Calculating Energy Loss in Inductors: Formula and Derivation

To express the energy loss in terms of v1, v2, i1, i2, L1, and L2, we need to consider the relationship between voltage and current in an inductor.

The voltage across an inductor is given by the formula: V = L * di/dt, where V is the voltage, L is the inductance, and di/dt is the rate of change of current.

Using this relationship, we can express the energy loss in terms of the given variables as follows:

1. Calculate the initial energy stored in each inductor:

   • Energy stored in inductor 1: W1 = 0.5 * L1 * (i1^2)
   • Energy stored in inductor 2: W2 = 0.5 * L2 * (i2^2)

2. Calculate the final energy stored in each inductor:

   • Energy stored in inductor 1 after the switch is opened: W1' = 0.5 * L1 * (i1'^2)
   • Energy stored in inductor 2 after the switch is opened: W2' = 0.5 * L2 * (i2'^2)

3. Calculate the energy losses:

   • The energy losses during the opening of the switch can be determined by subtracting the final energy stored from the initial energy stored for each inductor.
   • Energy losses = (W1 - W1') + (W2 - W2')
   • Energy losses = 0.5 * L1 * (i1^2 - i1'^2) + 0.5 * L2 * (i2^2 - i2'^2)

Now, let's express the currents (i1 and i2) in terms of the voltages (v1 and v2) and the inductances (L1 and L2):

Using Ohm's law (V = IR) and the voltage-current relationship in an inductor (V = L * di/dt), we can express the current in terms of the voltage and inductance:

i1 = (v1 / sqrt(L1)) (where sqrt denotes the square root) i2 = (v2 / sqrt(L2))

Substituting these expressions for i1 and i2 into the energy loss equation:

Energy losses = 0.5 * L1 * ((v1 / sqrt(L1))^2 - i1'^2) + 0.5 * L2 * ((v2 / sqrt(L2))^2 - i2'^2)

Simplifying further, we get the expression for energy loss in terms of v1, v2, i1, i2, L1, and L2:

Energy losses = 0.5 * (v1^2 - L1 * i1'^2) + 0.5 * (v2^2 - L2 * i2'^2)

Please note that this expression assumes ideal inductors without any resistance or other non-ideal factors. In practical circuits, additional factors may need to be considered.