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Chapter 30: Problem 74

Laboratory experiments with series RLC circuits require some care, as these circuits can produce large voltages at resonance. Suppose you have a 1.00 - \(\mathrm{H}\) inductor (not difficult to obtain) and a variety of resistors and capacitors. Design a series RLC circuit that will resonate at a frequency (not an angular frequency) of \(60.0 \mathrm{~Hz}\) and will produce at resonance a magnification of the voltage across the capacitor or the inductor by a factor of 20.0 times the input voltage or the voltage across the resistor.

Short Answer

Expert verified
Answer: To design a series RLC circuit that resonates at 60 Hz with a magnification factor of 20, we need a capacitance of approximately 7.056 x 10^-5 F and a resistance of approximately 37.662 Ohms.

Step by step solution

01

Resonance frequency formula

The resonance frequency, \(f_0\), can be determined using the following formula: $$f_0 = \frac{1}{2 \pi \sqrt{LC}}$$ Here, \(L\) is the inductor value (1 H), \(C\) is the capacitance, and \(f_0\) is the desired resonance frequency (60 Hz).
02

Q-factor formula

The Q-factor, or quality factor, of a series RLC circuit can be determined using the following formula: $$Q = \frac{1}{R} \sqrt{\frac{L}{C}}$$ Here, \(R\) is the resistance, and \(Q\) is the Q-factor.
03

Magnification factor formula

The magnification factor, \(M\), for a series RLC circuit is equal to the Q-factor: $$M = Q$$
04

Calculate the capacitance

Using the resonance frequency formula, we can solve for the capacitance: $$C = \frac{1}{(2 \pi f_0)^2 \cdot L}$$ Substitute the given values: $$C = \frac{1}{(2 \pi (60))^2 \cdot 1}$$ $$C \approx 7.056 \times 10^{-5} \mathrm{~F}$$
05

Calculate the Q-factor

Next, we will calculate the Q-factor using the magnification factor formula, since \(M = Q\): $$Q = M = 20$$
06

Calculate the resistance

Now, we can determine the resistance with the Q-factor formula: $$R = \frac{1}{Q} \sqrt{\frac{L}{C}}$$ Substitute the calculated values of \(L\), \(C\), and \(Q\): $$R = \frac{1}{20} \sqrt{\frac{1}{7.056 \times 10^{-5}}}$$ $$R \approx 37.662 \mathrm{~\Omega}$$
07

Design the circuit

Finally, we have determined the necessary values for the series RLC circuit that resonates at 60 Hz with a magnification factor of 20: - Inductor: \(L = 1.00 \mathrm{~H}\) - Capacitor: \(C \approx 7.056 \times 10^{-5} \mathrm{~F}\) - Resistor: \(R \approx 37.662 \mathrm{~\Omega}\) The student should now create a series RLC circuit using these values to achieve the desired resonance frequency and magnification factor.

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Most popular questions from this chapter

A transformer has 800 turns in the primary coil and 40 turns in the secondary coil. a) What happens if an AC voltage of \(100 . \mathrm{V}\) is across the primary coil? b) If the initial \(A C\) current is \(5.00 \mathrm{~A}\), what is the output current? c) What happens if a DC current at \(100 .\) V flows into the primary coil? d) If the initial DC current is \(5.00 \mathrm{~A}\), what is the output current?

A \(200-\Omega\) resistor, a \(40.0-\mathrm{mH}\) inductor and a \(3.0-\mu \mathrm{F}\) capacitor are connected in series with a time-varying source of emf that provides \(10.0 \mathrm{~V}\) at a frequency of \(1000 \mathrm{~Hz}\). What is the impedance of the circuit? a) \(200 \Omega\) b) \(228 \Omega\) c) \(342 \Omega\) d) \(282 \Omega\)

An AC power source with \(V_{\mathrm{m}}=220 \mathrm{~V}\) and \(f=60.0 \mathrm{~Hz}\) is connected in a series RLC circuit. The resistance, \(R\), inductance, \(L\), and capacitance, \(C\), of this circuit are, respectively, \(50.0 \Omega, 0.200 \mathrm{H},\) and \(0.040 \mathrm{mF}\). Find each of the following quantities: a) the inductive reactance b) the capacitive reactance c) the impedance of the circuit d) the maximum current through the circuit e) the maximum potential difference across each circuit element

A series RLC circuit has resistance \(R\), inductance \(L\), and capacitance \(C\). At what time does the energy in the circuit reach half of its initial value?

At what frequency will a \(10.0-\mu \mathrm{F}\) capacitor have reactance \(X_{C}=200 . \Omega ?\)
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