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A capacitor of capacitance C is connected to a alternating voltage source V (t) as shown in the
diagram below.
c
The voltage source alternatively charges and discharges the capacitor, generating an alternating
Current I(t) in the wires. wires are assumed to be of negligible resistance.
Complex numbers are useful in AC circuits to conveniently represent amplitudes (modulus)
and phase shifts (argument). Here we assume that the source generates a voltage
voe•w
(1)
with amplitude % and frequency co. In reality, V (t) is the real value of this complex-valued
function, but all calculations can be done with complex numbers and real values taken at the
end. The current generated during charge and discharge of the capacitor is
(2)
(a) Calculate the time derivative of V (t) in Equation (2) using Equation (l) and show that
the complex-valued impedance Zc of the capacitor, defined by V(t)= Zc I(t), is
Hint: Complex-valued constants are constants for differentiation.
(b) Find the current amplitude and show that I(t) is phase-shifted from V(t) by 71/2
(i.e., when V(t) has a maximum or minimum value, O, and vice versa).
Hint: write I(t) in exponential form.
(c) Find the real parts of V(t) and I(t).
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