Laboratory Activity: Exploring Differential Equations in RC Circuits Using Wolfram Alpha

Objective: Understand and solve differential equations for basic RC (Resistor-Capacitor) circuits using computational tools including Wolfram Alpha, MATLAB, and Python. Explore the theoretical foundations, numerical methods, and practical applications of RC circuit analysis.

Part 1: Theoretical Foundation of RC Circuits

1.1 Introduction to RC Circuit Theory

An RC circuit consists of a resistor (R) and a capacitor (C) connected in series or parallel. These fundamental circuits are ubiquitous in electronic systems, serving purposes such as filtering, timing, signal coupling, and power supply smoothing. Understanding RC circuits is essential for electrical engineering students as they demonstrate fundamental principles of energy storage, time-dependent behavior, and first-order system dynamics.

The capacitor stores electrical energy in an electric field between its plates, while the resistor dissipates energy as heat. The interaction between these components creates time-dependent voltage and current behaviors that are described mathematically by first-order ordinary differential equations (ODEs).

1.2 Derivation of the RC Circuit Differential Equation

For a series RC circuit connected to a voltage source, we apply Kirchhoff’s Voltage Law (KVL) around the loop:

$V_s = V_R + V_C$

where $V_s$ is the source voltage, $V_R$ is the voltage across the resistor, and $V_C$ is the voltage across the capacitor.

Using Ohm’s Law for the resistor and the capacitor voltage-current relationship:

$V_R = iR$

$i = C\frac{dV_C}{dt}$

Substituting these relationships into the KVL equation:

$V_s = RC\frac{dV_C}{dt} + V_C$

Rearranging to standard first-order ODE form:

$\frac{dV_C}{dt} + \frac{1}{RC}V_C = \frac{V_s}{RC}$

Or equivalently:

$\frac{dV_C}{dt} = \frac{1}{RC}(V_s - V_C)$

This is a first-order linear ordinary differential equation with constant coefficients, which has well-established analytical and numerical solution methods.

1.3 The Time Constant and Circuit Dynamics

The time constant $\tau$ (tau) is a fundamental parameter in RC circuit analysis, defined as:

$\tau = RC$

The time constant represents the time required for the capacitor voltage to reach approximately 63.2% of its final value during charging, or to decay to approximately 36.8% of its initial value during discharging. This emerges naturally from the solution to the differential equation.

For a charging capacitor with initial voltage $V_0$ and source voltage $V_s$ , the analytical solution is:

$V_C(t) = V_s + (V_0 - V_s)e^{-t/\tau}$

When $V_0 = 0$ (uncharged capacitor):

$V_C(t) = V_s(1 - e^{-t/\tau})$

For a discharging capacitor (with $V_s = 0$ ):

$V_C(t) = V_0 e^{-t/\tau}$

The time constant provides insight into circuit response speed and is crucial for designing timing circuits, filters, and signal processing applications.

Part 2: Laboratory Equipment and Setup

2.1 Required Equipment

Equipment	Specifications	Quantity
DC Power Supply	0-30V, 0-3A adjustable	1
Digital Multimeter (DMM)	4½ digit, voltage and current measurement	2
Oscilloscope	100 MHz, 2-channel minimum	1
Resistors	1kΩ, 10kΩ, 100kΩ (±5%, ½W)	3 each
Capacitors	1μF, 10μF, 100μF (electrolytic, 25V)	3 each
Breadboard	Standard solderless breadboard	1
Connecting Wires	22 AWG, various lengths	10+
SPDT Switch	Toggle or push-button	1
Stopwatch	Digital, 0.01s resolution	1

2.2 Safety Precautions

Verify power supply voltage before connecting to circuit
Observe correct polarity when connecting electrolytic capacitors
Discharge capacitors completely before handling or modifying circuit
Do not exceed component voltage and power ratings
Ensure all connections are secure before applying power
Use proper grounding techniques to prevent equipment damage

