Electrical Impedance

Electrical impedance is a quantity that describes the relationship between voltage and electric current at the terminals of an electrical component or circuit. In sinusoidal steady-state operation, it is defined as the ratio of the complex voltage amplitude to the complex current amplitude. It generalizes electrical resistance by accounting for both the relative magnitudes of voltage and current and their phase difference. Impedance is usually frequency-dependent and is expressed as a complex number. (ocw.mit.edu)

Definition and mathematical representation

For a linear, time-invariant circuit driven at angular frequency $\omega=2\pi f$, voltage and current in sinusoidal steady state can be written as

where $j^2=-1$, and $\tilde V$ and $\tilde I$ are phasors containing amplitude and phase information. Electrical engineering commonly uses $j$ rather than $i$ for the imaginary unit to avoid confusion with current. Impedance is

This is the frequency-domain generalization of Ohm’s law. It does not generally imply that the instantaneous ratio $v(t)/i(t)$ is constant. Both phasors must use the same amplitude convention, such as peak values or root-mean-square values. (ocw.mit.edu)

In rectangular and polar forms,

with

Here $R=\operatorname{Re}Z$ is resistance and $X=\operatorname{Im}Z$ is reactance. The phase angle $\phi$ is the voltage phase minus the current phase. Under the $e^{j\omega t}$ convention, positive reactance is inductive and negative reactance is capacitive. The unit is the ohm, symbol $\Omega$, equivalent to one volt per ampere in the International System of Units. (keysight.com)

Some elementary texts use “impedance” and the symbol $Z$ for the magnitude alone. In complex circuit analysis, however, $Z$ normally denotes the full complex quantity, while $|Z|$ denotes its magnitude. (openstax.org)

Resistance, reactance, and energy

Resistance and reactance describe different aspects of electrical response. In a passive resistive element, electrical energy is irreversibly converted into other forms, commonly heat. Ideal reactive elements instead store energy and return it to the circuit during another part of the cycle. A capacitor stores energy in an electric field, whereas an inductor stores energy in a magnetic field. (openstax.org)

Reactive behavior produces a phase difference between voltage and current. For an ideal inductor, current lags voltage by $90^\circ$; for an ideal capacitor, current leads voltage by $90^\circ$. Neither absorbs net energy over a complete steady-state cycle, although instantaneous power alternates between absorption and return. Real components also have losses, so their impedances generally contain both real and imaginary parts. (openstax.org)

Impedance of ideal circuit elements

The elementary impedance formulas follow from the time-domain relations for resistance, inductance, and capacitance. For sinusoidal signals, taking a derivative corresponds to multiplication of the phasor by $j\omega$. This converts the differential relations into algebraic ones. (ocw.mit.edu)

An ideal resistor’s impedance is independent of frequency. An ideal inductor’s impedance magnitude increases with frequency, while an ideal capacitor’s decreases. In the zero-frequency limit associated with direct current, an ideal inductor approaches a short circuit and an ideal capacitor approaches an open circuit. These statements describe limiting steady-state behavior, not the transient response immediately after switching. (ocw.mit.edu)

Actual components depart from these models because of winding resistance, leakage, dielectric losses, and unintended inductance or capacitance. Consequently, a physical capacitor can behave inductively above its self-resonant frequency, and an inductor can exhibit capacitive behavior at sufficiently high frequencies. (keysight.com.cn)

Combining impedances and admittance

For components connected in series, with no additional coupling between them,

For components connected in parallel,

These rules have the same form as the corresponding resistance rules, but the calculations use complex arithmetic and must be performed at the same frequency. (ocw.mit.edu)

The reciprocal of impedance is admittance:

where $G$ is conductance and $B$ is susceptance. Admittance is measured in siemens and is especially convenient for parallel networks. If $Z=R+jX$, then

Thus, conductance is not generally $1/R$; that simplification applies when reactance is zero. (keysight.com)

Frequency dependence and resonance

A series resistor–inductor–capacitor circuit has impedance

At the angular frequency

the inductive and capacitive reactances cancel. The total impedance becomes purely resistive, $Z(\omega_0)=R$, and its magnitude is minimized. For a fixed-amplitude voltage source and positive resistance, the current amplitude is therefore greatest at this frequency. This is series electrical resonance. (openstax.org)

