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

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

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

LR circuit: decay phase solution b 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

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

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

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

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:

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