1. RL Circuits – Current Growth And DecayBy
Dr. Vaibhav Jain
Associate Professor, Dept. of Physics,
D.A.V (PG) College, Bulandshahr, U.P. , India.

2. Inductors in Circuits—The RL Circuit
Basic series RL circuit:
Exhibits time-dependent behavior, reminiscent of RC circuit
A battery with EMF E drives a current around the loop
Changing current produces a back EMF or sustaining EMF EL in the inductor.
Derive circuit equations using Kirchhoff’s loop rule.
Convert to differential equations and solve
(as for RC circuits).
New for Kirchhoff rule:
When traversing an inductor in the same
direction as the assumed current insert:
Voltage drops by EL for growing current

3. Series LR circuit + - E i L R a b EL
Inductance & resistance + EMF
Find time dependent behavior
Use Loop Rule
NEW TERM FOR KIRCHHOFF LOOP RULE
Given E, R, L: Find i, EL, UL for inductor as functions of time
Growth phase, switch to “a”. Loop equation:
At t = 0, rapidly growing current but i = 0, EL= E
L acts like a broken wire
i through R is clockwise and growing EL opposes E
As t  infinity, current is large & stable, di/dt  0
Back EMF EL 0, i  E / R,
L acts like an ordinary wire
Energy is stored in L & dissipated in R
Decay phase, switch to “b”, exclude E, Loop equation:
Energy stored in L will be dissipated in R
EL acts like a battery maintaining previous current
At t = 0 current i = E / R, unchanged, CW
Current begins to collapse
Current  0 as t  infinity – energy is depleted

4. LR circuit: decay phase solutionb EL + -
After growth phase equilibrium, switch from a to b, battery out
Current i0 = E / R initially still flows CW through R
Inductance L tries to maintain current using stored energy
Polarity of EL reversed versus growth. Eventually EL 0
Loop Equation is :
Substitute :
Circuit Equation:
di/dt <0
during decay,
opposite to current
Current decays exponentially: t 2t 3t i0 i
First order differential equation with simple exponential decay solution
At t = 0+: large current implies large di / dt, so EL is large (now driving current)
As t  infinity: current stabilizes, di / dt and current i both  0
EMF EL and VR also decay exponentially:
Compare to RC circuit, decay

5. LR circuit: growth phase solution
Loop Equation is :
Substitute :
Circuit Equation:
Current starts from zero, grows as a saturating exponential.
First order differential equation again - saturating exponential solutions
At t = 0: current is small because di / dt is large. Back EMF opposes battery.
As t  infinity: current stabilizes at iinf = E / R. di / dt approaches zero, t 2t 3t
iinf i
i = 0 at t = 0 in above equation  di/dt = E/L
fastest rate of change, largest back EMF
Back EMF EL decays exponentially
Voltage drop across resistor VR= -iR
Compare to RC circuit, charging

6. Example: For growth phase find back EMF EL as a function of time
Use growth phase solution S - + i EL
At t = 0: current = 0
Back EMF is ~ to rate of change of current
Back EMF EL equals the battery potential
causing current i to be 0 at t = 0
iR drop across R = 0
L acts like a broken wire at t = 0 -E EL
After a very long (infinite) time:
Current stabilizes, back EMF=0
L acts like an ordinary wire at t = infinity

7. Summarizing RL circuits growth phase
Extra
Inductor acts like a wire.
Inductor acts like an open circuit.
When t / tL is large:
When t / tL is small: i = 0.
Capacitive time constant
Compare:
Inductive time constant
The current starts from zero and increases up to a maximum of with a time constant given by
The voltage across the resistor is
The voltage across the inductor is

8. Summarizing RL circuits decay phase
Extra
The switch is thrown from a to b
Kirchoff’s Loop Rule for growth was:
Now it is:
The current decays exponentially:
Voltage across resistor also decays:
VR (V)
Voltage across inductor:

9. Thank You