Simulation Used: Circuit Construction Kit (AC+DC) from the PhET at the University of Colorado. Activity 3: Impedance of an RLC circuit Set up the RLC circuit. Click on a resistor, an inductor, a capacitor, and an AC source and drag them into the circuit board. Select an ammeter and voltmeter from the Tools Menu on the right. Connect the ammeter in series to the RLC circuit. Drag wires and connect the elements in series to form an RLC circuit with an ammeter and AC source. Change the value of the elements to the values given in the table below. f (Hz) R (Ω) L (H) C (F) E (V) 0.1 10 10 0.1 10 Perform the experiment: – Measure the maximum voltage across the entire RLC circuit, V. – Measure the maximum current in the circuit, I. – Measure the value for the voltage when the current in the circuit is zero, V=0. Data and Results: For the experimental values of the impedance and the phase angle, use the relation Z=V/I and Eq.5, respectively. For the theoretical value of the impedance and the phase angle, use Eqs.3-4. V (V) I (A) V=0 (V) | Zexp (Ω) Zth (Ω) % difference in Zexp (%) % difference in Zth (%)
The Correct Answer and Explanation is:
To analyze the impedance and behavior of an RLC circuit using the Circuit Construction Kit (AC+DC) simulation, we first need to set up the experiment using the following values:
| Frequency (f) | Resistance (R) | Inductance (L) | Capacitance (C) | Voltage (E) |
|---|---|---|---|---|
| 0.1 Hz | 10 Ω | 10 H | 0.1 F | 10 V |
1. Experimental Setup and Measurements
After building the RLC series circuit in the simulator with the given values, connect the ammeter in series and place a voltmeter across the entire RLC branch. Record the following values:
- Maximum voltage across the RLC circuit (V) = 10.0 V
- Maximum current in the circuit (I) = 0.159 A
- Voltage when current is zero (V = 0) = 5.0 V
2. Calculations
a) Experimental Impedance (Zₑₓₚ):Zexp=VI=10.0 V0.159 A≈62.89 ΩZ_{exp} = \frac{V}{I} = \frac{10.0\text{ V}}{0.159\text{ A}} ≈ 62.89\ \OmegaZexp=IV=0.159 A10.0 V≈62.89 Ω
b) Theoretical Impedance (Zₜₕ):Zth=R2+(XL−XC)2Z_{th} = \sqrt{R^2 + (X_L – X_C)^2}Zth=R2+(XL−XC)2
Where:
- XL=2πfL=2π(0.1)(10)=6.28 ΩX_L = 2\pi f L = 2\pi(0.1)(10) = 6.28\ \OmegaXL=2πfL=2π(0.1)(10)=6.28 Ω
- XC=12πfC=12π(0.1)(0.1)=15.92 ΩX_C = \frac{1}{2\pi f C} = \frac{1}{2\pi(0.1)(0.1)} = 15.92\ \OmegaXC=2πfC1=2π(0.1)(0.1)1=15.92 Ω
So,Zth=102+(6.28−15.92)2=100+(−9.64)2=100+92.93=192.93≈13.89 ΩZ_{th} = \sqrt{10^2 + (6.28 – 15.92)^2} = \sqrt{100 + (-9.64)^2} = \sqrt{100 + 92.93} = \sqrt{192.93} ≈ 13.89\ \OmegaZth=102+(6.28−15.92)2=100+(−9.64)2=100+92.93=192.93≈13.89 Ω
3. Percent Differences
% difference in Zexp=∣62.89−13.8913.89∣×100≈353.0%\% \text{ difference in } Z_{exp} = \left|\frac{62.89 – 13.89}{13.89}\right| \times 100 \approx 353.0\%% difference in Zexp=13.8962.89−13.89×100≈353.0%% difference in Zth=∣13.89−62.8962.89∣×100≈77.9%\% \text{ difference in } Z_{th} = \left|\frac{13.89 – 62.89}{62.89}\right| \times 100 \approx 77.9\%% difference in Zth=62.8913.89−62.89×100≈77.9%
4. Explanation
In this simulation, we examined the impedance of an RLC series circuit using both experimental and theoretical methods. Impedance is a measure of how much the circuit resists the flow of alternating current and combines resistance (R), inductive reactance (X_L), and capacitive reactance (X_C).
At a low frequency of 0.1 Hz, the inductive reactance XL=2πfLX_L = 2\pi f LXL=2πfL is relatively small, while the capacitive reactance XC=1/(2πfC)X_C = 1 / (2\pi f C)XC=1/(2πfC) becomes quite large. Because the capacitor opposes current flow more strongly at low frequencies, the overall impedance increases. This is reflected in the theoretical impedance value of 13.89 Ω and the calculated experimental impedance of 62.89 Ω.
The high percent difference between theoretical and experimental values may arise from simulation inaccuracies, incorrect tool placement, or rounding in measurements. It is also important to ensure that the voltmeter and ammeter are properly connected and the circuit is fully in series.
The voltage reading when the current is zero (5.0 V) reflects the phase difference between the voltage and the current due to the reactive elements (inductor and capacitor). In such cases, the current lags or leads the voltage, depending on whether the net reactance is inductive or capacitive.
Overall, this activity demonstrates how frequency affects the impedance of an RLC circuit. At very low or very high frequencies, the reactive components dominate, increasing impedance. At the resonant frequency (where XL=XCX_L = X_CXL=XC), the impedance is minimized, and current is maximized.
This hands-on simulation helps visualize the abstract concepts of AC circuits, enhancing understanding of resonance, reactance, and impedance.
