Lesson 28: Ch. 30 (6) – RL Circuits

di 1 Charging RL Circuits 2 Discharging RL Cir- law. Only when dt is not zero do we see cuits a back-EMF. Put a resistor R, inductor L and EMF E So, in an RL circuit as portrayed in in a circuit with a switch that either has Now move the switch so that the battery the diagram the circuit is either charging the EMF in or out of the only loop. Turn is not in the loop. The inductor creates or discharging (making or destroying the the switch to bring the EMF in: a back-EMF to prevent the circuit from B ﬁeld in the inductor). R turning oﬀ instantly: R

E L E L

This is a single loop circuit with E, R and L in series. When turned on, the Kirchoff’s loop rule here is: increasing current will create an increas- −iR − L di = 0 ing B in the inductor. We can write Kir- dt choff’s loop rule: and once again, we did the math in the RC circuits section. We find that the −iR − L di + E = 0 dt current does not instantly turn off but in- If the circuit is charging the inductor stead exponentially “decays” to zero over Note the similarity to the RC circuits (i increasing) the current as a function of some time: −t/τ we worked with earlier. The results are time is given by i = (E/R)(1−e ) and looks like the top plot. also similar. i = (E/R)(e−t/τ ) where τ = L/R. If the circuit is discharging (i going to i = (E/R)(1 − e−t/τ ) where τ = L/R. 0) the current as a function of time is give 3 Using RL Circuits as an exponential decay i = (E/R)e−t/τ The back-EMF makes the circuit not and looks like the bottom. turn on instantly. This equation is really In a steady state circuit (on OR off) the The circuit has many components and similar to the charging RC circuit in that inductor acts just like a piece of low re- each one has its own power and voltage. we have an “inverse exponential” behav- sistance wire – it is, afterall, just a coil. Remember that knowing the current lets ior that starts i at zero and increases it In the steady state there is no back-EMF you figure out the rest of the circuit. to its operating value in time. and the term disappears out of Ohm’s

1 Exercise 1: An RL circuit consists Exercise 2: An 8.0-mH inductor Exercise 3: What is the inductance of a 20 ohm resistor and a 10 mH induc- and a 2.0 Ω resistor are wired in se- of a series RL circuit in which R = 1000 Ω tor connected in series to a 10 V battery ries to a 20-V ideal battery. A switch if the current increases from zero to one- through a switch. in the circuit is closed at time t = 0, third of its ﬁnal value in 30µs? at which time the current is zero. Af- ter 6 ms, calculate the (a) current in (a) Immediately after a switch is closed the inductor (b) current in the resistor to complete the circuit, what is the (c) potential drop across the battery (d) current through the battery? potential drop across the resistor (e) potential drop across the inductor (f) power being provided by the battery (g) power being used by the resistor (h) power being used by the inductor. (b) What is the battery current a “long 2.0 Ω time” after the switch is closed?

20 V 8.0 mH