Electric/Magnetic

# Electrical Impedance-What Lies Beneath

October 1, 2005

### Electrical Impedance—What Lies Beneath

You all remember Ohm's law and the definition of electrical resistance as the ability of a circuit to resist the flow of electrical current. Although useful, Ohm's law applies to only one circuit element and assumes it to be an ideal resistor. Real-world circuit elements are more complex and exhibit resistive, capacitive, and inductive behavior that together define its impedance.

## Impedance

Impedance differs from resistance in two significant aspects. First, it's an alternating current (AC) phenomenon, and second, it is usually specified at a particular frequency. If you measure impedance at a different frequency, you obtain a different impedance value. By measuring impedance across a number of frequencies, you can extract valuable data about an element. This is the basis of impedance spectroscopy, and it is the fundamental concept underlying many industrial, instrumentation, and automotive sensors.

The impedance of an electric element can consist of resistors, capacitors, or inductors—or more typically, a combination of these three units. You can model this effect using imaginary impedance values. Inductors have an impedance of jvL, and capacitors of 1 / jvC. In this case, j is the imaginary unit, and v is the angular frequency of the signal. The impedances of these components combine using complex number arithmetic. The imaginary component of an impedance is called reactance, and in general, Z = R + jX, where X is reactance and Z denotes impedance. When exposed to a signal of increasing frequency capacitive reactance, X C reduces, and inductive reactance X L increases, leading to changes in overall impedance as a function of frequency. The impedance of a pure resistor will not change with frequency.

## How to Analyze Impedance

To examine the impedance of an element when swept at different frequencies, you usually have to examine the response signal in either the time or the frequency domain. Analog signal analysis techniques, such as AC-coupled bridges, were commonly used to examine the signal in the frequency domain, but the advent of high-performance A/D converters has led to data collection in the time domain with subsequent conversion to the frequency domain.

You can use a number of integral transforms to convert data into the frequency domain, with Fourier analysis being a common approach. This technique takes a time-series representation of a signal and applies an integral transform to map the representation into its frequency spectrum. You can use the technique to provide a mathematical description of the relationship between any two signals. In impedance analysis, the relationship between the excitation current (input to the element) and the voltage response (output from the element) is of interest. If a system is linear, the ratio of the Fourier transforms of the measured time-domain voltage and current is equal to the impedance, and it is expressed as a complex number. The real and imaginary components of the resulting complex number form a key piece of the subsequent data analysis:

where:

E = system voltage

I = system current

t = time domain parameter

and

By converting the complex number to its polar form, you can determine both the magnitude of the response signal and the phase relative to the excitation signal at a particular frequency:

where:

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