Inductor in AC Circuits

Experiment: Phase Relationship in AC Inductive Circuits

1. Aim

To study the phase relationship between voltage and current in an AC circuit containing an inductor and a resistor, and to verify that voltage leads current in an inductive circuit.

2. Apparatus / Components Required

• SEELab3 or ExpEYES-17 unit
• One Resistor ($R = 100\text{ }\Omega$ or $1000\text{ }\Omega$)
• One Inductor (e.g., 10mH provided in the kit)
• Connecting wires
• PC or Smartphone with SEELab3 software

3. Theory & Principle

An inductor opposes changes in current by storing energy in a magnetic field. This property, known as self-inductance ($L$), causes the current to “lag” behind the applied voltage.

In a purely inductive circuit, the voltage leads the current by exactly $90^\circ$ ($\pi/2$ radians). In a series RL circuit, the phase angle $\phi$ is between $0^\circ$ and $90^\circ$, determined by the resistance ($R$) and the inductive reactance ($X_L$): \(X_L = 2\pi f L\) \(\phi = \tan^{-1}\left(\frac{X_L}{R}\right)\)

Current Measurement:

We measure the voltage across a series resistor ($R$) to represent the current waveform ($I = V_R / R$). Since $V_{total}$ is measured at A1 and $V_R$ is measured at A2, the voltage across the inductor is calculated mathematically as $V_L = A1 - A2$.

4. Circuit Diagram / Setup

1. Connect the Inductor ($L$) and Resistor ($R$) in series between WG and GND.
2. Connect A1 to WG (measures total voltage $V_{total}$).
3. Connect A2 to the junction between the Inductor and the Resistor.
4. In this configuration, A2 measures the voltage across the Resistor ($V_R$), which is in phase with the current ($I$).
5. The voltage across the Inductor ($V_L$) is obtained as A1 - A2.

5. Procedure

1. Launch the SEELab3 software and select the “AC through Inductor” experiment.
2. Set WG to a sine wave (e.g., $4000\text{ Hz}$ or $5000\text{ Hz}$ to get significant reactance).
3. Enable traces for A1 and A2.
4. Select a region of the graph. The software will extract the amplitudes and phases using mathematical curve fitting.
5. Observe that the peak of the voltage across the inductor ($A1-A2$) occurs before the peak of the current (A2). This confirms that voltage leads the current.
6. Note the phase difference $\phi$ and calculate the inductance $L$ using the formula $L = X_L / (2\pi f)$.

6. Observation Table

7. Error Analysis

The measurement of inductance $L$ is subject to errors from the reference resistor $R$ and the inductor’s own internal DC resistance ($R_L$).

The percentage error in $L$ can be estimated as: \(\frac{\Delta L}{L} \approx \frac{\Delta R}{R} + \frac{\Delta f}{f} + \frac{\Delta \phi}{\sin \phi \cos \phi}\)

Key Factors Affecting Accuracy:

• Inductor Resistance: Real inductors have a finite resistance $R_L$. If $R_L$ is not accounted for, the phase shift will be less than the theoretical $90^\circ$ even if the series resistor $R$ is small.
• Frequency Selection: At low frequencies, $X_L$ is small compared to $R_L + R$, making the phase shift difficult to measure accurately. Always use higher frequencies ($>2\text{ kHz}$) for small inductors.

8. Results and Discussion

• The voltage waveform across the inductor reaches its maximum value before the current waveform.
• The measured phase shift $\phi$ was approximately ____ degrees.
• As the frequency increases, the inductive reactance $X_L$ increases, which ____ (increases/decreases) the phase shift towards $90^\circ$.

9. Python Programming & Data

The SEELab3 software uses the capture2 function to retrieve the waveforms. The phase difference between the fitted sine waves $(\theta_1 - \theta_2)$ and the ratio of amplitudes allow for the calculation of Inductive Reactance ($X_L$) and Inductance ($L$).

• Download Python Program for Experiment
• Download Sample Data
• Download Analysis Script

10. Troubleshooting