Question

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

Short answer

Question: In a series RLC circuit connected to an AC power source with maximum voltage 220 V and frequency 60.0 Hz, the resistance is 50.0 Ω, inductance is 0.200 H, and capacitance is 0.040 mF. Calculate the following quantities: a) inductive reactance b) capacitive reactance c) impedance of the circuit d) maximum current through the circuit e) maximum potential difference across each circuit element Answer: a) Inductive reactance, \(X_L = 75.4 \Omega\). b) Capacitive reactance, \(X_C = 66.32 \Omega\). c) Impedance of the circuit, \(Z = 54.8 \Omega\). d) Maximum current through the circuit, \(I_{m} = 4.01 A\). e) Maximum potential difference across each circuit element: \(V_R = 200.5 V\), \(V_L = 302.3 V\), and \(V_C = 266.0 V\).

Step-by-step solution

Calculate inductive reactance \(X_L\)

First, we will find the inductive reactance \(X_L\) using the formula \(X_L=2\pi fL\). Plug in the given values of frequency and inductance to get \(X_L = 2\pi(60.0Hz)(0.200H)\).

Calculate capacitive reactance \(X_C\)

Next, we will find the capacitive reactance \(X_C\) using the formula \(X_C=\frac{1}{2\pi fC}\). Plug in the given values of frequency and capacitance to get \(X_C = \frac{1}{2\pi(60.0Hz)(0.040\times10^{-3}F)}\).

Calculate impedance of the circuit \(Z\)

Now, we will calculate the impedance \(Z\) using the formula \(Z=\sqrt{R^2+(X_L-X_C)^2}\). Plug in the values of resistances and reactances from Steps 1 and 2 to get \(Z = \sqrt{(50.0\Omega)^2+(X_L-X_C)^2}\).

Calculate maximum current \(I_{m}\)

Then, we will find the maximum current \(I_{m}\) through the circuit using the formula \(I_{m}=\frac{V_{m}}{Z}\). Plug in the values of maximum voltage and impedance from Steps 3 to get \(I_{m} = \frac{220V}{Z}\).

Calculate maximum potential difference across each circuit element

Finally, we will find the maximum potential difference across each circuit element using the formulas \(V_R=I_{m}R\), \(V_L=I_{m}X_L\), \(V_C=I_{m}X_C\). Plug in the values of maximum current, resistance, and reactances from Steps 1, 2, and 4 to get \(V_R = I_{m}(50.0\Omega)\), \(V_L = I_{m}X_L\), and \(V_C = I_{m}X_C\).

Key concepts

Inductive Reactance

Inductive reactance is a property of coils in AC circuits, symbolized by \(X_L\). It represents how much a coil resists the flow of alternating current. This resistance arises because the current change induces a magnetic field and opposing voltage in the coil.
You calculate inductive reactance using the formula \(X_L = 2\pi f L\), where \(f\) is the frequency, and \(L\) is the inductance in henrys (H).
In an RLC circuit:

• Higher frequency or larger inductance increases inductive reactance.
• At higher inductive reactance, the coil more effectively resists changes in current.

Understanding inductive reactance helps manage how much a coil will oppose the current at given frequencies.

Capacitive Reactance

Capacitive reactance, denoted as \(X_C\), signifies how much a capacitor resists the change in AC current. In alternating current, capacitors alternately charge and discharge, resisting current change at a rate dependent on frequency and capacitance.
The formula for capacitive reactance is \(X_C = \frac{1}{2\pi f C}\), with \(f\) as the frequency and \(C\) as the capacitance.
In analysis of AC circuits:

• Low frequency results in high capacitive reactance; the circuit resists current changes more.
• Small capacitance also increases \(X_C\), causing more resistance to current flow.

A good grasp of capacitive reactance enables better control over how capacitors influence current flow.

Impedance Calculation

Impedance, represented by \(Z\), is the total opposition to AC current flow in a circuit. It's a combination of resistance (\(R\)) and reactance (\(X\)). In RLC circuits, it combines resistive, inductive, and capacitive opposition.
Calculate it using: \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
Where:

• \(R\) is resistance.
• \(X_L\) is inductive reactance.
• \(X_C\) is capacitive reactance.

Impedance affects:

• Total current flow: High impedance reduces current.
• Phase angles between current and voltage.

Understanding impedance is critical for designing circuits and predicting behavior at different frequencies.

Maximum Current

The maximum current, \(I_m\), in an AC circuit occurs when voltage across the circuit reaches its peak value. Knowing the impedance helps determine this maximum current.
Use the formula:
\[ I_m = \frac{V_m}{Z} \]
where \(V_m\) is the maximum applied voltage, and \(Z\) is the impedance.
Key points:

• Higher impedance means lower maximum current.
• Voltage peaks cause the highest current flow.

Accurate maximum current calculations ensure the circuit components operate within safe limits.

AC Circuits

AC circuits are systems where alternating current (AC) is the source of electricity. Unlike direct current (DC), AC voltage and current alternate direction periodically, characterized by different frequencies and peak voltages.
Essential aspects of AC circuits:

• Reactance: Frequency-dependent resistance.
• Impedance: Combined opposition to current flow.
• Resonance: Circuit response at specific frequencies.

They find use in homes and industries due to ease of power generation and distribution. Understanding AC circuits involves recognizing how components like resistors, capacitors, and inductors affect current and voltage in various scenarios.