AC reactive circuit calculations

Electricity and Electronics

  • Question 1

    Determine the input frequency necessary to give the output voltage a phase shift of 75$^{o}$:

    Also, write an equation that solves for frequency ($f$), given all the other variables ($R$, $L$, and phase angle $\theta$).

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  • Question 2

    An AC electric motor under load can be considered as a parallel combination of resistance and inductance:

    Calculate the equivalent inductance ($L_{eq}$) if the measured source current is 27.5 amps and the motor’s equivalent resistance ($R_{eq}$) is 11.2 $\Omega$.

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  • Question 3

    An AC electric motor under load can be considered as a parallel combination of resistance and inductance:

    Calculate the current necessary to power this motor if the equivalent resistance and inductance is 20 $\Omega$ and 238 mH, respectively.

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  • Question 4

    Calculate the individual currents through the inductor and through the resistor, the total current, and the total circuit impedance:

    Also, draw a phasor diagram showing how the individual component currents relate to the total current.

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  • Question 5

    Calculate the total impedances (complete with phase angles) for each of the following capacitor-resistor circuits:

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  • Question 6

    Calculate the total impedances (complete with phase angles) for each of the following inductor-resistor circuits:

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  • Question 7

    Is this circuit’s overall behavior capacitive or inductive? In other words, from the perspective of the AC voltage source, does it ``appear’’ as though a capacitor is being powered, or an inductor?

    Now, suppose we take these same components and re-connect them in parallel rather than series. Does this change the circuit’s overall ``appearance’’ to the source? Does the source now ``see’’ an equivalent capacitor or an equivalent inductor? Explain your answer.

    {\bullet} Which component ``dominates’’ the behavior of a series LC circuit, the one with the least reactance or the one with the greatest reactance?
    {\bullet} Which component ``dominates’’ the behavior of a parallel LC circuit, the one with the least reactance or the one with the greatest reactance?

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  • Question 8

    Calculate the voltage dropped across the inductor, the capacitor, and the 8-ohm speaker in this sound system at the following frequencies, given a constant source voltage of 15 volts:

    {\bullet} $f =$ 200 Hz
    {\bullet} $f =$ 550 Hz
    {\bullet} $f =$ 900 Hz

    Regard the speaker as nothing more than an 8-ohm resistor.

    {\bullet} As part of an audio system, would this LC network tend to emphasize the {\it bass}, {\it treble}, or {\it mid-range} tones?

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  • Question 9

    Write an equation that solves for the impedance of this series circuit. The equation need not solve for the phase angle between voltage and current, but merely provide a scalar figure for impedance (in ohms):

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  • Question 10

    Determine the necessary resistor value to give the output voltage a phase shift of $-64^{o}$:

    Also, write an equation that solves for this resistance value ($R$), given all the other variables ($f$, $C$, and phase angle $\theta$).

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  • Question 11

    Determine the necessary resistor value to give the output voltage a phase shift of 58$^{o}$:

    Also, write an equation that solves for this resistance value ($R$), given all the other variables ($f$, $C$, and phase angle $\theta$).

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  • Question 12

    Determine the input frequency necessary to give the output voltage a phase shift of 25$^{o}$:

    Also, write an equation that solves for frequency ($f$), given all the other variables ($R$, $C$, and phase angle $\theta$).

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  • Question 13

    Determine the input frequency necessary to give the output voltage a phase shift of $-40^{o}$:

    Also, write an equation that solves for frequency ($f$), given all the other variables ($R$, $L$, and phase angle $\theta$).

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  • Question 14

    Calculate the total impedance offered by these two inductors to a sinusoidal signal with a frequency of 60 Hz:

    Show your work using two different problem-solving strategies:

    {\bullet} Calculating total inductance ($L_{total}$) first, then total impedance ($Z_{total}$).
    {\bullet} Calculating individual impedances first ($Z_{L1}$ and $Z_{L2}$), then total impedance ($Z_{total}$).

    Do these two strategies yield the same total impedance value? Why or why not?

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  • Question 15

    Determine the input frequency necessary to give the output voltage a phase shift of $-25^{o}$:

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  • Question 16

    Determine the input frequency necessary to give the output voltage a phase shift of 40$^{o}$:

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  • Question 17

    A student is asked to calculate the phase shift for the following circuit’s output voltage, relative to the phase of the source voltage:

    He recognizes this as a series circuit, and therefore realizes that a right triangle would be appropriate for representing component impedances and component voltage drops (because both impedance and voltage are quantities that add in series, and the triangle represents phasor addition):

    The problem now is, which angle does the student solve for in order to find the phase shift of $V_{out}$? The triangle contains two angles besides the 90$^{o}$ angle, $\theta$ and $\Phi$. Which one represents the output phase shift, and more importantly, {\it why}?

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  • Question 18

    Determine the total current and all voltage drops in this circuit, stating your answers the way a multimeter would register them:

    {\bullet} $C_1 = 125 \hbox{ pF}$
    {\bullet} $C_2 = 71 \hbox{ pF}$
    {\bullet} $R_1 = 6.8 \hbox{ k}\Omega$
    {\bullet} $R_2 = 1.2 \hbox{ k}\Omega$
    {\bullet} $V_{supply} = 20 \hbox{ V RMS}$
    {\bullet} $f_{supply} = 950 \hbox{ kHz}$

    Also, calculate the phase angle ($\theta$) between voltage and current in this circuit, and explain where and how you would connect an oscilloscope to measure that phase shift.

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  • Question 19

    Solve for all voltages and currents in this series RC circuit, and also calculate the phase angle of the total impedance:

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  • Question 20

    Solve for all voltages and currents in this series RC circuit:

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  • Question 21

    Solve for all voltages and currents in this series LR circuit, and also calculate the phase angle of the total impedance:

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  • Question 22

    Due to the effects of a changing electric field on the dielectric of a capacitor, some energy is dissipated in capacitors subjected to AC. Generally, this is not very much, but it is there. This dissipative behavior is typically modeled as a series-connected resistance:

    Calculate the magnitude and phase shift of the current through this capacitor, taking into consideration its equivalent series resistance (ESR):

    Compare this against the magnitude and phase shift of the current for an ideal 0.22 $\mu$F capacitor.

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  • Question 23

    Which component, the resistor or the capacitor, will drop more voltage in this circuit?

    Also, calculate the total impedance ($Z_{total}$) of this circuit, expressing it in both rectangular and polar forms.

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  • Question 24

    Calculate all voltages and currents in this circuit, as well as the total impedance:

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  • Question 25

    Use the ``impedance triangle’’ to calculate the necessary reactance of this series combination of resistance ($R$) and inductive reactance ($X$) to produce the desired total impedance of 145 $\Omega$:

    Explain what equation(s) you use to calculate $X$, and the algebra necessary to achieve this result from a more common formula.

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  • Question 26

    Use the ``impedance triangle’’ to calculate the impedance of this series combination of resistance ($R$) and inductive reactance ($X$):

    Explain what equation(s) you use to calculate $Z$.

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