2.3 Circuit Assembly Procedure

Measure and record actual component values using DMM (resistors and capacitors may vary from nominal values)
Insert resistor and capacitor into breadboard in series configuration
Connect SPDT switch to enable charging (position A: power supply connected) and discharging (position B: capacitor shorted through resistor)
Connect oscilloscope probes across capacitor terminals
Connect DMM in parallel with capacitor for voltage measurement
Verify all connections before applying power

Part 3: Computational Analysis Using Wolfram Alpha

3.1 Getting Started with Wolfram Alpha

Wolfram Alpha is a powerful computational knowledge engine that can solve differential equations, generate plots, and perform symbolic mathematics. Access it at www.wolframalpha.com.

3.2 Exercise 1: Charging Capacitor Analysis

Circuit Parameters:

Source voltage: $V_s = 12$ V
Resistance: $R = 2$ Ω
Capacitance: $C = 1$ F
Initial condition: $V_C(0) = 0$ V

Step 1: Calculate the time constant:

$\tau = RC = 2 \times 1 = 2 \text{ seconds}$

Step 2: Input the differential equation into Wolfram Alpha:

solve dV/dt = (1/2)*(12 - V), V(0) = 0

Step 3: Analyze the solution provided by Wolfram Alpha:

$V(t) = 12(1 - e^{-t/2})$

Step 4: Generate a plot by entering:

plot 12*(1 - e^(-t/2)) from t=0 to t=10

Step 5: Calculate key time points:

At $t = \tau = 2$ s: $V_C = 12(1 - e^{-1}) = 7.58$ V (63.2% of $V_s$ )
At $t = 2\tau = 4$ s: $V_C = 12(1 - e^{-2}) = 10.38$ V (86.5% of $V_s$ )
At $t = 3\tau = 6$ s: $V_C = 12(1 - e^{-3}) = 11.40$ V (95.0% of $V_s$ )
At $t = 5\tau = 10$ s: $V_C = 12(1 - e^{-5}) = 11.92$ V (99.3% of $V_s$ )

3.3 Exercise 2: Discharging Capacitor Analysis

Circuit Parameters:

Resistance: $R = 2$ Ω
Capacitance: $C = 1$ F
Initial condition: $V_C(0) = 12$ V (fully charged)

The differential equation for discharging:

$\frac{dV}{dt} = -\frac{1}{RC}V = -\frac{1}{2}V$

Wolfram Alpha input:

solve dV/dt = -V/2, V(0) = 12

Solution:

$V(t) = 12e^{-t/2}$

Plot command:

plot 12*e^(-t/2) from t=0 to t=10

3.4 Comparative Analysis

To visualize both charging and discharging on the same plot:

plot {12*(1-e^(-t/2)), 12*e^(-t/2)} from t=0 to t=10

Key Observations:

Both curves are exponential but mirror each other
Charging curve approaches $V_s$ asymptotically from below
Discharging curve approaches zero asymptotically from above
Both have the same time constant $\tau = 2$ s
Rate of change is maximum at $t = 0$ for both processes
After 5τ (10s), both processes are essentially complete (>99%)

Part 4: MATLAB and Python Implementation

4.1 MATLAB Code for RC Circuit Analysis

% RC Circuit Analysis - Charging and Discharging
% Parameters
R = 2;           % Resistance in Ohms
C = 1;           % Capacitance in Farads
Vs = 12;         % Source voltage in Volts
tau = R * C;     % Time constant

% Time vector
t = 0:0.01:10;   % 0 to 10 seconds

% Analytical solutions
Vc_charging = Vs * (1 - exp(-t/tau));
Vc_discharging = Vs * exp(-t/tau);

% Current during charging and discharging
Ic_charging = (Vs/R) * exp(-t/tau);
Ic_discharging = -(Vs/R) * exp(-t/tau);

% Create figure with subplots
figure('Position', [100, 100, 1200, 800]);

