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    Serial RC Circuit

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

    scheme

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

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

    The resistor's voltage:

    uR=Im⋅R⋅sin(ω⋅t)u_{R} = I_{m}\cdot R\cdot sin(\omega\cdot t)

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

    uC=Im⋅XC⋅sin(ω⋅t−π2)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):

    voltage triangle

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

    resistance triangle

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

    Z=R2+XC2Z = \sqrt{R^{2} + X_{C}^2}

    The total voltage lags behind current by phase angle φ:

    ϕ=−atan(XCR)\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:

    cosϕ=RZcos\phi = \frac{R}{Z}

    The effective current I, flowing through the circuit:

    I=UZI = \frac{U}{Z}

    The active power:

    P=I2⋅RP = I^{2}\cdot R

    The reactive power:

    QC=XC⋅I2Q_{C} = X_C\cdot I^{2}

    The total power:

    S=P2+QC2S = \sqrt{P^2 + Q_C^2}