created at June 16, 2021 (changed at July 2, 2021)

# Serial RL Circuit

A serial RL circuit contains a resistor and a coil connected in serial. A real coil has also an active resistance (R2 on the scheme below).

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

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

where Im - current amplitude, ω - angular frequency ω = 2·π·f. Let's consider the initial phase (in sinus) is zero.

The resistor's voltage:

$u_{1} = I_{m}\cdot R_{1}\cdot sin(\omega\cdot t)$

The voltage on the active resistance of the coil:

$u_{1} = I_{m}\cdot R_{1}\cdot sin(\omega\cdot t)$

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

$u_{L} = I_{m}\cdot X_{L}\cdot sin(\omega\cdot t + \frac{\pi}{2})$

where XL = ω·L - inductive resistance. One can write the formula for effective voltage:

$U_{L} = I\cdot X_{L}$

where:

$U_{L} = \frac{U_{mL}}{\sqrt{2}};\quad I = \frac{I_{m}}{\sqrt{2}}$

The vector diagram for all voltages (voltage triangle):

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

where R1 + R2 - the total active resistance, XL - inductive resistance, Z - total resistance (or impedance). The total resistance is:

$Z = \sqrt{(R_{1} + R_{2})^{2} + X_{L}^2}$

The total resistance of the coil (includes its active and inductive resistances):

$Z_L = \sqrt{R_{2}^{2} + X_{L}^2}$

The voltage on the real coil (with active resistance) outstrips the current I by the phase angle φ2, which is a bit smaller than π/2:

$\phi_{2} = atan\left ( \frac{X_{L}}{R_{2}} \right )$

The total voltage outstrips the current by the phase angle φ:

$\phi = atan\left ( \frac{X_{L}}{R_{1} + R_{2}}\right )$

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

$cos\phi = \frac{R_{1} + R_{2}}{Z}$

The effective current I, flowing through the circuit:

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

The instant current:

$i(t) = I_{m}\cdot sin(\omega\cdot t);\quad I_{m} = I\cdot \sqrt{2}$

The total active power:

$P = I^{2}\cdot (R_{1} + R_{2})$

The reactive power:

$Q_{L} = X_L\cdot I^{2}$

The total power:

$S = U\cdot I = \sqrt{P^2 + Q_{L}^2}$