Question

A vacuum cleaner motor can be viewed as an inductor with an inductance of \(100 . \mathrm{mH} .\) For a \(60.0-\mathrm{Hz} \mathrm{AC}\) voltage, \(V_{\mathrm{rms}}=115 \mathrm{~V}\), what capacitance must be in series with the motor to maximize the power output of the vacuum cleaner?

Short answer

To maximize the power output of a vacuum cleaner with a motor having an inductance of 100 mH and operating at a frequency of 60 Hz, with an RMS voltage of 115 V, a capacitance of approximately 70.5 µF should be in series with the motor.

Step-by-step solution

Calculate the inductive reactance (X_L)

First, we need to find the inductive reactance of the motor. The formula for inductive reactance is: $$X_L = 2\pi fL$$ Where \(X_L\) is the inductive reactance, \(f\) is the frequency (60 Hz), and \(L\) is the inductance in henry (100 mH = 0.1 H). Plugging in the given values, we get: $$X_L = 2\pi (60)(0.1)$$ $$X_L \approx 37.7 \ \Omega$$

Calculate the capacitive reactance (X_C)

To minimize the impedance, we need the capacitive reactance to be equal to the inductive reactance. Thus, we can set \(X_C = X_L\) and use the formula for capacitive reactance: $$X_C = \frac{1}{2\pi fC}$$ Where \(X_C\) is the capacitive reactance, \(f\) is the frequency (60 Hz), and \(C\) is the capacitance. Since \(X_C = X_L\), we can rewrite the equation as: $$\frac{1}{2\pi fC} = 37.7$$

Solve for capacitance (C)

Now we can isolate the capacitance (C) in the equation and solve for it: $$C = \frac{1}{2\pi fX_L}$$ Plugging in the values for frequency (60 Hz) and inductive reactance (37.7 Ω), we get: $$C = \frac{1}{2\pi (60)(37.7)}$$ $$C \approx 70.5 \ \mu\text{F}$$ So, a capacitance of approximately 70.5 µF in series with the motor will maximize the power output of the vacuum cleaner.

Key concepts

Inductive Reactance

Inductive reactance (\(X_L\)) describes the opposition a coil or inductor in an AC circuit presents to the current, depending on the frequency of the AC voltage applied. The higher the frequency or the greater the inductance, the larger the reactance. It's given by the formula \(X_L = 2\pi fL\), with \(f\) representing the frequency of the oscillations and \(L\) the inductance measured in henrys. In our vacuum cleaner motor example, a 100 mH inductor facing a 60 Hz AC supply has an inductive reactance of approximately \(37.7 \Omega\). Understanding inductive reactance is crucial for AC circuit analysis and goes hand in hand with capacitive reactance for circuit optimization.

Behind this concept is Faraday's law of electromagnetic induction, which explains how the varying magnetic field within an inductor induces a voltage opposing the source. This induced voltage is what creates inductive reactance, acting analogously to resistance in a DC circuit but crucially, only affecting AC circuits.

AC Circuit Optimization

Optimizing AC circuits involves adjusting components such as resistors, capacitors, and inductors to achieve desired performance, like maximizing power output as in our vacuum cleaner motor example. For optimization, one key factor is impedance, which is a measure of the total opposition to current flow. Ideally, AC circuits are most efficient when the impedance is minimized. This is achieved when inductive reactance and capacitive reactance are balanced, leading to a condition known as resonance, which we will explore more in the next section.

To achieve optimization, you need to understand the role of each component. Capacitors store energy in the form of an electric field and release it, while inductors store energy in a magnetic field. Altering capacitance values can fine-tune the circuit's response to different frequencies, harmonize with the inductive reactance, and thus optimize the circuit for a variety of applications. The math used to find the correct capacitance involves equating capacitive and inductive reactance and solving for the desired component value.

Series Resonance

Series resonance occurs in an AC circuit when the inductive reactance \(X_L\) and the capacitive reactance \(X_C\) are equal and opposite in phase, effectively canceling each other out. This leads to a minimum impedance condition, thereby maximizing the current flow given a constant voltage supply. The condition of series resonance is achieved at a specific frequency, known as the resonant frequency, which for our vacuum cleaner's motor can be determined with the provided formulas.

At resonance, the circuit behaves purely resistive as if it only contains resistive elements, allowing power to be delivered efficiently. For the vacuum cleaner motor, introducing the correct capacitance in series creates this resonance, optimizing its operation and power output. The calculated capacitance \(70.5 \mu\text{F}\) would ensure that the motor operates at peak efficiency at the given frequency of 60 Hz. This principle of series resonance is widely applied in electronic devices, such as radio tuners, where it helps in selecting the desired signal amidst a myriad of different frequencies.