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 XC = 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, XC - 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:

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