LR Series Circuits

The LR Series Circuit

In the first tutorial about Inductors, we looked briefly at the time constant of an inductor stating that the current flowing through an inductor could not change instantaneously but would increase at a constant rate determined by the self-induced emf in the inductor. In other words, an inductor in a circuit opposes the flow of current, ( i ) through it. While this is perfectly correct, we made the assumption in the tutorial that it was an ideal inductor which had no resistance or capacitance associated with its windings. However, in the real world "ALL" coils whether they are chokes, solenoids, relays or any wound component will always have a certain amount of resistance no matter how small associated with the coils turns of wire being used to make it as the copper wire will have a resistive value. Then we can consider our simple coil as being a resistance in series with an inductance, in other words an LR Series Circuit.

An LR Series Circuit consists basically of an inductor of inductance L connected in series with a resistor of resistance R. The resistance R is the DC resistive value of the wire turns or loops that go to make up the inductors coil. Consider the LR series circuit below.

The LR Series Circuit

The above LR series circuit is connected across a constant voltage source, (the battery) and a switch. Assume that the switch, S is open until it is closed at a time t = 0, and then remains permantly closed producing a "Step Response" type voltage input. The current, i begins to flow through the circuit but does not rise rapidly to its maximum value of Imax as determined by the ratio of V / R (Ohms Law). This limiting factor is due to the presence of the self induced emf within the inductor as a result of the growth of magnetic flux, (Lenz's Law). We can use Kirchoffs Voltage Law, (KVL) to define the individual voltage drops that exist around the circuit and then hopefully use it to give us an expression for the flow of current.

Kirchoffs voltage law gives us:

The voltage drop across the resistor, R is IR (Ohms Law).

The voltage drop across the inductor, L is by now our familiar expression L = di/dt

Then the final expression for the individual voltage drops around the circuit can be given as:

We can see that the voltage drop across the resistor depends upon the current, I while the voltage drop across the inductor depends upon the rate of change of the current, dI/dt. When the current is equal to zero, ( i = 0 ) at time t = 0 the above expression, which is also a first order differential equation, can be rewritten to give the value of the current at any instant of time as:

Expression for the Current in an LR Series Circuit

• Where: V is in volts, R is in ohms, L is in henries, t is in seconds and
      e is the base of the Natural Logarithm = 2.71828

The L/R term in the above equation is known commonly as the Time Constant, (τ) of the LR series circuit and V/R also represents the final steady state current value in the circuit. Once the current reaches this maximum steady state value at 5τ, the inductance of the coil has reduced to zero acting more like a short circuit. Therefore the current flowing through the coil is limited only by the resistive element in Ohms of the coils windings. A graphical representation of the current growth representing the voltage/time characteristics of the circuit can be presented as.

Transient Curves for a LR Series Circuit

Since the voltage drop across the resistor, V_R is equal to IxR (Ohms Law), it will have the same exponential growth and shape as the current. However, the voltage drop across the inductor, V_L will have a value equal to:  Ve^(-Rt/L). Then the voltage across the inductor, V_L will have an initial value equal to the battery voltage at time t = 0 or when the switch is first closed and then decays exponentially to zero as represented in the above curves.

The time required for the current flowing in the circuit to reach its maximum steady state value is equivalent to about 5 time constants or 5T. This time constant T, is measured by τ = L/R, in seconds, were R is the value of the resistor in ohms and L is the value of the inductor in Henrys. This then forms the basis of an LR charging circuit were 5T can also be thought of as "5 x L/R" or the transient time of the circuit.

The transient time of any inductive circuit is determined by the relationship between the inductance and the resistance. For example, for a fixed value resistance the larger the inductance the longer will be the transient time and therefore the time constant of the LR series circuit. Likewise, for a fixed value inductance the smaller the resistance value the longer the transient time. However, for a fixed value inductance, by increasing the resistance value the transient time and therefore the time constant of the circuit becomes shorter. This is because as the resistance increases the circuit becomes more and more resistive as the value of the inductance becomes negligible compared to the resistance. If the value of the resistance is increased sufficiently large compared to the inductance the transient time would effectively be reduced to zero.

Example No1.

A coil which has an inductance of 40mH and a resistance of 2Ωs is connected to a 20V d.c. supply.

a) What will be the final steady state value of the current.

b) What will be the time constant of the circuit.

c) What will be the transient time of the circuit.

c) What will be the value of the induced emf after 10mS.

d) What will be the value of the circuit current one time constant after the switch is closed.

  The Time Constant, τ of the circuit was calculated in question b) as being 20mS. Then the circuit current at this time is given as:

You may notice that the answer for question (d) which gives a value of 6.32 Amps at one time constant, is equal to 63.2% of the final steady state current value of 10 Amps we calculated in question (a). This value of 63.2% or 0.632 x I_MAX also corresponds with the transisent curves shown above.