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When confronted with projects that require impedance analysis, the engineer has no better friend than Microsoft Excel. Here, I'll show how to make sense of impedance magnitude and phase angle measurements and how Excel can be used to generate a Nyquist plot like the one shown above. Note that measuring the complex impedance of a loudspeaker system, across a range of frequencies, requires dedicated measurement hardware. This brief doesn't consider how to measure the impedance but assumes that frequency dependent impedance magnitude and phase angle data is available for analysis.

Where impedance analysis has value is in modeling loudspeaker system behavior. For instance, a simulation model of a loudspeaker system can be assessed for accuracy by comparing a Nyquist plot of the simulated response to the same measured experimentally.

Audio amplifiers are AC voltage sources. They provide a voltage across the loudspeaker terminals and the loudspeaker responds by drawing current from the amplifier in proportion to the magnitude of the applied voltage. The amplitude of the sourced voltage should be "stiff" meaning the voltage amplitude does not change regardless of the load presented to the terminals. Stating that a bit differently, the impedance presented to the amplifier by the load must be significantly larger than the internal source impedance of the amplifier. For reactive loads that store energy such as inductors, capacitors, loudspeakers and crossover networks, the instantaneous current amplitude will change as a function of frequency as does the phase relationship between the applied voltage and the current provided. The relationship between voltage sourced and current drawn is determined by the impedance. The current can either lead or lag the voltage and the magnitude of the lead or lag is measured by the phase angle. Referenced to voltage, a positive phase angle indicates an inductive reactance whilst a negative phase angle indicates a capacitive reactance.

Here's an example. The plot shown below is a simulation of an audio amplifier output with a reactive load across its output terminals. The black trace is a 50Hz sine wave with an 18V amplitude applied to the load. The current waveform provided by the amplifier is shown in blue. After a few milliseconds of being turned on, the current waveform stabilizes to lead the voltage by 2.15ms. Given that it takes 20ms for a 50Hz sine wave to sweep through a single cycle (=360°), a 2.15ms leading current waveform corresponds to approximately a +39° (=+0.69 radians) phase angle between the two when referenced to the voltage.

where

In the plot shown above, I'll show how the impedance of the load at 50Hz is determined. But first, the Nyquist plot is explained.

Let's restate the exponential form of impedance:

.

Using the Euler relationship, the exponential function of phase angle can be expanded:

and, when substituted back into the exponential form and multiplied by I

which demonstrates the exponential form of the impedance consists of the sum of a real and imaginary number. Thus, by observation and comparison to the rectangular form:

and

.

Thus, the rectangular form of the impedance is equal to:

or

and that's it for the equations.

Consider the

For example, a reactance of -

In reality, capacitors also present a resistance that's independent of frequency but let's ignore that.

In the second plot, the magnitude vector is capacitive and has a magnitude of 17 Ohm, the resistance is 15.9 Ohm and the reactance is -

Now consider the impedance of the amplifier simulation shown earlier. Below is the simulated Nyquist plot of the impedance across the amplifier terminals between 50 and 1000Hz. At 50Hz the resistive part is 11.2 Ohm and the reactive part is +

With

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