% Plot 1: Voltage comparison
subplot(2,2,1);
plot(t, Vc_charging, 'b-', 'LineWidth', 2);
hold on;
plot(t, Vc_discharging, 'r-', 'LineWidth', 2);
plot([tau tau], [0 Vs], 'k--', 'LineWidth', 1);
xlabel('Time (s)', 'FontSize', 12);
ylabel('Voltage (V)', 'FontSize', 12);
title('RC Circuit: Voltage vs Time', 'FontSize', 14);
legend('Charging', 'Discharging', '\tau = RC', 'Location', 'best');
grid on;

% Plot 2: Current comparison
subplot(2,2,2);
plot(t, Ic_charging*1000, 'b-', 'LineWidth', 2);
hold on;
plot(t, Ic_discharging*1000, 'r-', 'LineWidth', 2);
xlabel('Time (s)', 'FontSize', 12);
ylabel('Current (mA)', 'FontSize', 12);
title('RC Circuit: Current vs Time', 'FontSize', 14);
legend('Charging', 'Discharging', 'Location', 'best');
grid on;

% Plot 3: Phase plane (V vs I)
subplot(2,2,3);
plot(Vc_charging, Ic_charging*1000, 'b-', 'LineWidth', 2);
hold on;
plot(Vc_discharging, abs(Ic_discharging)*1000, 'r-', 'LineWidth', 2);
xlabel('Voltage (V)', 'FontSize', 12);
ylabel('Current (mA)', 'FontSize', 12);
title('Phase Plane: Current vs Voltage', 'FontSize', 14);
legend('Charging', 'Discharging', 'Location', 'best');
grid on;

% Plot 4: Energy stored in capacitor
E_charging = 0.5 * C * Vc_charging.^2;
E_discharging = 0.5 * C * Vc_discharging.^2;
subplot(2,2,4);
plot(t, E_charging, 'b-', 'LineWidth', 2);
hold on;
plot(t, E_discharging, 'r-', 'LineWidth', 2);
xlabel('Time (s)', 'FontSize', 12);
ylabel('Energy (J)', 'FontSize', 12);
title('Energy Stored in Capacitor', 'FontSize', 14);
legend('Charging', 'Discharging', 'Location', 'best');
grid on;

% Display key values
fprintf('Time Constant (tau): %.2f seconds\n', tau);
fprintf('Voltage at t=tau (charging): %.2f V (%.1f%% of Vs)\n', ...
    Vs*(1-exp(-1)), 100*(1-exp(-1)));
fprintf('Voltage at t=tau (discharging): %.2f V (%.1f%% of Vs)\n', ...
    Vs*exp(-1), 100*exp(-1));
fprintf('Time to reach 95%% of Vs: %.2f seconds (%.2f * tau)\n', ...
    -tau*log(0.05), -log(0.05));
fprintf('Time to reach 99%% of Vs: %.2f seconds (%.2f * tau)\n', ...
    -tau*log(0.01), -log(0.01));

% Numerical solution using ODE45
[t_ode, V_ode] = ode45(@(t,V) (Vs-V)/(R*C), [0 10], 0);
fprintf('\nNumerical vs Analytical Solution Error (RMS): %.2e V\n', ...
    sqrt(mean((interp1(t_ode, V_ode, t) - Vc_charging).^2)));

4.2 Python Code for RC Circuit Analysis

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
from scipy.optimize import curve_fit

# RC Circuit Analysis - Charging and Discharging

# Parameters
R = 2.0          # Resistance in Ohms
C = 1.0          # Capacitance in Farads
Vs = 12.0        # Source voltage in Volts
tau = R * C      # Time constant

# Time vector
t = np.linspace(0, 10, 1000)

# Analytical solutions
Vc_charging = Vs * (1 - np.exp(-t/tau))
Vc_discharging = Vs * np.exp(-t/tau)

# Current during charging and discharging
Ic_charging = (Vs/R) * np.exp(-t/tau)
Ic_discharging = -(Vs/R) * np.exp(-t/tau)

