created at July 2, 2021

Serial RLC Circuit

A serial RLC circuit contains a resistor, a coil, 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=Imsin(ωt)(1)i = I_{m}\cdot sin(\omega\cdot t)\quad (1)

The resistor's voltage:

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

The voltage on the inductive part of the circuit outstrips the current by phase angle π/2:

uL=ImXLsin(ωt+π2)u_{L} = I_{m}\cdot X_{L}\cdot sin \left ( \omega\cdot t + \frac{\pi}{2} \right )

where XL = ω·L - inductive resistance.

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

uC=ImXCsin(ωtπ2)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, XL - the inductive resistance, XC - the capacitive resistance, Z - the total resistance (or impedance). The total resistance is:

Z=R2+(XLXC)2Z = \sqrt{R^{2} + \left ( X_{L} - X_{C} \right ) ^2}

The total reactive resistance depends on the inductive and capacitive resistances:

X=XLXCX = X_{L} - X_{C}

The phase angle φ between the current I and total voltage U is:

ϕ=atan(XR)\phi = atan\left ( \frac{X}{R}\right )

When inductive resistance XL exceeds capacitive resistance XC, the total reactive resistance X is positive, and φ > 0. In this case the circuit has a resistance of active-inductive quality, and the total voltage U outstrips the current by phase angle φ. When capacitive resistance exceeds inductive one, the angle is φ < 0, the circuit is of active-capacitive quality, and the voltage U lags behind the current by phase angle φ.

When inductive and capacitive resistances are equal, the total reactive resistance X is zero. Then the impedance Z reaches its minimum, and the current I reaches the maximum. The circuit works in resonance mode. The voltages on inductive and capacitive circuit parts are quite high, however they compensate each other, as their phases are opposite.

Since XL = ω·L and XC = 1/ω·C, we can achieve the resonance mode by varying as one of: inductance L, capacity C, the frequency ω. When increasing the ω, the XL increases, and XC decreases. The resonance occurs when XL = XC, therefore:

ω0=1LC\omega_{0} = \frac{1}{\sqrt{L\cdot C}}

After calculating the values for current I for different ω values, one can build the resonance curve:

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=I2RP = I^{2}\cdot R

The reactive power:

Q=(XLXC)I2Q = (X_L - X_C)\cdot I^{2}

The total power:

S=P2+Q2S = \sqrt{P^{2} + Q^{2}}