Lab AC Circuits Use The PhET RLC Simulation For Lab Goals ✓ Solved

Lab AC circuits. Use the PhET RLC simulation to: Lab Goals:

Lab AC circuits. Use the PhET RLC simulation to: Lab Goals: To find the frequency of resonance of a RLC series circuit; to study the phase shift between two signals; to investigate power in AC circuits.

Part 1 – RLC Circuit

1. Create a circuit with an AC voltage source in series with a resistor, inductor, and capacitor.

2. Note the AC source voltage and frequency.

3. Place voltage charts around each element and the AC source with consistent polarity; capture the waveforms.

4. Mark maximum/minimum voltages on each graph; qualitatively compare graphs, identify which are in phase, which are not, and note sign of amplitudes.

5. Using peak-to-RMS relation (VRMS = V0/√2) show that the RMS voltage of the source equals VRMS = √[ (VRMS,R)^2 + (VRMS,L - VRMS,C)^2 ].

6. Calculate the circuit impedance Z from R, L, C, and frequency.

7. Place a current chart after the source, note peak current, and using IRMS show consistency VRMS = IRMS · Z.

8. Change the frequency to resonance, let dynamics settle, and repeat steps 4–7; compare results and note similarities and differences.

Part 2 – Average Power

1. Create an RLC circuit with a chosen resonant frequency. For R = 10 Ω, record for at least seven frequencies (including points below, at, and above resonance): f [Hz], ω [rad/s], Z [Ω], IRMS [A], VRMS [V], PAVE [W], using IRMS and VRMS leaving the source and calculating average power.

2. Plot average power versus ω with labeled axes and units.

3. Repeat step 1–2 with R = 100 Ω and compare the two power curves; discuss the role of resistance in an RLC circuit.

Paper For Above Instructions

Introduction and Objectives

This laboratory exercise uses the PhET RLC simulation to explore three core topics in AC circuit analysis: resonance frequency of a series RLC circuit, phase relationships among voltages and current, and average power delivery. The objectives are to (1) locate the resonant frequency ω0 = 1/√(LC), (2) analyze waveform phase shifts between the source and element voltages, and (3) measure and plot average power as a function of angular frequency ω for two resistance values (R = 10 Ω and R = 100 Ω).

Part 1 — RLC Series Circuit: procedure and analysis

Construct a series circuit with an AC voltage source, resistor R, inductor L, and capacitor C. Record the nominal source amplitude V0 and operating frequency f (ω = 2πf). Display voltage charts across the source, resistor, inductor, and capacitor, ensuring consistent polarity so phase relationships are unambiguous (Alexander & Sadiku, 2003).

Qualitative waveform comparison: When plotted simultaneously, the resistor voltage VR is in phase with current i(t); the inductor voltage VL leads i(t) by +90°; the capacitor voltage VC lags i(t) by −90° (Nilsson & Riedel, 2019). Therefore, VR and i(t) peaks coincide; VL and VC show opposite phase tendencies. At non-resonant frequencies the vector sum of voltages satisfies Kirchhoff’s voltage law; signs of amplitudes indicate whether inductive or capacitive reactance dominates (Serway & Jewett, 2018).

RMS relations and phasor consistency: Using peak amplitudes V0 for each element, convert to RMS via VRMS = V0/√2. The theoretical RMS relation for a series RLC source voltage is:

Vsource,RMS = √[ (VRMS,R)^2 + (VRMS,L − VRMS,C)^2 ].

This expression follows from phasor addition: VR is in quadrature with (VL − VC), so the magnitude of the sum is the square root of sum-of-squares (Hayt & Kemmerly, 1986). Verify numerically by extracting peak values from charts and converting to RMS; discrepancies beyond simulation rounding indicate measurement or polarity errors.

Impedance calculation: For a frequency ω, compute reactances XL = ωL and XC = 1/(ωC). The circuit impedance magnitude is

|Z| = √[ R^2 + (XL − XC)^2 ].

Using measured IRMS from the current chart, check Ohm’s law VRMS ≈ IRMS · |Z|. At resonance (ω = ω0), XL = XC so |Z| = R and the current reaches its maximum value for a given VRMS (Sedra & Smith, 2015).