# Numerical solution using ODE solver
def rc_charging(V, t):
    return (Vs - V) / (R * C)

def rc_discharging(V, t):
    return -V / (R * C)

V_charging_numerical = odeint(rc_charging, 0, t)
V_discharging_numerical = odeint(rc_discharging, Vs, t)

# Create comprehensive plots
fig, axes = plt.subplots(2, 2, figsize=(15, 10))

# Plot 1: Voltage comparison
axes[0, 0].plot(t, Vc_charging, 'b-', linewidth=2, label='Charging (Analytical)')
axes[0, 0].plot(t, Vc_discharging, 'r-', linewidth=2, label='Discharging (Analytical)')
axes[0, 0].plot(t, V_charging_numerical, 'b--', linewidth=1, label='Charging (Numerical)')
axes[0, 0].plot(t, V_discharging_numerical, 'r--', linewidth=1, label='Discharging (Numerical)')
axes[0, 0].axvline(x=tau, color='k', linestyle='--', linewidth=1, label=f'τ = {tau}s')
axes[0, 0].axhline(y=Vs*0.632, color='gray', linestyle=':', linewidth=1)
axes[0, 0].set_xlabel('Time (s)', fontsize=12)
axes[0, 0].set_ylabel('Voltage (V)', fontsize=12)
axes[0, 0].set_title('RC Circuit: Voltage vs Time', fontsize=14, fontweight='bold')
axes[0, 0].legend(loc='best')
axes[0, 0].grid(True, alpha=0.3)

# Plot 2: Current comparison
axes[0, 1].plot(t, Ic_charging*1000, 'b-', linewidth=2, label='Charging')
axes[0, 1].plot(t, Ic_discharging*1000, 'r-', linewidth=2, label='Discharging')
axes[0, 1].set_xlabel('Time (s)', fontsize=12)
axes[0, 1].set_ylabel('Current (mA)', fontsize=12)
axes[0, 1].set_title('RC Circuit: Current vs Time', fontsize=14, fontweight='bold')
axes[0, 1].legend(loc='best')
axes[0, 1].grid(True, alpha=0.3)

# Plot 3: Power dissipation in resistor
P_charging = Ic_charging**2 * R
P_discharging = Ic_discharging**2 * R
axes[1, 0].plot(t, P_charging, 'b-', linewidth=2, label='Charging')
axes[1, 0].plot(t, P_discharging, 'r-', linewidth=2, label='Discharging')
axes[1, 0].set_xlabel('Time (s)', fontsize=12)
axes[1, 0].set_ylabel('Power (W)', fontsize=12)
axes[1, 0].set_title('Power Dissipation in Resistor', fontsize=14, fontweight='bold')
axes[1, 0].legend(loc='best')
axes[1, 0].grid(True, alpha=0.3)

# Plot 4: Energy stored in capacitor
E_charging = 0.5 * C * Vc_charging**2
E_discharging = 0.5 * C * Vc_discharging**2
axes[1, 1].plot(t, E_charging, 'b-', linewidth=2, label='Charging')
axes[1, 1].plot(t, E_discharging, 'r-', linewidth=2, label='Discharging')
axes[1, 1].set_xlabel('Time (s)', fontsize=12)
axes[1, 1].set_ylabel('Energy (J)', fontsize=12)
axes[1, 1].set_title('Energy Stored in Capacitor', fontsize=14, fontweight='bold')
axes[1, 1].legend(loc='best')
axes[1, 1].grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('rc_circuit_analysis.png', dpi=300, bbox_inches='tight')
plt.show()

# Print key calculations
print(f"Time Constant (τ): {tau:.2f} seconds")
print(f"Voltage at t=τ (charging): {Vs*(1-np.exp(-1)):.2f} V ({100*(1-np.exp(-1)):.1f}% of Vs)")
print(f"Voltage at t=τ (discharging): {Vs*np.exp(-1):.2f} V ({100*np.exp(-1):.1f}% of Vs)")
print(f"Time to reach 95% of Vs: {-tau*np.log(0.05):.2f} seconds ({-np.log(0.05):.2f} × τ)")
print(f"Time to reach 99% of Vs: {-tau*np.log(0.01):.2f} seconds ({-np.log(0.01):.2f} × τ)")

