Series LCR Circuit: Impedance and Phasor Relation

Calculate XL and XC if needed. Find net reactance XL - XC with sign. Compute Z using the square-root formula. Use I = V/Z. Decide the circuit nature from the sign of XL - XC. Draw phasors using current as the reference.

The phasor diagram should show I and VR along the horizontal axis, VL upward, VC downward, net reactive voltage VL - VC, resultant supply voltage V, and phase angle phi between V and I. The circuit diagram should label R, L, C, and AC source in series.

Adding all three as Z = R + XL + XC gives a larger and physically wrong impedance because it ignores phase.

A series LCR circuit contains a resistor, inductor, and capacitor in series with an AC source; its total opposition to current is impedance Z = sqrt(R^2 + (XL - XC)^2).

A series LCR circuit contains a resistor, inductor, and capacitor in series with an AC source; its total opposition to current is impedance Z = sqrt(R^2 + (XL - XC)^2). Z = sqrt(R^2 + (XL - XC)^2), tan phi = (XL - XC)/R, I = Vrms/Z. R, XL, XC, and Z are in ohm. phi is positive for inductive behaviour and negative for capacitive behaviour if current is taken as reference. Use RMS values for circuit current and power calculations. If XL is greater than XC, the circuit behaves inductively and current lags the applied voltage. If XC is greater than XL, the circuit behaves capacitively and current leads the applied voltage.

In a series LCR circuit, the same current flows through R, L, and C. The resistor voltage VR is in phase with current, the inductor voltage VL leads current by pi/2, and the capacitor voltage VC lags current by pi/2. Since VL and VC are in opposite directions in the phasor diagram, the effective reactive voltage is VL - VC. The applied voltage is the vector sum of VR and the net reactive voltage, giving V = sqrt(VR^2 + (VL - VC)^2). Dividing by current gives Z = sqrt(R^2 + (XL - XC)^2). The phase angle is tan phi = (XL - XC)/R. Z = sqrt(R^2 + (XL - XC)^2), tan phi = (XL - XC)/R, I = Vrms/Z. R, XL, XC, and Z are in ohm. phi is positive for inductive behaviour and negative for capacitive behaviour if current is taken as reference. Use RMS values for circuit current and power calculations. A series LCR circuit has R = 30 ohm, XL = 80 ohm, and XC = 40 ohm. Net reactance = XL - XC = 40 ohm. Z = sqrt(30^2 + 40^2) = sqrt(2500) = 50 ohm. If Vrms = 100 V, Irms = V/Z = 100/50 = 2 A. tan phi = 40/30 = 4/3, so the circuit is inductive and current lags voltage. Questions often ask for impedance, phase angle, current, voltage across each element, phasor diagram, or whether the circuit is inductive, capacitive, or resistive. Adding all three as Z = R + XL + XC gives a larger and physically wrong impedance because it ignores phase.