Impedance

The questions below are due on Tuesday November 12, 2024; 10:00:00 PM.
 
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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.

For a resistor with resistance R, \tilde{Z} =~

For a capacitor with capacitance C, \tilde{Z} =~

For an inductor with inductance L, \tilde{Z} =~

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.

As \omega\to 0, what value does \tilde{Z}_C approach?

As \omega\to \infty, what value does \tilde{Z}_C approach?

As \omega\to 0, what value does \tilde{Z}_L approach?

As \omega\to \infty, what value does \tilde{Z}_L approach?