# Error analysis
error_charging = np.sqrt(np.mean((V_charging_numerical.flatten() - Vc_charging)**2))
error_discharging = np.sqrt(np.mean((V_discharging_numerical.flatten() - Vc_discharging)**2))
print(f"\nNumerical vs Analytical Solution Error (RMS):")
print(f"  Charging: {error_charging:.2e} V")
print(f"  Discharging: {error_discharging:.2e} V")

# Total energy dissipated
E_dissipated_charging = np.trapz(P_charging, t)
E_dissipated_discharging = np.trapz(P_discharging, t)
print(f"\nTotal Energy Dissipated in Resistor:")
print(f"  Charging: {E_dissipated_charging:.2f} J")
print(f"  Discharging: {E_dissipated_discharging:.2f} J")

Part 5: LTSpice Circuit Simulation

5.1 LTSpice Setup and Simulation

LTSpice is a powerful SPICE-based analog circuit simulator. To simulate the RC circuit:

Circuit Netlist for Charging:

* RC Circuit Charging Analysis
V1 N001 0 PULSE(0 12 0 1n 1n 10 20)
R1 N001 N002 2
C1 N002 0 1 IC=0
.tran 0 10 0 0.01
.backanno
.end

Circuit Netlist for Discharging:

* RC Circuit Discharging Analysis
V1 N001 0 PULSE(12 0 0 1n 1n 10 20)
R1 N001 N002 2
C1 N002 0 1 IC=12
.tran 0 10 0 0.01
.backanno
.end

5.2 Simulation Directives

.tran: Transient analysis from 0 to 10 seconds
PULSE: Voltage source that switches between levels
IC: Initial condition for capacitor voltage
Add voltage and current probes to visualize waveforms
Use .meas commands to automatically extract rise/fall times

Measurement Commands:

.meas TRAN Vtau FIND V(N002) AT {2}
.meas TRAN T63 WHEN V(N002)=7.58
.meas TRAN T95 WHEN V(N002)=11.4

Part 6: Experimental Scenarios and Analysis

Scenario 1: Effect of Varying Resistance

Objective: Investigate how resistance affects charging/discharging time.

Procedure:

Fix capacitance at C = 10μF
Test with R = 1kΩ, 10kΩ, 100kΩ
Measure time to reach 63.2% of Vs for each resistor
Record voltage vs. time data

Expected Results:

R (kΩ)	C (μF)	τ (ms)	Time to 63.2% Vs
1	10	10	~10 ms
10	10	100	~100 ms
100	10	1000	~1 s

Scenario 2: Effect of Varying Capacitance

Objective: Analyze capacitance impact on circuit dynamics.

Procedure:

Fix resistance at R = 10kΩ
Test with C = 1μF, 10μF, 100μF
Measure time constant for each configuration
Compare theoretical vs. experimental values

Scenario 3: Temperature Effects on Components

Objective: Study temperature coefficient effects on RC behavior.

Procedure:

Measure component values at room temperature (25°C)
Heat components to 50°C (using heat gun or oven)
Remeasure component values and time constant
Calculate temperature coefficient from measurements

Theory: Electrolytic capacitors typically have negative temperature coefficients (-2% to -5% per °C), while resistors vary based on composition.

Scenario 4: Non-Ideal Capacitor Behavior (ESR)

Objective: Investigate effects of equivalent series resistance (ESR).

Modified Circuit Model: Real capacitors have internal resistance (ESR) that affects performance, especially at high frequencies.

Procedure:

Measure ESR of electrolytic capacitor using LCR meter
Include ESR in simulation model
Compare ideal vs. non-ideal capacitor response
Calculate power dissipation in ESR

Scenario 5: Multiple Time Constant Circuits

Objective: Analyze cascaded RC networks with different time constants.

