LR Series Circuit

All coils, inductors, chokes and transformers create a magnetic field around themselves
consist of an Inductance in series with a Resistance forming an LR Series Circuit

The current flowing through an inductor does 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 an electrical 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 coil 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. This
is because the actual coils turns of wire being used to make it uses copper wire which has a
resistive value.

Then for real world purposes we can consider our simple coil as being an “Inductance”, L in
series with a “Resistance”, R. In other words forming an LR Series Circuit.

A 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 goes into making 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
permanently 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). After a time the voltage source
neutralizes the effect of the self induced emf, the current flow becomes constant and the
induced current and field are reduced to zero.

We can use Kirchhoff’s 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.

Kirchhoff’s voltage law (KVL) gives us:

The voltage drop across the resistor, R is I*R (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 LR series 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
• e is the base of the Natural Logarithm = 2.71828

The Time Constant, ( τ ) of the LR series circuit is given as L/R and in which V/R represents
the final steady state current value after five time constant values. 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 and effectively removing it from the 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 an LR Series Circuit

Since the voltage drop across the resistor, V R is equal to I*R (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 LR series circuit to reach its maximum steady
state value is equivalent to about 5 time constants or 5τ. This time constant τ, is measured by
τ = L/R, in seconds, where R is the value of the resistor in ohms and L is the value of the
inductor in Henries. This then forms the basis of an RL charging circuit were 5τ can also be
thought of as “5*(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 slower will be the transient time and therefore a longer time constant for 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 almost zero.

LR Series Circuit Example No1

A coil which has an inductance of 40mH and a resistance of 2Ω is connected together to form
a LR series circuit. If they are connected to a 20V DC supply.
a). What will be the final steady state value of the current.

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

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

d) What will be the value of the induced emf after 10ms.

e) 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 have noticed that the answer for question (e) 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
transient curves shown above.

Power in an LR Series Circuit

Then from above, the instantaneous rate at which the voltage source delivers power to the
circuit is given as:

The instantaneous rate at which power is dissipated by the resistor in the form of heat is given
as:

The rate at which energy is stored in the inductor in the form of magnetic potential energy is
given as:

Then we can find the total power in a RL series circuit by multiplying by i and is therefore:

Where the first I 2 R term represents the power dissipated by the resistor in heat, and the
second term represents the power absorbed by the inductor, its magnetic energy.