Transformer in a circuit with a constant current source

Question

I'm trying to understand the behavior of inductors and transformers and I was trying out some circuits. I'm having trouble understanding what happens when we use a current source and a transformer. For the circuit below, the current source is constant in time: $i_1(t) = I_1$.

The KVL equations for this system are as follows: $$L_1 \frac{di_1(t)}{dt} + M \frac{di_2(t)}{dt} + R_1 i_1(t) = 0$$ $$L_2 \frac{di_2(t)}{dt} + M \frac{di_1(t)}{dt} + R_2 i_2(t) = 0$$ which reduces to: $$M \frac{di_2(t)}{dt} + R_1 I_1 = 0$$ $$L_2 \frac{di_2(t)}{dt} + R_2 i_2(t) = 0$$ Now, both equations give different solutions for $i_2(t)$ and there is no solution for the entire system. When using the initial value $i_2(0)=0$, the first equations gives $i_2(t) = \dfrac{-R_1 I_1}{M}t$ and the second equation gives $i_2(t)=0$. This second solution makes more sense because a constant current through the left inductor doesn't cause an emf in the right inductor. But why do I get two conflicting equations and what am I doing wrong?

Thanks in advance.

Answer 1

There are a number of mistakes in your assumptions of the circuit:

1. You are neglecting the voltage of the current source. The right way to write the first equation would be:

$$L_1 \frac{di_1(t)}{dt} - M \frac{di_2(t)}{dt} + R_1 i_1(t) = V_S$$ $\qquad$where $V_S$ is the voltage of the source (in your diagram, $V_+-V_-$). Note that the sign for the mutual inductance is also incorrect.

2. The second equation also has the wrong sign for the mutual inductance term, the correct thing would be:

$$L_2 \frac{di_2(t)}{dt} - M \frac{di_1(t)}{dt} + R_2 i_2(t) = 0$$

3. You are neglecting the time dependence of the first current, but there has to be a moment where you "switch it on". So the correct expression would be (if it is instantaneous, you could more realistically switch it on gradually):

$$i_1=I_1~\theta(t)$$

$\qquad$ where $\theta(t)$ is Heaviside's step function (the derivative is a Dirac delta "function").

So the reason for the contradictions is twofold. Firstly, the equations have mistakes. Secondly, (and I think this is the most important reason) you are assuming that the primary circuit is in a steady state but you want to calculate the transient solution for the secondary circuit... this is wrong because the systems are linked together.

Answer 2

Hint: If the current is constant in the primary what does that say about the voltage across the primary winding and therefore the voltage induced in the secondary winding?

Hope this helps.