Circuit: Two RC stages in series (R1-C1 followed by R2-C2)

Procedure:

Design circuit with τ1 = 10ms and τ2 = 100ms
Apply step input and measure voltage at both capacitors
Compare single vs. dual time constant response
Analyze frequency response using Bode plots

Scenario 6: RC Circuit as Low-Pass Filter

Objective: Demonstrate RC circuit filtering properties.

Cutoff Frequency:

$f_c = \frac{1}{2\pi RC}$

Procedure:

Calculate theoretical cutoff frequency
Apply sinusoidal input at various frequencies (0.1fc to 10fc)
Measure output amplitude and phase shift
Plot frequency response (magnitude and phase)
Verify -3dB point at cutoff frequency
Confirm -20dB/decade roll-off in stopband

Part 7: Error Analysis and Uncertainty

7.1 Sources of Experimental Error

Component Tolerances:
- Resistors: Typically ±5% (color code: gold band)
- Capacitors: ±10% to ±20% (especially electrolytic types)
- Impact: Time constant can vary by ±15% to ±25%
Measurement Instrument Errors:
- DMM accuracy: ±(0.5% of reading + 2 digits)
- Oscilloscope timebase: ±0.01%
- Probe loading effects on high-impedance circuits
Environmental Factors:
- Temperature variations affecting component values
- Humidity effects on capacitor performance
- Power supply noise and ripple
Parasitic Effects:
- Stray capacitance in breadboard (~2-5pF per connection)
- Lead inductance (~1nH/mm)
- Contact resistance in breadboard connections

7.2 Uncertainty Calculations

Time Constant Uncertainty:

Using propagation of uncertainty for τ = RC:

$\frac{\delta\tau}{\tau} = \sqrt{\left(\frac{\delta R}{R}\right)^2 + \left(\frac{\delta C}{C}\right)^2}$

Example: For R = 10kΩ ±5% and C = 10μF ±10%:

$\frac{\delta\tau}{\tau} = \sqrt{(0.05)^2 + (0.10)^2} = 0.112 = 11.2\%$

$\tau = 100\text{ms} \pm 11.2\text{ms}$

7.3 Minimizing Experimental Errors

Use precision resistors (±1% or better) when accuracy is critical
Measure actual component values before circuit assembly
Allow components to stabilize at ambient temperature
Use short leads to minimize parasitic effects
Account for oscilloscope probe input capacitance (~15pF)
Repeat measurements multiple times and average results
Compare experimental results with simulation to identify systematic errors

Part 8: Report Writing and Documentation

8.1 Laboratory Report Structure

Your comprehensive report should include the following sections:

Title Page: Include lab title, your name, date, course information
Abstract: 150-200 word summary of objectives, methods, and key findings
Introduction: Background on RC circuits, theoretical foundation, objectives
Theory: Derivation of differential equations, analytical solutions, key equations
Experimental Setup: Equipment list, circuit diagrams, measurement procedures
Results: Data tables, graphs, computational outputs, simulation screenshots
Analysis: Comparison of theoretical, computational, and experimental results
Error Analysis: Sources of error, uncertainty calculations, improvement suggestions
Discussion: Interpretation of results, practical applications, limitations
Conclusion: Summary of findings, learning outcomes, future work
References: IEEE format citations
Appendices: Raw data, code listings, additional plots

8.2 Data Presentation Guidelines

Use consistent units throughout (SI preferred)
Include error bars on experimental data points
Label all axes with quantities and units
Use descriptive captions for figures and tables
Number equations, figures, and tables sequentially
Reference all figures and tables in text

8.3 Sample Data Table

Time (s)	Vc Theoretical (V)	Vc Experimental (V)	Error (%)
0.00	0.00	0.02	–
0.05	3.58	3.52	1.68
0.10	5.65	5.59	1.06
0.20	7.97	7.88	1.13
0.50	10.78	10.65	1.21
1.00	11.75	11.68	0.60

