Q.Define the impedance of an AC circuit. For a series LR circuit connected to an AC source, derive the expression for the impedance $Z$ and the phase angle $\phi$ by which the current lags the voltage.

Defining impedance. Impedance $Z$ generalises resistance/reactance to a circuit with more than one element in series, defined through the Ohm's-law-like relation $V=IZ$ between rms (or peak) voltage and current, measured in ohms.

Setting up the LR phasor diagram. With the shared current $I$ as reference, $V_R=IR$ is in phase with $I$, while $V_L=IX_L$ leads $I$ by $90^\circ$, so its phasor is drawn perpendicular to $V_R$'s.

Adding the phasors. The resultant voltage is the hypotenuse of the right triangle formed by $V_R$ and $V_L$:

Comparing with $V=IZ$:

Phase angle. From the same triangle, …

Define impedance via $V=IZ$, draw the LR phasor diagram with $V_R$ and $V_L$ at right angles, then read off $ …

• Adding $R$ and $X_L$ directly (arithmetically) instead of combining them as perpendicular phasors via the Pythagorean theorem. …