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Consider the circuit diagram, containing an AC sinusoidal voltage source and an unknown passive element (K). The element K will be a *resistance*, only if the voltage across it and the current flowing through it, are in phase with each other.

Let the equation for alternating voltage is

$$\mathrm{v=V_{m}\sin\omega\:t\:\:\:...(1)}$$

As a result of this voltage, an alternating current i will flow through the element. Now, the applied voltage has to overcome the drop in the element only, i.e.

$$\mathrm{v=i\times\:k}$$

$$\mathrm{\Rightarrow\:i=\frac{v}{k}=\frac{V_{m}\sin\omega\:t}{k}\:\:\:...(2)}$$

The value of current will be maximum, when *sin(ωt) = 1*.

$$\mathrm{\therefore\:I_{m}=\frac{V_{m}}{K}}$$

Thus, the equation (2) becomes,

$$\mathrm{i=I_{m}\sin\omega\:t\:\:\:...(3)}$$

It is clear from the eqns. (1) and (3) that the applied voltage and resulting current are in phase with each other. Therefore, the unknown element is resistance, i.e.

$$\mathrm{K=R\:\:\:...(4)}$$

If an AC circuit containing both resistive and reactive components, then the total opposition offered by the circuit in the flow of electric current is known as *impedance of the AC circuit.*

It is denoted by letter ‘Z’ and measured in ohms (Ω).

Mathematically, the impedance is expressed as,

$$\mathrm{Impedance,Z=R+jX\:\:\:...(5)}$$

**Case 1 – Impedance of a Series R-L Circuit**

$$\mathrm{Z=R+jX_{L}=R+j\omega\:L\:\:\:...(6)}$$

**Case 2 – Impedance of a Series R-C Circuit**

$$\mathrm{Z=R+jX_{C}=R-j\frac{1}{\omega\:C}\:\:\:...(7)}$$

**Case 3 – Impedance of a Series RLC Circuit**

$$\mathrm{Z=R+j(X_{L}-X_{C})=R+j(\omega\:L-\frac{1}{\omega\:C})\:\:\:...(8)}$$

**Case 4 – Impedance in a Parallel AC Circuit**

In case of a parallel AC circuit, the impedance being given in terms of *admittance,* i.e.

$$\mathrm{Impdance=\frac{1}{Admittance}}$$

$$\mathrm{\Rightarrow\:Z=\frac{1}{Y}=\frac{1}{G+jB}\:\:\:....(9)}$$

Where,

**G**= 1/R, is called conductance**B**= 1/X, is called susceptance

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