Problems

Section 3.10 Problems

Calculate the energy (in Joules) stored in a \(2000\mu\text{F}\) capacitor charged to 5V.
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Calculate the impedance \(Z_{AB}\) and provide your answer in the form \(Z_{AB}=a+ib\) and \(Z_{AB}=\left|z\right|e^{i\theta}\) for each of the circuits in Figure 3.10.1-Figure 3.10.3.

Figure 3.10.1.
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Figure 3.10.2.
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Figure 3.10.3.
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Design a low-pass RC filter that will attenuate a 60-Hz sinusoidal voltage by 12 dB relative to the dc gain. Use a \(100\Omega\) resistance. Explain in words why the low-pass RC filter attenuates the high frequencies.
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The circuit shown in Figure 3.10.4is used to trigger a device connected between terminals A and B. The device turns ON when \(V_{AB}\ge 6.0\text{V}\) and turns OFF when \(V_{AB}\le 2.0\text{V}\text{.}\) (Assume that connecting the device between terminals A and B has no effect o nthe rest of the circuit. In other words, the device effectively has almost infinite output resistance.)
1. Assume that the switch has been in position Y for a long time. Then, at \(t=0\text{s}\text{,}\) the switch moves to position X. How long after the switch is turned to position X does the device turn ON?
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2. After being left at position X for over one minute, the switch is turned to position Y. How long does it take the device to turn off after this change?
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3. Use Python to plot your charging and discharging voltage curves (as a function of time).
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Figure 3.10.4.
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Use phasors to derive an expression for the voltage gain and phase shift for the following LR circuit.

Figure 3.10.5.
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Sketch a graph of the magnitude of the impedance versus frequency for (a) a series RLC circuit and (b) a parallel RLC circuit. In each case, determine the phase of the impedance as the frequency passes through resonance.
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Use Python to plot the gain-versus-frequency curve and phase-versus-frequency curve for the circuit in Figure 3.10.6. I’d strongly recommend using Python to solve your system of equations.

Figure 3.10.6.
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Use phasors to answer the following questions about the circuit in Figure 3.10.7.
1. Show that the peak current through the resistor is
  
  \begin{equation*} I=\frac{V_0}{\sqrt{R^2 + \left(X_L^{-1}+X_C^{-1}\right)^{-2}}}\text{.} \end{equation*}
  
  (Hint: Find an equivalent impedance for the capacitor and inductor, and then examine the circuit behavior at a time when the resistor current \(I_R\) is purely real.)
  
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2. What is \(I_R\) in the limits \(\omega\rightarrow 0\) and \(\omega\rightarrow \infty\text{?}\)
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3. Find an expression for the resonance frequency \(\omega_0\text{?}\) What is \(I_R\) at this frequency?
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  Figure 3.10.7.
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