Problem 24 - College Physics

Question

\(\bullet\) In a series \(R-L-C\) circuit, \(R=400 \Omega, L=0.350 \mathrm{H},\) and \(C=0.0120 \mu \mathrm{F}\) (a) What is the resonance angular frequency of the circuit? (b) The capacitor can withstand a peak voltage of 550 \(\mathrm{V}\) . If the voltage source operates at the resonance frequency, what maximum voltage amplitude can it have if the maximum capacitor voltage is not exceeded?

Step-by-Step Solution

Answer

(a) \( \omega_0 \approx 4885.8 \, \text{rad/s} \); (b) Maximum voltage amplitude is approximately 2351.25 V.

Step 1: Understand the Resonant Frequency Formula

The resonance angular frequency \( \omega_0 \) of a series \( R-L-C \) circuit is given by the formula \[ \omega_0 = \frac{1}{\sqrt{LC}} \]where \( L \) is the inductance and \( C \) is the capacitance. This formula describes the frequency at which the impedance of the circuit is minimized.

Step 2: Substitute the Values

Substitute the given values \( L = 0.350 \, \text{H} \) and \( C = 0.0120 \, \mu \text{F} = 0.0120 \times 10^{-6} \, \text{F} \) into the formula.\[ \omega_0 = \frac{1}{\sqrt{0.350 \times 0.0120 \times 10^{-6}}} \]

Step 3: Calculate the Resonant Frequency

Calculate using the substitution from the previous step:\[ \omega_0 = \frac{1}{\sqrt{0.350 \times 0.012 \times 10^{-6}}} = \frac{1}{\sqrt{4.2 \times 10^{-9}}} \]Then simplify to find:\[ \omega_0 \approx \frac{1}{2.049 \times 10^{-4}} \approx 4885.8 \, \text{rad/s} \]

Step 4: Maximum Voltage Amplitude Calculation

Given that the capacitor can withstand a peak voltage of 550 V and that the circuit is at resonance, the maximum source voltage amplitude \( V_0 \) can be determined by:\[ V_c = \frac{V_0}{Q} = 550 \]where \( Q \) is the quality factor calculated by:\[ Q = \frac{\omega_0 L}{R} \]

Step 5: Compute the Quality Factor

Substitute the known values into the quality factor equation:\[ Q = \frac{4885.8 \times 0.350}{400} \approx \frac{1710.03}{400} \approx 4.275 \]

Step 6: Solve for Maximum Voltage Amplitude

Solving the equation from Step 4 using the value of \( Q \),\[ 550 = \frac{V_0}{4.275} \]Solving for \( V_0 \) gives:\[ V_0 = 550 \times 4.275 \approx 2351.25 \, \text{V} \]

Key Concepts

Series RLC Circuit

A series RLC circuit is a type of electrical circuit that combines three fundamental components: a resistor (R), an inductor (L), and a capacitor (C), all connected in a series alignment. This configuration is crucial in the study of resonance in electrical circuits.

Understanding the series RLC circuit is vital because it is the foundation for many applications, including filters and oscillators. In such circuits, the impedance, or total resistance to the flow of current, can be described by the combination of individual impedances offered by the resistor, inductor, and capacitor.

At resonance, the circuit behaves uniquely. The inductive and capacitive reactances cancel each other out, making the impedance minimal. This means the circuit can efficiently transfer energy with minimal resistance from the inductor and capacitor.

Series RLC circuits are valued for their ability to control current flow at certain frequencies, making them indispensable in tuning circuits for radios and other communication devices.

Quality Factor

The quality factor, often represented by the letter \(Q\), is a dimensionless parameter that describes the sharpness of the resonance of the circuit. It is linked to how efficiently a series RLC circuit can store and transfer energy. Let's dive deeper into this concept.

The quality factor is determined by the ratio of the resonance frequency times the inductance to the resistance, given by the formula:
\[ Q = \frac{\omega_0 L}{R} \]
where:

• \(\omega_0\) is the resonance angular frequency
• \(L\) is the inductance
• \(R\) is the resistance

When the \(Q\) factor is high, it signifies that the circuit has a narrow bandwidth, meaning it will resonate sharply at the resonance frequency. This results in lower energy losses, ideal for filters and oscillators where precise frequency control is needed.

On the other hand, a low \(Q\) factor indicates wider bandwidth and less resonance sharpness, which can be beneficial in applications requiring the passage of a range of frequencies. Understanding the quality factor helps in designing circuits to meet specific performance criteria.

Maximum Voltage Amplitude

The maximum voltage amplitude for a series RLC circuit is an important concept because it determines the highest voltage that the circuit can handle while still maintaining optimal performance.

In resonance conditions, the voltage across the inductor and capacitor can increase substantially. The relationship between the maximum source voltage amplitude \(V_0\) and the peak voltage \(V_c\) of the capacitor is defined as:
\[ V_c = \frac{V_0}{Q} \]
This formula helps ensure that the source voltage amplitude does not exceed what the capacitor can withstand. Here's why it's essential to manage this:

• Preventing over-voltage conditions can help avoid damage to the capacitor, leading to circuit failure.
• Ensuring the maximum voltage amplitude stays within limits keeps the circuit operating effectively and efficiently.

Managing the maximum voltage helps in protecting not just the capacitor but the entire circuit, ensuring longevity and reliability.

Capacitor Voltage

In a series RLC circuit, the capacitor voltage is significant, especially when running at its resonance frequency. At resonance, the voltages across the inductor and capacitor can be much larger than the source voltage.

This phenomenon occurs due to the low net impedance at resonance, leading to higher currents and subsequently higher voltages across these reactive components. However, it's crucial to ensure that the capacitor's voltage does not exceed its threshold, as this can result in the failure of the capacitor.

To maintain safety and performance, one must calculate and monitor the capacitor voltage using the quality factor and resonance facts:

• Understand the relationship: \( V_c = \frac{V_0}{Q} \)
• Assess the component ratings to prevent over-voltage.
• Implement safeguards, such as protective circuits, to maintain voltages within safe levels.

In summary, careful management of the capacitor voltage ensures efficient operation while protecting components from potential damage.