RL Circuits – Current Growth And Decay

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

2. 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 ELfor growing current Inductors in Circuits—The RL Circuit

3. Series LR circuit i a R + E b - EL L i • 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 Growth phase, switch to “a”. Loop equation: Decay phase, switch to “b”, exclude E, Loop equation: • Energy stored in L will be dissipated in R • EL actslike 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 • 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

4. R - b EL L + i Circuit Equation: Loop Equation is : di/dt <0 during decay, opposite to current Substitute : • 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 Current decays exponentially: i0 i t 2t 3t t EMF EL and VR also decay exponentially: Compare to RC circuit, decay LR circuit: decay phase solution • 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

5. Circuit Equation: Loop Equation is : Substitute : • 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, Current starts from zero, grows as a saturating exponential. iinf i • i = 0 at t = 0 in above equation  di/dt = E/L • fastest rate of change, largest back EMF t 2t 3t Back EMF EL decays exponentially t Compare to RC circuit, charging Voltage drop across resistor VR= -iR LR circuit: growth phase solution

6. S + EL i - 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 Use growth phase solution EL -E • After a very long (infinite) time: • Current stabilizes, back EMF=0 • L acts like an ordinary wire at t = infinity Example: For growth phase find back EMF EL as a function of time At t = 0: current = 0

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

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

9. Thank You