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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.

If the applied voltage is constant, capacitors act like insulators and inductors act like conductors; the impedance is then due to resistors alone and is a real number equal to the component's resistance R.

If the applied voltage is changing over time (as in an AC circuit), then the component may affect both the phase and the amplitude of the current, due to inductors and capacitors inside the component. In this case, the impedance is a complex number (this is a mathematically convenient way of describing the amplitude ratio and the phase difference together in a single number). It is composed of the resistance R, the inductive reactance XL and the capacitive reactance XC according to the formula

<math>Z=R+j(X_L-X_C)</math>

where j is the imaginary unit, the square root of -1. Inductive reactance and capacitive reactance can be lumped together in a single quantity called reactance, X = XL - XC, so that we have

<math>X=R+jX.</math>

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

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

<math>u(t)=ue^{2\pi jft}</math>
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
<math>Z=u(t)/i(t),</math>
an equation quite similar to Ohm's law.

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.

The notion of impedance can be useful even when the voltage/current is normally constant (as in many DC circuits), 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.

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.

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

<math>\left|Z\right|=\sqrt{R^2+X^2}.</math>
It is equal to the ratio of RMS voltage (VRMS) to RMS current (IRMS):
<math>\left|Z\right|=V_{RMS}/I_{RMS}.</math>

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.

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.

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.

Further reading:


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.



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