 scope probe, as shown.

The better approach is to use a 50 W -terminated cable, as shown below. Even better, especially for chip resistors, is to use a GR line insertion unit, to preserve the 50 W cable environment.

3. The oscilloscope "probe" should be a 50 W coaxial cable terminated at the scope end in 50 W . (See channel 1 below.) The generator must be terminated in its source resistance at the end of a cable with the same characteristic impedance (50 W ). This termination should be at the oscilloscope input. This results in an equivalent source resistance of 25 W – still 1000 times that of the 25 mW sense resistor.

Despite these measurement precautions (which are worth knowing anyway), this frequency-response method is of limited applicability. The value of L_s in the above formula depends on R_s, which increases with frequency once the skin depth is less than the conductor radius or thickness. For 18-gauge wire, the resistance is already around several times the dc value at 100 kHz. For thin-film resistors, which are becoming more common, the technique has some merit, but is still affected by R_s variation.

A test setup using the 25 mW Manganin resistor (shown above) resulted in a ´ 3 increase in amplitude at 550 kHz. That calculates to be a value of L_s of 2.4 nH – a suspiciously low value. (I expect it to be closer to 25 nH based on geometry.) It is reasonable to expect the resistance of R_s to be about 30 times larger, or 0.78 W at 550 kHz. By taking the skin effect into account, this method shows some promise. Skin-effect compensation can be calculated from the following equation, taken from Grounding and Shielding Techniques in Instrumentation, 2nd Ed., by Ralph Morrison (Wiley, 1977, p. 126):

and, for various wire gages, k is given in the following table:

Ó Dennis L. Feucht, 2000