Electrical impedance
In 1950, Slurzberg and Osterheld published a textbook that defined impedance as the opposition to alternating current presented by resistance and reactance. This definition marked a shift from simple DC resistance to a complex ratio of voltage to current in AC circuits. The concept extends beyond magnitude alone to include phase differences between voltage and current signals. Engineers use this ratio to analyze how circuits behave when driven by sinusoidal voltages at specific frequencies. The unit for this measurement is the ohm, which remains consistent with standard resistance units despite the added complexity. A circuit element's impedance depends heavily on the frequency of the applied voltage source. Without understanding this frequency dependence, engineers cannot accurately predict circuit performance under varying conditions.
Johann Victor Wietlisbach utilized complex numbers in circuit analysis during 1879 while working on the Maxwell bridge. He avoided differential equations by expressing AC currents and voltages as exponential functions containing imaginary exponents. Although he multiplied current by a complex number to find required voltage, he did not identify this value as a general parameter. Oliver Heaviside coined the term impedance in July 1886 within his operational calculus framework. He recognized that the resistance operator was actually a complex number representing an AC equivalent to Ohm's law. Arthur Kennelly published an influential paper on impedance in 1893 using a more direct method than Wietlisbach. Kennelly followed graphical representations developed by John Ambrose Fleming in 1889 showing resistance, reactance, and impedance as sides of a right angle triangle. Charles Proteus Steinmetz generalized Kennelly's work to all AC circuits later that same year. Steinmetz represented impedances, voltages, and currents by complex numbers to express AC equivalents of DC laws like Kirchhoff's laws. His work spread the technique widely among engineers throughout the early twentieth century.
Engineers represent impedance as a complex quantity Z with both magnitude and phase characteristics. The polar form captures these traits as Z equals r times e to the power of j theta. Here r represents the ratio of voltage difference amplitude to current amplitude while theta gives the phase difference between them. Electrical engineering uses the letter j for the imaginary unit instead of i to avoid confusion with electric current I. Cartesian form defines impedance as R plus jX where R is resistance and X is reactance. Adding or subtracting impedances becomes easier when using the cartesian form. Multiplying or dividing quantities simplifies calculations if the polar form is used first. Circuit analysis often requires converting between forms multiple times during a single calculation. Conversion follows standard rules for complex numbers found in mathematics textbooks. Phasors simplify computations involving sinusoids by reducing differential equation problems to algebraic ones. This method allows engineers to analyze steady state behavior without solving time domain equations repeatedly.
An ideal resistor exhibits purely real resistive impedance where voltage and current waveforms remain proportional and in phase. Inductors and capacitors possess purely imaginary reactive impedance that varies inversely with frequency changes. An inductor's impedance increases as frequency rises because its back electromotive force opposes changing currents. A capacitor's impedance decreases as frequency rises since less charge accumulates before potential differences change sign. In an inductor, current lags voltage by ninety degrees while in a capacitor, current leads voltage by the same amount. These components alternately absorb energy from the circuit and then return it to the system. Pure reactance does not dissipate power but instead stores and releases energy cyclically. The total reactance combines capacitive and inductive effects into a single value X equal to XL minus XC. This combination determines how much opposition exists at any given signal frequency within the network.
Calculating total impedance for simple networks uses series and parallel rules identical to those for resistance except numbers are complex. Components connected in series share the same current so their total impedance equals the sum of individual component impedances. Z total becomes Z1 plus Z2 plus Z3 and so on through every element in the chain. Parallel connections maintain the same voltage across each element so inverse total impedance equals the sum of inverses of component impedances. For two elements this ratio simplifies to Z1 times Z2 divided by Z1 plus Z2. Equivalent impedance can be calculated in terms of equivalent series resistance and reactance values. General cases require additional transforms beyond basic series or parallel combinations to handle multiple ports. Engineers apply these algebraic procedures to determine how signals propagate through complex electrical systems. The rules allow prediction of circuit behavior without solving differential equations for every new configuration.
Impedance analyzers measure the electrical impedance of devices and transmission lines at specific frequencies or over ranges. Bridge methods similar to the Wheatstone bridge adjust a calibrated reference impedance to balance out device effects. Measuring impedance requires determining magnitude of voltage and current along with phase difference between them. Applying a sinusoidal voltage to a device in series with a resistor allows calculation via complex division of measured voltages. Sweeping applied signal frequencies provides both impedance phase and magnitude data points for analysis. Fast Fourier transform combined with impulse response testing rapidly measures electrical impedance of various devices. LCR meters measure inductance, capacitance, and resistance from which impedance at any frequency can be derived directly. Standing wave ratio or reflection coefficient may prove more useful than raw impedance values in radio antenna applications. Simultaneous measurement and power provision become necessary when testing power electronic devices under operating conditions.
Up Next
Common questions
Who coined the term impedance and when did this happen?
Oliver Heaviside coined the term impedance in July 1886 within his operational calculus framework. He recognized that the resistance operator was actually a complex number representing an AC equivalent to Ohm's law.
What is the unit for electrical impedance measurement?
The unit for this measurement is the ohm, which remains consistent with standard resistance units despite the added complexity. Engineers use this ratio to analyze how circuits behave when driven by sinusoidal voltages at specific frequencies.
How does frequency affect the impedance of an inductor?
An inductor's impedance increases as frequency rises because its back electromotive force opposes changing currents. In an inductor, current lags voltage by ninety degrees while in a capacitor, current leads voltage by the same amount.
When did Charles Proteus Steinmetz generalize work on impedance to all AC circuits?
Charles Proteus Steinmetz generalized Kennelly's work to all AC circuits later that same year after Arthur Kennelly published an influential paper on impedance in 1893. Steinmetz represented impedances, voltages, and currents by complex numbers to express AC equivalents of DC laws like Kirchhoff's laws.
Why do engineers use the letter j instead of i for the imaginary unit?
Electrical engineering uses the letter j for the imaginary unit instead of i to avoid confusion with electric current I. Cartesian form defines impedance as R plus jX where R is resistance and X is reactance.