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A Series L-R-C Circuit: The Phasor Approach


Part A

Find the impedance Z (defined by Z=frac{V_0}{I_0}) of the circuitusing thephasor diagram shown. Notice that in this series circuit,the current is same for all elements of the circuit:.

You may find the following reminders helpful:

  • In a series circuit, the overall voltage is the sum of theindividual voltages.
  • In an AC circuit, the voltage across a capacitor lags behindthe current, whereas the voltage across an inductor leads thecurrent.
  • The reactance (effective resistance) of an inductor in an ACcircuit is given by X_L=omega l.
  • The reactance (effective resistance) of the capacitor in an ACcircuit is given by X_C=frac{1}{omega c}.
The answer should be expressed interms of R, L, C and omega.
Part B
Now find the tangent of the phaseangle phi between thecurrent and the overallvoltage in this circuit.Express tan(phi) in terms of R, L, C, and omega.

Part C
magine that the parameters R, L, C, and the amplitude of the voltage V_0 arefixed, but the frequency of thevoltage source is changeable. If the frequency of the source ischanged from a very low one to a very high one, the currentamplitudeI_0 will also change. The frequency at whichI_0 is at a maximum is calledresonance. Find the frequencyomega_0 at which the circuitreaches resonance.Theanswer should be expressed in terms ofany or all of R, L, and C.

Part D
What is the phase angle phi between the voltageand the current whenresonance is reached?

Part E
Now imagine that the parametersR, L, omega, and the amplitude of thevoltage V_0 are fixed but that the value of C can be changed. This isone of the easiestparameters to change when "tuning" such (radio frequency) circuitsin order to make them resonate. This is because the capacitance canbe changedjust by moving the capacitor plates closer or fartherapart. Find the resonance value of capacitance C_0.The answer should be expressed interms of L and omega.
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Homework Answers

Answer #1
a)The formula for the impedance is b)c)At resonance So at the resonance the resonance frequency isd)The phase angle at the resonance is given by
answered by: Psalm91
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Answer #2
where is part E?
answered by: vikash
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Answer #3

e) C_0 = 1 / Lω^2

source: Rearrange ω equation at resonance; Mastering Physics
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