RL CIRCUIT Transient Response, Power Factor and power factor correction.

Cuthbert Nyack

The RL circuit is shown schematically above with the alternating source replaced by a fixed source. If a unit step voltage is applied to the RL circuit, then the voltage as a function of time is given by V(R) = 1 - e^-Rt/L. and V(L) = e^-Rt/L.

If the voltage is reduced to zero after time T, (switch moved from a to b) then the response is shown below. For 0 £ t £ T the response is V_r = 1 - e^-Rt/L
For T < t £ ¥ the response is V_r = e^-Rt/L ´ (e^RT/L - 1 )
For t < T , V_l + V_r = 1 and
fot t > T , V_l + V_r = 0.

There is a similarity between the RL and RC circuits as illustrated below:-
For RC Vr/Vi = st/(st + 1), Vc/Vi = 1/(st + 1) where t = CR.

For RL Vl/Vi = st/(st + 1), Vr/Vi = 1/(st + 1) where t = L/R.
The resistor voltage in the RL circuit behaves similarly to the capacitor voltage in the RC circuit and the inductor voltage in the RL circuit behaves similarly to the resistor voltage in the RC circuit.

The applet below shows the transient response of an RL circuit for different inputs.
Different inputs can be seen by changing the parameter Fn which is controlled by scrollbar (0).

Fn = 0 shows the step response for 2 time constants.
Fn = 1 shows the pulse response.
Fn = 2 shows the response to a rectangular wave whose asymmetry can be changed by scrollbar (7).
Fn = 3,4 and 5 show special cases of the rectangular wave response.
Fn = 6 shows the response to a signal which can be changed from a triangle to a square.
Fn = 7,8 and 9 show special cases of this signal.
Fn = 10 shows the response to a signal which can be changed from a triangle to a ramp to a square.
Fn = 11, 12 show special cases of 10.
Fn = 13 shows the response to a signal which can be changed from from a triangle to a parabola.
Fn = 14, 15 show special cases of 13.
Fn = 16 shows the transient response to sinusoid whose positive and negative part can be limited.
Fn = 17, 18 and 19 show special cases of 16.
Fn = 20 shows the response to 2 sinsusoids with an average level.
Fn = 21 show a special case of 20.
Fn = 22,23 show the response to PWL signals which can be adjusted by scrollbars 15 to 24.
Fn = 24 shows the response of the RL series circuit to a current input.
Fn = 25,26 shows the response of the parallel RL circuit to current and voltage inputs.
Fn = 27, 28 shows the power relations for a series RL circuit with voltage and current inputs.
The average power in the inductor is zero. When the current is increasing, power goes into the inductor to increase the energy stored in the magnetic field of the inductor. When the current is decreasing energy comes out of the magnetic field and is returned to the source.
At any value of time, the power supplied by the source is equal to the sum of the power dissipated in the resistor and the power going into/out of the inductor.
The power factor is the cosine of the phase difference between the voltage and current. In each case the power factor and the different powers are shown.
Fn = 29, 30 shows the power relations for a parallel RL circuit with voltage and current inputs.
Fn = 31 shows the power factor being improved by adding a capacitor across the RL circuit. When a capacitor is added across the RL circuit, the phase difference between the current and voltage is reduced and the current is reduced. The average power is unaffected but the apparent power is reduced. Current is shown as pink without the capacitor and as brown with the capacitor.
Fn = 32 show a special case of 31. Here the power factor is increased to ~ 1 and the apparent power is reduced by ~40%. This requires a large capacitor, in practice a compromise would be made with a smaller capacitor.

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