Hints for Assignment 11

The hints below are very detailed, since there are many new things to learn here. If you get stuck following these suggestions, please send me a note telling me where and how. I will try to get you unstuck and modify the hints to help everyone else as well.

I will use "w" for omega everywhere in these hints.

55. What should be plotted is the absolute magnitude of the complex impedance Z , which we call |Z| . There are 3 different circuits involved here, composed of elements whose impedances are:

resistor ZR = R    [real]

inductor ZL = iwL     [pure imaginary]

capacitor ZC = -1/(wC)    [pure imaginary]

For each circuit, answer the following questions in order:

  1. What is the (complex) impedance of a series combination of these elements?
    [Compare the formula for the effective resistance of 2-3 resistors in series].

  2. Mentally: What is the real part of this impedance? What is the imaginary part?

  3. What is the absolute magnitude of this impedance?

  4. Now plot |Z| as a function of w [omega].
Compare the behavior of the plots, especially the LRC one. Do the plots make sense?

56.

  1. Draw a diagram of the circuit.

  2. You are given the frequency of the oscillation of the voltage, 60/sec. What is the angular frequency w?

  3. Write down expressions for the voltage drops across the resistor, the inductor, and the two of them together using V=IZ (omitting the exp(iwt) factor common to each if you wish).

  4. Since we do not need any phases, we can take advantage of the relations

    V = |V| cos(wt+phi)
    = |I| |Z| cos(wt+phi)

    to get the amplitudes of each of the voltage drops. Do so.

  5. Given the absolute magnitude of the voltage across the inductor, find the absolute magnitude of the current. Then use the value for the current to find (the absolute magnitude of) the resistance.

  6. Now you know R and L and so you know the effective impedance of the combination. Use this Z to find the absolute value of the voltage across the two. This voltage must be the amplitude of the input voltage.

  7. The peak voltage across both elements will not be the sum of the peak voltage across the resistor added to the peak voltage across the inductor, because the peaks across the two elements occur at different times. Make sure you understand why the time of the peak voltage is different for the two devices.

57. Note: TWO circuits, one of them DC. In the DC circuit, there are two resistances, the explicit R and the resistance of the coil. The impedance of the coil is R+iwL but w=0 so the inductance does not contribute. Find the resistance of the coil.

Now analyze the AC circuit as in the problems above (there are still two resistances, but you know both of them). In particular get the total impedance of the circuit and the absolute magnitude of the total impedance. |V|=|I| |Z| can be used to find L .

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Last revised 5/2/97