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LR circuit analysis

RL Circuit Transient: Inductor Charging and Discharging

Practically, the inductor has some resistive factor which is very minute and are ignored. It is represented by a series resistor and inductor and referred as RL Circuit. Suppose the following RL circuit where a toggle switch can connect and disconnect to circuit source.

Inductor charging in RL circuit:

Suppose the inductor has no energy stored initially. At some point in time, the switch is moved to position 1, the moment is called time t=0. As the switch closes the source voltage will appear across the inductor and will try to pass current (I=V/R) abruptly through the inductor. But according to the Lenz Law, the inductor will oppose the change in current. The current will gradually increase unless it reaches its final value of current (I=V/R). At the same time, the voltage across the inductor will decrease unless it reaches zero.

$i_{L}=\frac{E}{R} (1-e^{-\frac{t}{\tau }}) $

where $\tau =L/R$

$v_{L}=L\frac{di}{dt}=E(e^{-\frac{t}{\tau }})$

It’s worth mentioning that the current reaches its final value at $5\tau $ as well as voltage reach at that time to zero. The graph of current and voltage transients are shown below.

Inductor RL Series circuit storage phase
RL series circuit for inductor charging
Inductor voltage in rl series circuit during charging phase
Inductor voltage during charging phase
Inductor current during charging phase
Inductor current during charging phase

Inductor discharging in RL circuit:

Suppose the above inductor is charged (has stored energy in the magnetic field around it) and has been disconnected from the voltage source. Now connected to the resistive load i.e. the switch is moved to position 2 at the time t=0. The energy stored will be discharged to a resistive load and will be dissipated in the resistor. The current will continue to flow in the same direction and will gradually decrease to zero as well as the voltage across the inductor. But if the inductor is disconnected and not connected to any load, so current will stop abruptly because of no closed path. According to the equations above, it will cause the huge voltage across the inductor and you will observe in the form of spark at switch terminals. The same phenomenon is used for car engine ignition.

Inductor RL Series circuit discharging phase
RL sereis circuit during decay phase
Inductor voltage during discharging phase in RL circuit
Inductor voltage during discharging phase
Inductor current during decay phase
Inductor current during discharging phase

conclusion :

The above discussion showed the following key point in detail.

  1. Inductor doesn’t dissipate energy, it only stores it.
  2. Inductor changes current gradually rather than abruptly.
  3. Inductor reaches maximum or minimum voltage and current just in five time constants.
  4. An inductor behaves like a short circuit in DC network after five time constants.
  5. Inductor provides zero resistance after five time constants.
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About Shahab Yousafzai

Shahab Yousafzai is a professional electrical engineer, who has a history with the field since 2008. He is happy to share his experience and thoughts over the forum. He is interested in Power Electronics, Analog Electronics, and Electric Machines.

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  1. Hey there… I have a small confusion regarding the discharging of the inductor. I was wondering about the polarity of voltage across inductor while discharging. Why is it’s polarity negative?

  2. As an inductor works as a load during the charging phase and works source in the discharging phase. And the direction of the current flow in inductor remains the same. Therefore the direction voltage across the inductor changes as per the KVL.

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