Using phasor diagram for a series LCR circuit, obtain an expression for impedance. Define admittance.

Answer: \(Z = \sqrt{R^2 + (X_L - X_C)^2}\); admittance \(Y = 1/Z\).

A resistor \(R\), inductor \(L\) and capacitor \(C\) are connected in series with an a.c. source \(e = e_0\sin\omega t\). The same current \(i = i_0\sin(\omega t + \phi)\) flows through all three. Let the peak voltages be:

• \(V_R = i_0R\), in phase with the current;
• \(V_L = i_0X_L\) (\(X_L = \omega L\)), leading the current by \(\pi/2\);
• \(V_C = i_0X_C\) (\(X_C = 1/\omega C\)), lagging the current by \(\pi/2\).

Phasor diagram: [Figure: current phasor \(i_0\) along OX. \(V_R\) along OX; \(V_L\) vertically upward along OY; \(V_C\) vertically downward. Taking \(V_L \gt V_C\), the resultant of \(V_L\) and \(V_C\) is \((V_L - V_C)\) upward. The resultant of \(V_R\) and \((V_L - V_C)\) is the source phasor \(e_0\), making angle \(\phi\) with the current phasor.]

Since \(V_R\) and \((V_L - V_C)\) are perpendicular,

\(Z\) is the impedance of the circuit (unit ohm). The phase angle is given by \(\tan\phi = \dfrac{X_L - X_C}{R}\).

Admittance: The reciprocal of impedance is called admittance, \(Y = \dfrac{1}{Z}\). Its SI unit is the siemens (S) or mho.