# Impedance

In electrical engineering, impedance is a measure for the manner and degree a component resists the flow of electrical current if a given voltage is applied. It is denoted by the symbol Z and is measured in ohms. Impedance differs from simple resistance in that it takes into account possible phase offset (see below). The concept of impedance also has significance outside of electrical systems in the discussion of any driven oscillator. In July of 1886, Oliver Heaviside coined this term.

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## Constant signals

If the applied signal is constant, such as a DC signal, the circuit is in a steady-state. In this state, capacitors are considered 'open' and inductors are considered 'closed'. This makes sense because capacitors in steady-state are modeled as two plates separated by a gap and inductors are a coil of wire (which has negligible resistance); the former allows for no current flow, whereas the latter does. The impedance is then due to resistors alone and is a real number equal to the component's resistance R.

The notion of impedance remains useful in such a circuit, in order to study what happens at the instant when the constant voltage is switched on or off: generally, inductors cause the change in current to be gradual, while capacitors can cause large peaks in current.

## Sine function signals

If the applied signal is a sine function, or a sum of many such functions, then capacitors and inductors can produce a phase shift (time delay) from the source. This effect can be modelled by assigning those components imaginary impedance values. Inductors have an impedance of [itex]j\omega L[itex] and capacitors [itex]1/j\omega C[itex], where j is the imaginary unit (instead of i, to avoid confusion with current) and ω is the angular frequency of the signal. The impedances of components then combine according to the same formulas used for resistances, using complex number arithmetic.

The imaginary component of an impedance is called reactance, X. In general,

[itex]Z=R+jX \,[itex]

Note that the reactance depends on the frequency f of the applied voltage: the higher the frequency, the smaller the capacitive reactance XC and the larger the inductive reactance XL.

## Fixed frequency signals

If the applied voltage is periodically changing with a fixed frequency f, according to a sine curve, it is represented as the real part of a function of the form:

[itex]u(t)=ue^{2\pi jft} \,[itex]

where

u is a complex number that encodes the phase and amplitude (see Euler's formula).

If the current is represented in an analogous manner as the real value of a function i(t), then the relation between current and voltage is given by

[itex]Z=\frac {u(t)} {i(t)} \,[itex]

an equation quite similar to Ohm's law.

## Variable frequency signals

If the voltage is not a sine curve of fixed frequency, then one first has to perform Fourier analysis to find the signal components at the various frequencies. Each one is then represented as the real part of a complex function as above and divided by the impedance at the respective frequency. Adding the resulting current components yields a function i(t) whose real part is the current.

If the internal structure of a component is known, its impedance can be computed using the same laws that are used for resistances: the total impedance of subcomponents connected in series is the sum of the subcomponents' impedances; the reciprocal of the total impedance of subcomponents connected in parallel is the sum of the reciprocals of the subcomponents' impedances. These simple rules are the main reason for using the formalism of complex numbers.

## Magnitude

Often it is enough to know only the magnitude of the impedance:

[itex]\left|Z\right|=\sqrt{R^2+X^2} \,[itex]

It is equal to the ratio of RMS voltage (VRMS) to RMS current (IRMS):

[itex]\left|Z\right|=\frac {V_{RMS}} {I_{RMS}}[itex]

The word "impedance" is often used for this magnitude; it is however important to realize that in order to compute this magnitude, one first computes the complex impedance as explained above and then takes the magnitude of the result. There are no simple rules that allow one to compute |Z| directly.

## Matched impedances

When fitting components together to carry electromagnetic signals, it is important to match impedance, which can be achieved with various matching devices. Failing to do so is known as impedance mismatch and results in signal loss and reflections. The existence of reflections allows the use of a time-domain reflectometer to locate mismatches in a transmission system.

For example, a conventional radio frequency antenna for carrying broadcast television in North America was standardized to 300 ohms, using balanced, unshielded, flat wiring. However cable television systems introduced the use of 75 ohm unbalanced, shielded, circular wiring, which could not be plugged into most TV sets of the era. To use the newer wiring on an older TV, small devices known as baluns were widely available. Today most TVs simply standardize on 75-ohm feeds instead.

## Inverse quantities

The reciprocal of a non-reactive resistance is called conductance. Similarly, the reciprocal of an impedance is called admittance. The conductance is the real part of the admittance, and the imaginary part is called the susceptance. Conductance and susceptance are not the reciprocals of resistance and reactance in general, but only for impedances that are purely resistive or purely reactive.

## Acoustic impedance & data-transfer impedance

In complete analogy to the electrical impedance discussed here, one also defines acoustic impedance, a complex number which describes how a medium absorbs sound by relating the amplitude and phase of an applied sound pressure to the amplitude and phase of the resulting sound flux.

Another analogous coinage is the use of impedance by computer programmers to describe how easy or difficult it is to pass data and flow of control between parts of a system, commonly ones written in different languages. The common usage is to describe two programs or languages/environments as having a low or high 'impedance mismatch'.

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