# Serial RL Circuit

A serial RL circuit contains a resistor and a coil connected in serial. A real coil has also an active resistance (R_{2} on the scheme below).

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

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

The resistor's voltage:

The voltage on the active resistance of the coil:

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

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

where:

The vector diagram for all voltages (voltage triangle):

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

where R_{1} + R_{2} - the total active resistance, X_{L} - inductive resistance, Z - total resistance (or impedance). The total resistance is:

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

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:

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

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

The effective current I, flowing through the circuit:

The instant current:

The total active power:

The reactive power:

The total power: