At , an LRC circuit has a capacitive reactance of , an inductive reactance of , and a resistance of .

1. (3 pts) What is the minimum possible impedance of the circuit, and at what angular frequency does it occur?

   The minimum impedance is Z = R, which occurs at resonance, when . Here that means This occurs at the resonant , when . To find this with a minimum of work (that is, without computing L and C), note that if is a factor x times the given frequency,

   

   then the inductive and capacitive reactances at resonance can be related to the given values at :

   

   Setting these equal gives x, which gives :

   

   Thus x=3, giving

2. (3 pts) What are and at the resonant frequency?

   Using our solution

   

   

   These are equal as promised.

3. (6 pts) Draw the phasor diagram for this circuit at , at the instant when the voltage across the capacitor is at its maximum positive value. Take , and show the phasors and , explicitly displaying the magnitude and direction for each, as well as the total voltage phasor , showing its magnitude and phase angle .

   First we calculate the phasor magnitudes , and , and the phase angle . Remember that phasor magnitudes are always the peak values, so that their maximum vertical projection (maximum instantaneous value) is the peak value. Then we use the information given about the instantaneous voltage across the capacitor to draw the phasor diagram, with the phasors at the indicated instantaneous position.

   Calculations:

   

   Thus lags by .

   To assemble these values together into an instantaneous phasor diagram, we must note the following facts:

   1. leads and , which are in phase, by ; it also leads by .
   2. lags by .
   3. The instantaneous is the positive peak value, . Thus the vertical projection of is its full length, meaning that must be vertical. Since the projection is positive, must point upwards.

   Together these 3 facts determine the instantaneous phasor diagram

   

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