Forced sinusoidal regime and complex impedance

RLC circuit driven by a sinusoidal source

The series RLC circuit is now driven by a voltage generator e(t) = Emcos(omegat). In the steady state (established regime), the current is written i(t) = Imcos(omegat - phi): the same angular frequency as the excitation, but the amplitude Im and phase phi depend on omega.

Complex impedance

In complex notation, each component has an impedance:

• Resistor: Z_R = R
• Coil: Z_L = jLomega
• Capacitor: Z_C = 1/(jComega) = -j/(C*omega)

The total impedance of the series circuit is the sum:

Z = R + j*(Lomega - 1/(Comega))

The modulus |Z| = sqrt(R^2 + (Lomega - 1/(Comega))^2) relates the amplitudes: Im = Em/|Z|.

Current resonance

The current is maximal when |Z| is minimal, that is, when the imaginary part of Z vanishes:

Lomega - 1/(Comega) = 0 -> omega = omega0 = 1/sqrt(L*C)

At resonance, Z = R (purely resistive), the current is in phase with the supply voltage, and Im = Em/R, the absolute maximum value.

Example

With L = 1 H, C = 1 uF, R = 100 ohms, Em = 10 V: omega0 = 1000 rad/s. At this angular frequency, Im = 10/100 = 0.1 A, whereas at omega = 100 rad/s, |Z| is much larger and Im is markedly lower.

Common pitfall

Do not forget that current resonance always occurs exactly at omega0, regardless of R: R only affects the height and width of the resonance peak, never its position.