Circuits with inductors

We will begin to analyze circuits with inductors in addition to resistors, capacitors and batteries. The same procedure developed in a previous chapter will be used, together with the new information we just learned about inductors. Roughly, this is the procedure again: Wait! What about state variables??? Before we had one state variable for each capacitor. What about inductors? Going back to the analogy with mechanical systems, just like the velocity of the mass is a state variable, the current in the inductor is also a state variable. So, for electrical systems we will have one state variable for each capacitor and each inductor.

Here is an example of a simple circuit that we will analyze:

In the circuit above, C = 5 mF, L = 20 mH (milli-henries), and Vb = 9 V (volts).

Initially, the capacitor is uncharged: qc(t=0) = 0. The switch is open. At this point, the current in the inductor must be zero and the voltages across each element is zero, except for the battery of course.

Now close the switch:

 

KCL: iB= iC= iL= i 

KVL: -VB+ VC+ VL= 0 

Const. Laws: VC= qC /C 

VL= L i'L
 
we see that VC(t=0) =0. Thus by KVL, VL(t=0) =VB(t=0) .
and we see that iL(t=0) =0. Thus by KCL, i(t=0) = 0
 
 
 

What do you think the solution for this circuit will look like as a function of time? That is, how will the voltage in the capacitor and the current in the inductor vary with time? Remember that for a RC circuit, the capacitor charges up to the voltage of the battery in an exponential fashion. Once it reaches the voltage of the battery, the current in the system stops and there is a steady state. What about for this LC circuit?

Remember that in a steady state (V'C=0, i'L=0), the capacitor wants to have a constant voltage and zero current through it, while the inductor wants to have a zero voltage and constant current. These conflicting demands cause oscillatory behavior! This should not be too surprising if we make the analogy to mechanical systems: an LC circuit is analgous to a spring-mass system, which indeed will oscillate. Here is the solution for the following parameters: C=5*10^(-6); Vb=9; L=20*10^(-3).

 

 

So, what is happening?