Below resonance, this circuit is predominantly capacitive; above resonance, it is predominantly inductive. Parallel resonant circuits can instead exhibit a maximum in impedance. Resonance is therefore a property of the complete network, not simply the presence of an inductor and capacitor. Frequency-dependent impedance provides the basis for tuning circuits and frequency-selective electrical filters used in signal processing. (ocw.mit.edu)

Impedance and electrical power

For sinusoidal operation, the average power absorbed by a load is

The factor $\cos\phi$ is the sinusoidal power factor. A phase difference reduces average power relative to the product of the RMS voltage and current. For a purely reactive ideal load, $\cos\phi=0$, so average power is zero even though current flows. (openstax.org)

Impedance is also central to impedance matching. For a linear source represented by an ideal voltage source in series with a fixed impedance $Z_S$, with $\operatorname{Re}Z_S>0$, maximum average power reaches an adjustable load when

The asterisk denotes complex conjugation: load resistance equals source resistance, while load reactance cancels source reactance. This maximum-power condition is distinct from maximizing efficiency or minimizing signal reflections. (eng.libretexts.org)

Transmission lines and characteristic impedance

When propagation along a conductor cannot be neglected, the system requires a transmission-line description rather than a simple lumped circuit model. A line’s characteristic impedance $Z_0$ is the voltage-to-current ratio for a single traveling wave. It is not necessarily the impedance measured at the input of a finite line; input impedance also depends on the termination, line length, and frequency. (ocw.mit.edu)

At a load $Z_L$, the voltage reflection coefficient is

When $Z_L=Z_0$, the load produces no reflected wave. A lossless line can have a real characteristic impedance even though it dissipates no energy: energy travels along the line rather than being converted into heat locally. This distinguishes characteristic impedance from the resistance of an ordinary resistor. (ocw.mit.edu)

Measurement and applications

Impedance measurement determines both the amplitude ratio and the phase difference between an applied electrical signal and the response. Instruments include LCR meters, impedance analyzers, bridge circuits, and network analyzers. The appropriate method depends on frequency, impedance range, and required accuracy. Test leads and fixtures contribute their own impedance, making calibration and compensation important. (keysight.com.cn)

Frequency sweeps are commonly displayed in two forms:

• Bode plots: impedance magnitude and phase plotted against frequency.
• Nyquist plots: imaginary impedance plotted against real impedance; in electrochemistry, the vertical axis commonly represents $-\operatorname{Im}Z$.

Each point in a Nyquist plot corresponds to a frequency, but frequency is not itself an axis. (gamry.com)

In electrochemical impedance spectroscopy, a small alternating perturbation probes an electrochemical cell around an operating condition. Its frequency-dependent response can reveal contributions from electrolyte resistance, interfacial capacitance, charge-transfer processes, and diffusion. Equivalent-circuit models are used to interpret these responses in studies of batteries, corrosion, coatings, and electrode reactions. (gamry.com)

Scope and limitations

A single frequency-dependent impedance function assumes a linear, time-invariant relationship between voltage and current. Nonlinear devices may generate harmonics, while changing operating conditions can make the response vary during measurement. Small-signal impedance can approximate a nonlinear system near a specified operating point, but it should not automatically be extended to large disturbances. (arxiv.org)

Equivalent-circuit interpretation also has limits. Different circuit models can fit similar impedance spectra, and a successful numerical fit does not by itself establish a unique physical mechanism. Measurement bandwidth, noise, system drift, and parameter correlations can restrict what can be inferred from the data. Physical justification and independent evidence are therefore important when assigning meaning to fitted circuit elements. (gamry.com)

References

1. ES.1803 S24: Reading: Topic Enrichment: Complex Impedanceocw.mit.edu
2. E4980A Operation Manualkeysight.com
3. sss_phsor_impdce.pdf | Introduction to Electronics, Signals, and Measurementocw.mit.edu
4. Basics of Electrochemical Impedance Spectroscopygamry.com
5. 03SCF11 text: Impedanceocw.mit.edu
6. 4 Power in an AC Circuit - University Physics Volume 2openstax.org
7. Impedance Measurement Handbookkeysight.com.cn
8. 5 Resonance in an AC Circuit - University Physics Volume 2openstax.org
9. 6: Reflections at Interfaceseng.libretexts.org
10. 013 Electromagnetics and Applications, Course Notesocw.mit.edu
11. Equivalent Circuit Modeling Using the Gamry Electrochemical Impedance Spectroscopy Softwaregamry.com
12. Electrochemical impedance spectroscopy beyond linearity and stationarity - a critical reviewarxiv.org