created at July 2, 2021

Serial RC Circuit

A serial RC circuit contains a resistor and a capacitor connected in serial.

The alternating voltage U is applied to the circuit, so the alternating current is flowing through the branch:

i = I_{m}\cdot sin(\omega\cdot t)\quad (1)

The resistor's voltage:

u_{R} = I_{m}\cdot R\cdot sin(\omega\cdot t)

The capacitor's voltage lags behind the current by phase angle π/2:

u_{C} = I_{m}\cdot X_{C}\cdot sin \left ( \omega\cdot t - \frac{\pi}{2} \right )

where X_C = 1/ω·C - capacitive resistance.

The vector diagram for all voltages (voltage triangle):

After we divide all voltages by current I, we get the resistance triangle:

where R - the active resistance, X_C - the capacitive resistance, Z - the total resistance (or impedance). The total resistance is:

Z = \sqrt{R^{2} + X_{C}^2}

The total voltage lags behind current by phase angle φ:

\phi = -atan \left ( \frac{X_C}{R} \right )

The angle goes in clockwise direction (from current vector to voltage vector), hence it is negative here.

The power coefficient is defined as an active and reactive resistance ratio:

cos\phi = \frac{R}{Z}

The effective current I, flowing through the circuit:

I = \frac{U}{Z}

The active power:

P = I^{2}\cdot R

The reactive power:

Q_{C} = X_C\cdot I^{2}

The total power:

S = \sqrt{P^2 + Q_C^2}