The switch in the circuit as shown below has been open for a long time. The volt
ID: 1812804 • Letter: T
Question
The switch in the circuit as shown below has been open for a long time. The voltage of the sinusoidal source is vg=Vm*sin(%u03C9t+%u03D5). The switch closes at t = 0. Note that the angle %u03D5 in the voltage expression determines the value of voltage at the moment when the switch closes, that is, vg(0)=Vm*sin(%u03D5)
a) Use the Laplace transform method to find i for t > 0
b) Using the expression derived in a), write the expression for current after the switch has been closed for a long time
c) Using the expression derived in a), write the expression for the transient component of i.
d) Find the steady-state expression for i using the phasor method. Verify that your expression is equivalent to that obtained in b).
e) Specify the value of %u03D5 so that the circuit passes directly into steady-state operation when the switch is closed.
Explanation / Answer
To productively use the Laplace Transform, we need to be able to transform functions from the time domain to the Laplace domain. We can do this by applying the definition of the Laplace Transform but this quickly becomes tedious. Our goal is to avoid having to evaluate the integral by finding the Laplace Transform of many useful functions and compiling them in a table. Thereafter the Laplace Transform of functions can almost always be looked by using the tables without any need to integrate. A table of Laplace Transform of functions is available here. The Unit Step Function The unit step function is defined as Some notes about this function: Most references use u(t) instead of ?(t). However, u(t) has some other common uses, so we will use ?(t) to avoid confusion (and because it's Laplace Transform ?(s) looks a little like a step input). Some references change the definition of the step so that the bottom inequality becomes a strict inequality (t>0) and either leave ?(0) undefined or say that ?(0)=
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