Part 9: Learning Outcomes and Assessment

9.1 Expected Learning Outcomes

Upon completion of this laboratory activity, students should be able to:

Derive and solve first-order differential equations for RC circuits
Calculate and interpret the time constant τ = RC
Use computational tools (Wolfram Alpha, MATLAB, Python) for circuit analysis
Design and build RC circuits for specific time constant requirements
Perform accurate experimental measurements using oscilloscope and DMM
Simulate circuits using SPICE-based software (LTSpice)
Analyze sources of error and calculate experimental uncertainty
Compare theoretical, computational, and experimental results
Document findings in professional technical report format
Apply RC circuit principles to practical applications (filters, timers, coupling)

9.2 Assessment Rubric

Category	Weight	Criteria
Theoretical Understanding	20%	Correct derivation of ODEs, understanding of time constant, analytical solutions
Computational Analysis	20%	Proper use of Wolfram Alpha, MATLAB/Python code functionality, simulation accuracy
Experimental Work	20%	Circuit assembly, measurement technique, data collection quality
Data Analysis	15%	Comparison of results, error analysis, uncertainty calculations
Report Quality	15%	Organization, clarity, graphs/tables, professional presentation
Discussion & Conclusions	10%	Insight, critical thinking, practical applications, limitations

References

[1] A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. Upper Saddle River, NJ: Prentice Hall, 1997.

[2] C. K. Alexander and M. N. O. Sadiku, Fundamentals of Electric Circuits, 6th ed. New York, NY: McGraw-Hill, 2017.

[3] R. C. Dorf and J. A. Svoboda, Introduction to Electric Circuits, 9th ed. Hoboken, NJ: Wiley, 2014.

[4] P. Horowitz and W. Hill, The Art of Electronics, 3rd ed. New York, NY: Cambridge University Press, 2015.

[5] J. W. Nilsson and S. A. Riedel, Electric Circuits, 11th ed. Upper Saddle River, NJ: Pearson, 2019.

[6] IEEE Standard for Transitions, Pulses, and Related Waveforms, IEEE Std 181-2011, IEEE, New York, NY, 2011.

[7] R. L. Burden and J. D. Faires, Numerical Analysis, 9th ed. Boston, MA: Brooks/Cole, 2011.

[8] S. Wolfram, The Mathematica Book, 5th ed. Champaign, IL: Wolfram Media, 2003.

[9] MATLAB Documentation, MathWorks, [Online]. Available: https://www.mathworks.com/help/matlab/

[10] LTspice XVII User Guide, Analog Devices, 2021. [Online]. Available: https://www.analog.com/en/design-center/design-tools-and-calculators/ltspice-simulator.html

Appendix A: Quick Reference Formulas

Time Constant: $\tau = RC$

Charging (V₀ = 0): $V_C(t) = V_s(1 - e^{-t/\tau})$

Discharging: $V_C(t) = V_0 e^{-t/\tau}$

Current (charging): $i(t) = \frac{V_s}{R}e^{-t/\tau}$

Energy in capacitor: $E = \frac{1}{2}CV_C^2$

Power in resistor: $P = i^2R$

Cutoff frequency: $f_c = \frac{1}{2\pi RC}$

Time to reach X%: $t = -\tau \ln(1 - X/100)$

Appendix B: Troubleshooting Guide

Problem	Possible Cause	Solution
Capacitor not charging	Reversed polarity (electrolytic)	Check polarity markings, reconnect correctly
Time constant too different from calculated	Component values outside tolerance	Measure actual R and C values, recalculate τ
Oscilloscope shows no waveform	Incorrect trigger settings	Set trigger to edge mode, adjust trigger level
Voltage exceeds supply voltage	Capacitor initially charged	Discharge capacitor completely before starting
Erratic measurements	Poor breadboard connections	Ensure all connections are firm and secure

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Laboratory Activity: Exploring Differential Equations in RC Circuits Using Wolfram Alpha