R-L Circuits

R-L Circuits? What does the “L” stand for? Good Question! “L” stands for the self-inductance of an inductor measured in Henrys (H). So…What is an inductor? An inductor is an electronic device that is put into a circuit to prevent rapid changes in current. It is basically a coil of wire which uses the basic principles of electromagnetism and Lenz’s Law to store magnetic energy within the circuit for the purposes of stabilizing the current in that circuit. The voltage drop across an inductor depends on the inductance value L and the rate of change of the current di/dt.

R-L Circuits The set up and initial conditions S1S1 S2S2 ε L R a bc Assume an ideal source (r=0) An R-L circuit is any circuit that contains both a resistor and an inductor. Initial conditions: At time t = 0…when S 1 is closed…i = 0 and At time t = 0, we will close switch S1 to create a series circuit that includes the battery. The current will grow to a “steady-state” constant value at which the device will operate until powered off (i.e. the battery is removed)

R-L Circuits Current Growth S1S1 S2S2 ε L R a bc i Note: we will use lower-case letters to represent time-varying quantities. At time t = 0, S 1 is closed, current, i, will grow at a rate that depends on the value of L until it reaches it’s final steady-state value, I If we apply Kirchoff’s Law to this circuit and do a little algebra we get… As “i” increases, “iR/L” also increases, so “di/dt” decreases until it reaches zero. At this time, the current has reached it’s final “steady-state” value “I”.

R-L Circuits Steady-State Current S1S1 S2S2 ε L R a bc i When the current reaches its final “steady-state” value, I, then di/dt = 0. Solving this equation for I… Do you recognize this? It is Ohm’s Law!!! So…when the current is at steady-state, the circuit behaves like the inductor is not there…unless it tries to change current values quickly! The steady- state current does NOT depend on L!

R-L Circuits Current as a function of time during Growth The calculus and the algebra! S1S1 S2S2 ε L R a bc i Let’s start with the equation we derived earlier from Kirchoff’s Law… Rearrange and integrate… Solve for i…

R-L Circuits The time constant! S1S1 S2S2 ε L R a bc i The time constant is the time at which the power of the “e” function is “-1”. Therefore, time constant is L/R At time t = 2 time-constants, i = 0.86 I, and at time t = 5 time-constants, i = I Therefore, after approximately 5 time-constant intervals have passed, the circuit reaches its steady-state current.

R-L Circuits Energy and Power S1S1 S2S2 ε L R a bc i