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Phasor Mathematics
13 questions By Tony R. Kuphaldt
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Question 13 of 13
∫f(x) dx Calculus alert!
Impedance is defined as the complex ratio of voltage and current:Z = V IWe may determine the complex definition of impedance provided by an capacitor if we consider the equation relating capacitor voltage and capacitor current:
i = C d dt[ v(t) ] First, substitute a phasor expression of voltage into the above equation, then differentiate with respect to time to obtain an expression for current:
v(t) = ej ωt i = C d dt[ ej ωt ] Then, divide the two phasor expressions to obtain an expression for capacitive impedance.
Reveal answerZC = 1 j ωCor ZC = −j 1 ωCFollow-up question: explain why the following expression for capacitive impedance is equivalent to the one shown above:
ZC = 1 ωCej (−π/ 2) Notes:In order to solve this problem, your students must remember the basic rule for differentiating exponential functions:
d dx[ eax ] = a eax This is one of the beauties of representing sinusoidal voltages and currents in complex exponential (phasor) form: it makes differentiation and integration relatively easy! In this sense, the Euler relation of ejx = cosx j sinx is a transform function, transforming one type of mathematical problem into a different (easier) type.
