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Chapter 28: Problem 12

If you measure the rms voltages across the resistor, capacitor, and inductor in a series \(R L C\) circuit, will they add to the rms generator voltage?

Short Answer

Expert verified
No, the rms voltages across the resistor, inductor, and capacitor in a series RLC circuit do not add up directly to the rms voltage of the generator due to phase differences.

Step by step solution

01

Understanding the RLC Circuit

In a series RLC circuit where all the components are connected in series, the same current flows through each component. However, voltage across each component (resistor, inductor, and capacitor) is different. The voltage across each component is given by \(V = I \cdot Z\) where \(I\) is the current and \(Z\) is the impedance of the component.
02

Kirchhoff's Voltage Law (KVL) in AC Circuits

In an AC circuit, according to Kirchhoff's Voltage Law (KVL), the sum of the emf (or potential differences) in any closed loop or mesh in a network is always equal to the sum of the potential drops in that loop. However, in AC circuits, this doesn't simply apply as vector addition due to the presence of phase angles.
03

Calculation of Voltages

The voltage across the resistor is \(V_R=I \cdot R\), the voltage across the inductor is \(V_L=I \cdot X_L\) and across the capacitor, it's \(V_C=I \cdot X_C\). However, owing to the phase shifts because of the inductive and capacitive reactance, the voltages across L and C are not in phase with the current, and hence they do not just directly add up.
04

Final Conclusion

Therefore, in an RLC circuit, the rms voltages across the resistor, capacitor, and inductor will not simply add to the rms generator voltage. The correct relationship in an RLC series circuit is \(V = \sqrt{V_R^2 + (V_L - V_C)^2}\)

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Most popular questions from this chapter

A \(40-\) W fluorescent lamp has power factor 0.85 and operates from the \(120-\mathrm{V}\) rms AC power line. How much current does it draw?

You're planning a semester in China, so you want to purchase a transformer to step the 220 -V Chinese power down to \(120 \mathrm{V}\) to power your stereo. (a) If the transformer's primary has 660 turns, how many should the secondary have? (b) You can save money with a transformer whose maximum primary current is \(1.4 \mathrm{A} .\) If your stereo draws \(2.9 \mathrm{A},\) will this transformer work?

The same AC voltage appears across a capacitor and a resistor, and the same rms current flows in each. Is the power dissipation the same in each?

(a) A \(2.2-\mathrm{H}\) inductor is connected across \(120-\mathrm{V}\) rms, \(60-\mathrm{Hz}\) power. Find the rms inductor current. (b) Repeat if the same inductor is connected across the \(230-\mathrm{V}\) rms, \(50-\mathrm{Hz}\) power commonly used in Europe.

A 50 -mH inductor is connected across a \(10-\mathrm{V}\) rms AC generator, and a 2.0 -mA rms current flows. What's the generator frequency?
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