Electrical impedance

• 1. Introduction

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  Electrical impedance is the reason your headphones sound different plugged into a phone than into a studio amplifier. It is the reason radio antennas need to be carefully tuned before they can broadcast. And it is the reason a capacitor, placed in a circuit, will block a steady battery current entirely while letting a high-frequency signal sail right through.

  At its core, impedance is the opposition a circuit presents to alternating current. Unlike plain resistance, which simply slows current down, impedance has two parts working together: resistance and something called reactance. One part deals with energy lost as heat; the other deals with energy temporarily stored and returned by coils and capacitors. Together they produce a quantity that is richer, and stranger, than resistance alone.

  To describe it properly, engineers had to reach for a branch of mathematics that, in the nineteenth century, many considered purely abstract: complex numbers. The story of how impedance was identified, named, and turned into a working tool for electrical engineers spans several decades and involves a handful of brilliant, overlapping contributions. Oliver Heaviside coined the term in July 1886. Charles Proteus Steinmetz made it useful for all AC circuits just a few years later. And the mathematical machinery that makes it tractable is still taught in every electrical engineering course today.

• 2. Resistance and Reactance

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  Resistance is the simpler of the two components of impedance. In a purely resistive circuit, voltage and current rise and fall in perfect step with each other. There is no lag, no lead, no shift in time. All the opposition a resistor offers is expressed as a straightforward ratio between voltage amplitude and current amplitude.

  Reactance is where impedance becomes genuinely strange. A capacitor consists of two conductors separated by an insulating material called a dielectric. When a steady voltage is applied, charge builds up on one side until the electric field exactly balances the applied voltage and current drops to zero. A capacitor, in other words, blocks direct current entirely. Under an alternating signal, though, the voltage keeps reversing before the charge has time to fully accumulate. The higher the frequency, the less charge builds up, and the smaller the opposition. This is capacitive reactance, and it is inversely proportional to frequency.

  An inductor behaves in the opposite way. It is a coil of wire, and when current flows through it, a magnetic field builds up around the coil. By Faraday's law of electromagnetic induction, any change in that magnetic flux generates a voltage that opposes the change in current. A steady direct current produces no change in flux, so an inductor presents almost no opposition to DC. But as frequency rises, the rate of change of current rises with it, the opposing voltage grows, and the inductor's reactance increases with frequency.

  Because capacitive reactance and inductive reactance behave as opposites, they can, under the right conditions, cancel each other out. The total reactance in a circuit is the sum of the two, and when they balance, the circuit hits a resonant condition that has practical consequences in everything from radio tuners to power supplies.

• 3. Complex Numbers Enter the Circuit

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  Johann Victor Wietlisbach was among the first to use complex numbers in circuit analysis, doing so in 1879 while studying the Maxwell bridge. Wietlisbach avoided differential equations by expressing alternating currents and voltages as exponential functions with imaginary exponents. He found that the required voltage could be obtained by multiplying the current by a complex number, though he did not yet identify that complex number as a general property worth naming.

  The naming fell to Oliver Heaviside, who coined the word "impedance" in July 1886. Heaviside was working on his operational calculus and recognised that what he had been calling the "resistance operator" was in fact a complex number. The following year, he showed that this operator obeyed an AC version of Ohm's law, the fundamental rule that voltage equals current times resistance.

  Arthur Kennelly published an influential paper on the subject in 1893. He arrived at the complex number representation more directly than Wietlisbach had. Kennelly drew on a graphical method developed by John Ambrose Fleming in 1889, in which resistance, reactance, and impedance were shown as the three sides of a right-angled triangle. Kennelly realised this triangle map was directly analogous to the Argand diagram used to plot complex numbers, which meant impedances could be added as vectors and problems could be solved with algebra.

  Later that same year, Charles Proteus Steinmetz extended Kennelly's work to all AC circuits. Steinmetz went further: he applied the complex representation not just to impedance but to voltages and currents themselves. That step allowed him to express AC equivalents of Ohm's law and Kirchhoff's laws, the standard rules governing how voltages and currents distribute themselves across a network. Steinmetz's formulation spread the technique widely among practicing engineers.

