Impedance
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1) Impedance for Different Devices
Consider the respective constitutive equations for resistors, capacitors, and inductors:
Resistor | Capacitor | Inductor |
---|---|---|
v(t) = Ri(t) | i(t) = C{{\rm d}\over {\rm d}t}v(t) | v(t) = L{{\rm d}\over {\rm d}t} i(t) |
If we assume that v(t) = \tilde{A}e^{j\omega t}, where \tilde{A} is a complex
number and \omega is a real-valued frequency, each of the equations above can
be re-written to look like v(t) = \tilde{Z}i(t), where \tilde{Z} is a complex
number. Find these relationships and enter your answers below as Python
expressions, using j
to represent the imaginary unit and use omega
to represent
\omega.
The beautiful thing about this representation is that, under the assumption that the input is a complex exponential, all of these relationships are simple linear relationships (rather than differential equations). And, in fact, each of these looks a lot like the equation for resistors, just with R replaced with the complex-valued \tilde{Z}, which will depend on the frequency \omega for some devices (we'll refer to \tilde{Z} as the impedance of that device). This transformation will let us solve some kinds of circuits as though they only consisted of resistors, which can dramatically simplify analysis (as we'll see in the next problem).
2) Limiting Cases
Let \tilde{Z}_{\rm L} refer to the impedance of an inductor and \tilde{Z}_{\rm C} refer to the impedance of a capacitor. Note that both of these depend on the frequency \omega at which they're being driven. It can often be useful to think about the limiting cases of both of these values.