What is Electrical Impedance?

Electrical impedance is the measure of opposition produced by circuit elements in the path of alternating current. Therefore, the electrical impedance is the parameter related to the AC circuit. The electrical impedance is usually denoted by the symbol ‘Z’ and is measured in ohms $\mathrm{(\Omega)}$.

The electrical impedance is a complex quantity, i.e. it has both magnitude and phase. The electrical impedance is basically the extended concept of resistance. It is the function of frequency of alternating current flowing in the circuit. It because the impedance comes into picture only when there is a frequency dependent element like inductor or capacitor exist in the circuit. Although, sometime we treat the resistance as impedance with a phase angle of zero degree.

When we analyze an AC circuit, we come across two terms that represent the opposition in the flow of alternating current. These are resistance and reactance. Where, the resistance opposes the current due to interatomic collisions, while the reactance is the opposition in the flow of electric current due to frequency dependent elements like inductor or capacitor. If the reactance in the circuit is due to inductor, then it is called inductive reactance, and when it is due to presence of capacitor, it is known as capacitive reactance.

In a practical AC circuit, the combined opposition due to resistance and reactance in the flow of alternating current is known as impedance. Mathematically, the impedance of an AC circuit is given by,

$$\mathrm{Z= R\pm\;jX}$$

Where, R is the resistance and X is the reactance. The inductive reactance is taken as positive and the capacitive reactance is taken as negative. The above expression is the rectangular form representation of the electrical impedance. As we can see it is a complex quantity. Thus, the magnitude of the impedance is given by,

$$\mathrm{|Z|=\sqrt{{R^2+X^2}}}$$

And, the phase angle of the impedance, also called impedance angle, is given by,

$$\mathrm{\angle Z=\theta=\tan^{-1}\left(\begin{array}{c}X\ R\end{array}\right)}$$

The electrical impedance may also be represented in the polar form as,

$$\mathrm{Z=|Z|\angle\theta}$$

Now, we shall learn about the electrical impedance and its mathematical expression for different AC electric circuit.

Electrical Impedance of Series RL Circuit

A series RL circuit is one which consists a resistor connected in series with an inductor. Then, if the resistance is given by R and the inductive reactance is given by $\mathrm{X_{L}}$, then the impedance of the circuit is given by,

$$\mathrm{Z=R+jX_{L}=R+jwL}$$

The magnitude of the impedance of the series RL circuit is,

$$\mathrm{|Z|=\sqrt{{R^2+w^2L^2}}}$$

Electrical Impedance of Parallel RL Circuit

A parallel RL circuit is one in which the resistor and inductor are connected in parallel. In case of parallel RL circuit, the reciprocal of the electrical impedance is equal to the sum of the reciprocal of the resistance and the reciprocal of inductive reactance, i.e

$$\mathrm{\frac{1}{Z}=\frac{1}{R}+\frac{1}{jX_{L}}=\frac{1}{R}-j\frac{1}{wL}}$$

Here, the magnitude of the impedance is given by,

$$\mathrm{|Z|=\left(\sqrt{\frac{1}{R^2}+\frac{1}{w^2L^2}}\right)^{-1}}$$

Electrical Impedance of Series RC Circuit

A series RC circuit is one in which a resistor of resistance R is connected in series with a capacitor of capacitance C and capacitive reactance X_C. For this circuit, the expression of the electrical impedance is given by,

$$\mathrm{Z=R+jX_{c}=R-j\frac{1}{wC}}$$

The magnitude of the impedance of the series RC circuit is,

$$\mathrm{|Z|=\sqrt{{R^2}+\frac{1}{w^2C^2}}}$$

Electrical Impedance of Parallel RC Circuit

When a resistor of resistance R is connected in parallel with a capacitor of capacitance C and capacitive reactance X_C. Then, the impedance of this circuit is given by,

$$\mathrm{\frac{1}{Z}=\frac{1}{R}+j\frac{1}{X_{C}}=\frac{1}{R}+jwC}$$

Thus, the magnitude of the impedance of the parallel RC circuit is,

$$\mathrm{|Z|=\left(\sqrt{\frac{1}{R^2}+jwC}\right)^{-1}}$$

Electrical Impedance of Series RLC Circuit

The AC circuit in which a resistor of resistance R, an inductor of reactance X_L and a capacitor of reactance X_C are connected in series, it is called a series RLC circuit. The expression of the electrical impedance of the series RLC circuit is given by,

$$\mathrm{Z=R+j(X_{L}-X_{C})=R+j( wL-\frac{1}{wC})}$$

The magnitude of impedance of the series RLC circuit is,

$$\mathrm{|Z|=\left(\sqrt{{R^2}+(wL-\frac{1}{wC}}\right)^{-2}}$$

Electrical Impedance of Parallel RLC Circuit

A parallel RLC circuit is one in which a resistor, an inductor and a capacitor are connected in parallel is known as a parallel RLC circuit. The impedance of a parallel RLC circuit is given by the following expression,

$$\mathrm{\frac{1}{Z}=\frac{1}{R}+j\frac{1}{X}}$$

Where,

$$\mathrm{X=wL\lVert(\frac{1}{wC})}$$

Conclusion

In this article, we discussed about the term electrical impedance and its mathematical expression for different circuit configurations. From the above discussion, it is clear that the electrical impedance is a cumulative effect of resistance and reactance present in an AC circuit. Where, the reactance is the opposition in the flow of alternating current due to presence of inductor or capacitor in the circuit.