• 4. Measuring What You Cannot See

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  Measuring impedance is a practical challenge because the quantity is not directly visible on a meter. It requires measuring both the magnitude of voltage and current and the phase difference between them, which is the time offset between when each wave peaks.

  One classical approach uses a bridge method similar to the Wheatstone bridge used for DC resistance. A calibrated reference impedance is adjusted until it exactly balances the unknown impedance under test, a condition detected by the absence of a signal across the bridge. This class of measurement has been a standard technique in radio technology and related fields for well over a century.

  A more modern instrument, the LCR meter, directly measures a component's inductance, capacitance, and resistance. From those three values, the impedance at any desired frequency can be calculated. A related method applies a sinusoidal voltage in series with a known resistor and measures the voltage across each element. Sweeping across a range of frequencies using this approach yields the full impedance profile as a function of frequency.

  For rapid characterisation, engineers can use an impulse response combined with the fast Fourier transform. A short impulse contains energy spread across a wide range of frequencies simultaneously. Transforming the measured response reveals how the device behaves across all those frequencies in a single measurement pass.

  In some applications, impedance itself is not the most convenient quantity to display. In a radio antenna, for example, the standing wave ratio or the reflection coefficient may communicate the relevant information more directly than the raw impedance value in ohms. Impedance measurement in power electronic devices adds another layer of difficulty: the device may need to be supplied with operating power at the same time as the measurement is being taken.

• 5. Extending Impedance Beyond Steady State

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  The definitions built on sinusoidal signals apply cleanly only when the circuit has settled into a steady rhythm. The moment a source is switched on or off, that steady-state assumption breaks down. A purely sinusoidal analysis cannot capture what happens in that transitional window.

  To handle transient behaviour, the concept of impedance can be extended using the Laplace transform. Instead of representing signals as functions of ordinary frequency, the Laplace approach uses a complex frequency parameter, commonly denoted by the letter s. In this more general notation, a resistor's impedance is still just its resistance. An inductor's impedance becomes the product of s and its inductance. A capacitor's impedance becomes the reciprocal of the product of s and its capacitance.

  When the circuit settles to a steady AC state, the Laplace parameter s reduces to a purely imaginary value, and the general expressions collapse back to the familiar sinusoidal formulas. For DC circuits specifically, the frequency is zero, which means every inductor in the circuit behaves as a short circuit and every capacitor behaves as an open circuit.

  This generalisation matters for any circuit that must respond to changing conditions rather than just a continuous tone. The varicap diodes used in radio tuners, for instance, exhibit a voltage-to-current ratio that looks linear and time-invariant when signals are small and the observation window is short. Engineers treat them as having a time-varying impedance, but the source makes clear this is an approximation. Over large signal swings or long time windows, the relationship stops being linear and the impedance description loses its validity.

• 6. Networks, Matrices, and Admittance

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  For circuits with only two terminals, impedance is a single number. For circuits with multiple ports, that single number is inadequate. The voltages and currents at each port are still related to one another in a linear way, but that relationship must be expressed as an impedance matrix, a rectangular array of complex numbers capturing how a signal at one port influences every other port.

  Within single networks, the rules for combining impedances in series and parallel are the same as those for combining resistances, except that the values involved are complex numbers. Components in series share the same current, and their impedances add directly. Components in parallel share the same voltage, and their inverse impedances, rather than the impedances themselves, add together.

  The reciprocal of impedance has its own name: admittance. Where impedance measures opposition, admittance measures how readily a circuit allows current to flow. Its SI unit is the siemens. Admittance is often the more natural quantity when dealing with parallel circuits, because admittances in parallel simply add, just as resistances in series do.

  The instruments built to characterise impedance directly are called impedance analyzers. They combine the measurement of voltage, current, and phase into a single readout, making it possible to profile a component's behaviour across a sweep of frequencies. The parallel LC tank circuit is a textbook example of what such a measurement reveals: there exists a specific resonant angular frequency at which the circuit's net impedance drops to zero, a point that corresponds to the circuit oscillating at its natural frequency.