Question 7.5: A time-varying inductor in series with a voltage source and ...

Use Kirchhoff’s voltage law to write

e_{s} =e_{L}+Ri_{L}.          (7.77)

Substituting from Eq. (7.76)  e_{L} =L\left(t\right) \frac{di_{L} }{dt} +i_{L} \frac{dL}{dt}.   for e_{L} in Eq. (7.77) gives 

e_{s} =L\left(t\right) \frac{di_{L} }{dt} +i_{L} \frac{dL}{dt} +Ri_{L} .          (7.78)

If i_{L} is selected as the state variable, the state-variable equation takes the following form:

\frac{di_{L} }{dt} =-\frac{1}{L\left(t\right) } \left(R+\frac{dL}{dt} \right) i_{L} +\frac{1}{L\left(t\right) } e_{s} .          (7.79)

This equation would be very difficult to solve because of the presence of the derivative of inductance. The state model of the circuit can be greatly simplified if a flux linkage λ is used as the state variable instead of current i_{L} Substituting from Eq. (7.75)  e_{L} =\frac{d\lambda }{dt}, for e_{L} and i_{L} yields

e_{s}= \frac{d\lambda }{dt}+\frac{R}{L\left(t\right) } \lambda .          (7.80)

Hence the state-variable equation in a standard form is

\frac{d\lambda }{dt}=-\frac{R}{L\left(t\right) } \lambda +e_{s}          (7.81)

The output equation relating e_{o} to the state variable λ is

e_{o}=\frac{R}{L\left(t\right) }\lambda .          (7.82)

This example clearly demonstrates the benefits of selecting flux linkage as the state variable in circuits involving time-varying inductors.