Expected resonance behavior and phase

At resonance the reactive voltages across L and C are equal in magnitude and opposite in phase; they cancel in the series sum, leaving only VR such that the source voltage is in phase with current (Purcell & Morin, 2013). Off-resonance, either inductive or capacitive reactance dominates: if XL > XC the circuit is net inductive and current lags the source; if XL

Part 2 — Average power versus frequency

For the average power study, choose component values L and C such that the resonant frequency ω0 = 1/√(LC) falls within a convenient range for the PhET controls. With R = 10 Ω, sample at least seven frequencies: three below, one at, and three above resonance. For each frequency compute ω = 2πf, calculate |Z| = √[R^2 + (ωL − 1/(ωC))^2], measure IRMS from the simulation, record VRMS at the source, and compute average power Pave. Two equivalent formulas may be used:

Pave = VRMS · IRMS · cosφ = I_rms^2 · R,

where cosφ = R/|Z| is the power factor (Nilsson & Riedel, 2019). The simplest and most robust is Pave = I_rms^2 R because only the resistive element dissipates real power.

Plot Pave versus ω. The expected curve is a Lorentzian-like peak centered at ω0. For low R (10 Ω) the peak will be sharp and high because the circuit Q (quality factor Q = ω0L/R) is large; the power is concentrated near resonance. For R = 100 Ω the peak will be lower and broader because increased damping (lower Q) reduces the resonant amplification of current (Kinsler et al., 2000).

Comparison and role of resistance

Comparing the R = 10 Ω and R = 100 Ω plots, expect three main differences: (1) peak Pave is much higher for lower R (since I_rms,max ≈ V_rms/R at resonance); (2) the bandwidth Δω (width at half-power) increases with R, so the resonance sharpness decreases; (3) the power factor near resonance approaches unity in both cases, but off resonance the larger R reduces sensitivity of Pave to frequency. Thus resistance controls both the amplitude of dissipated power and the sharpness of resonance (Bracewell, 2000).

Practical measurement notes

When extracting data from the PhET simulation, ensure charts have consistent polarity markers and allow the simulation to reach steady-state before recording peak values. Convert peaks to RMS (V0/√2) for consistency with analytical formulas. Use spreadsheet software to compute ω, |Z|, I_rms_predicted = V_rms / |Z|, and Pave = I_rms^2 R, and overlay measured and predicted curves to assess experimental accuracy.

Conclusion

By following the described procedure, one verifies fundamental AC circuit relationships: phasor voltage addition, impedance expression |Z| = √[R^2 + (ωL − 1/(ωC))^2], resonance at ω0 = 1/√(LC), and average power Pave = I_rms^2 R. The R-dependence of the power curve demonstrates how resistance damps resonance and controls dissipated energy. These behaviors are central to filter design, tuned circuits, and power-delivery systems (Alexander & Sadiku, 2003; Sedra & Smith, 2015).

References

• Alexander, C. K., & Sadiku, M. N. O. (2003). Fundamentals of Electric Circuits (3rd ed.). McGraw-Hill.
• Nilsson, J. W., & Riedel, S. A. (2019). Electric Circuits (11th ed.). Pearson.
• Sedra, A. S., & Smith, K. C. (2015). Microelectronic Circuits (7th ed.). Oxford University Press.
• Hayt, W. H., Kemmerly, J. E., & Durbin, S. M. (1986). Engineering Circuit Analysis (5th ed.). McGraw-Hill.
• Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers (10th ed.). Cengage Learning.
• PhET Interactive Simulations. (n.d.). RLC Circuits Simulation. University of Colorado Boulder. https://phet.colorado.edu
• Purcell, E. M., & Morin, D. J. (2013). Electricity and Magnetism (3rd ed.). Cambridge University Press.
• Kinsler, L. E., Frey, A. R., Coppens, A. B., & Sanders, J. V. (2000). Fundamentals of Acoustics (4th ed.). Wiley. (for resonance and Q-factor discussion)
• Griffiths, D. J. (2017). Introduction to Electrodynamics (4th ed.). Cambridge University Press.
• HyperPhysics. (n.d.). RLC circuits and resonance. Georgia State University. http://hyperphysics.phy-astr.gsu.edu