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Q101PE

Expert-verifiedFound in: Page 864

Book edition
1st Edition

Author(s)
Paul Peter Urone

Pages
1272 pages

ISBN
9781938168000

**An RLC series circuit has a $2.50\mathrm{\xce\copyright}$ ****resistor, a $100\mathrm{\xce\xbc}H$**** inductor, and a $80.0\mathrm{\xce\xbc}F$ ****capacitor. (a) Find the circuit's impedance at $120Hz$ ****. (b) Find the circuit's impedance at $5.00kHz$ ****. (c) If the voltage source has** ${V}_{rms}=5.60V$ **, what is ${I}_{rms}$**** at each frequency? (d) What is the resonant frequency of the circuit? (e) What is ${I}_{rms}$** **at resonance?**

- The circuit's impedance at
**$120Hz$**is $16.7\mathrm{\xce\copyright}$ . - The circuit's impedance at $5kHz$
- The ${I}_{rms}$ at 120 Hz is 0.335 A and at 5 kHz is $1.51A$ .
- The resonant frequency of the circuit is 1780 Hz .
- The ${I}_{rms}$ at resonance frequency is $2.24A$

**A capacitor is a two-terminal electrical component that may store energy in the form of an electric charge. It is made up of two electrical wires separated by a certain distance. The space between the conductors can be filled with vacuum or a dielectric, which is an insulating substance.**

The resistance of the RLC circuit is $R=2.50\mathrm{\xce\copyright}$

The capacitance of the RLC circuit is $C=80.0\mathrm{\xce\xbc}F\left(\frac{{10}^{-6}F}{1\mathrm{\xce\xbc}F}\right)=8.00\xc3\u2014{10}^{-5}F$

The inductance of the RLC circuit is $L=100\mathrm{\xce\xbc}H\left(\frac{{10}^{-6}H}{1\mathrm{\xce\xbc}H}\right)=1.00\xc3\u2014{10}^{-4}H$

The reactance of an RLC circuit can be expressed as,

$Z=\sqrt{{R}^{2}+{\left({X}_{L}-{X}_{C}\right)}^{2}}$â€¦â€¦â€¦â€¦â€¦â€¦(1)

This means we'll need to figure out the capacitor and inductor's active resistances, which can be given as,

${X}_{C}=\frac{1}{2\mathrm{\xcf\u20ac}fC}$â€¦â€¦â€¦â€¦â€¦â€¦â€¦.(2)

${X}_{L}=2\mathrm{\xcf\u20ac}fL$â€¦â€¦â€¦â€¦â€¦â€¦â€¦..(3)

We have ${f}_{1}=120Hz$, therefore substituting the given data in equations (2) and (3), we get,

${X}_{C}=\frac{1}{2\mathrm{\xcf\u20ac}\xc3\u2014120Hz\xc3\u20148.00\xc3\u2014{10}^{-5}F}\phantom{\rule{0ex}{0ex}}=16.6\mathrm{\xce\copyright}\phantom{\rule{0ex}{0ex}}{X}_{L}=2\mathrm{\xcf\u20ac}\xc3\u2014120Hz\xc3\u20141.00\xc3\u2014{10}^{-4}H\phantom{\rule{0ex}{0ex}}=0.075\mathrm{\xce\copyright}$

Now, we use the above-obtained values of X_{C} and X_{L} in equation (3), and we get

$Z=\sqrt{{\left(2.5\mathrm{\xce\copyright}\right)}^{2}+{\left(0.075\mathrm{\xce\copyright}-16.6\mathrm{\xce\copyright}\right)}^{2}}\phantom{\rule{0ex}{0ex}}=16.7\mathrm{\xce\copyright}$

Therefore,** **the circuit's impedance at** 120 Hz **is $16.7\mathrm{\xce\copyright}$.

We have ${f}_{2}=5kHz\left(\frac{{10}^{3}Hz}{1kHz}\right)=5\xc3\u2014{10}^{3}Hz$, therefore substituting the given data in equations (2) and (3), we get,

${X}_{C}=\frac{1}{2\mathrm{\xcf\u20ac}\xc3\u20145\xc3\u2014{10}^{3}Hz\xc3\u20148.00\xc3\u2014{10}^{-5}F}\phantom{\rule{0ex}{0ex}}=0.4\mathrm{\xce\copyright}\phantom{\rule{0ex}{0ex}}{X}_{L}=2\mathrm{\xcf\u20ac}\xc3\u20145\xc3\u2014{10}^{3}Hz\xc3\u20141.00\xc3\u2014{10}^{-4}H\phantom{\rule{0ex}{0ex}}=3.14\mathrm{\xce\copyright}$

Now, we use the above-obtained values of X_{C} and X_{L} in equation (3), and we get

$Z=\sqrt{{\left(2.5\mathrm{\xce\copyright}\right)}^{2}+{\left(3.14\mathrm{\xce\copyright}-0.4\mathrm{\xce\copyright}\right)}^{2}}\phantom{\rule{0ex}{0ex}}=3.7\mathrm{\xce\copyright}$

Therefore,** **the circuit's impedance at** 5 kHz **is $3.7\mathrm{\xce\copyright}$ .

According to Ohm's law, the root-mean-square intensity may be calculated using the reactance and the rms potential difference such that,

${I}_{rms}=\frac{{V}_{rms}}{Z}$â€¦â€¦â€¦â€¦â€¦.(4)

For the frequency 120 Hz , the value of impedance is $Z=16.7\mathrm{\xce\copyright}$, therefore

${I}_{rms}=\frac{5.6V}{16.7\mathrm{\xce\copyright}}\phantom{\rule{0ex}{0ex}}=0.335A$

For the frequency 5 kHz , the value of impedance is $Z=3.7\mathrm{\xce\copyright}$, therefore

${I}_{rms}=\frac{5.6V}{3.7\mathrm{\xce\copyright}}\phantom{\rule{0ex}{0ex}}=1.52A$

Therefore, the ${I}_{rms}$ at 120 Hz is 0.335 A and at 5 kHz is 1.51 A .

The resonant frequency can be calculated using the expression,

${f}_{0}=\frac{1}{2\mathrm{\xcf\u20ac}\sqrt{LC}}$â€¦â€¦â€¦â€¦â€¦(5)

That indicates that in our case, the end outcome will be

${f}_{0}=\frac{1}{2\mathrm{\xcf\u20ac}\sqrt{\left(1.00\xc3\u2014{10}^{-4}H\right)\xc3\u2014\left(8.00\xc3\u2014{10}^{-5}F\right)}}\phantom{\rule{0ex}{0ex}}=1780Hz$

Therefore,** **the resonant frequency of the circuit is 1780 Hz .

Remember that the resonant frequency occurs when the capacitor and inductor reactance are equal. If we look at the total reactance formula, this suggests that our resistance will be the same as the resistor. To put it another way, the rms current will be,

${I}_{rms}=\frac{5.6V}{2.5\mathrm{\xce\copyright}}\phantom{\rule{0ex}{0ex}}=2.24A$

Therefore, the ${I}_{rms}$ at resonance is $2.24A$ .

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