Experiment: Steady-State Response of a Series RL Circuit

1. Aim

To study the behavior of a series RL circuit under a sinusoidal (AC) voltage, and to measure the voltage amplitudes across each element and the phase difference between the applied voltage and the current.

2. Apparatus / Components Required

3. Theory & Principle

When a sinusoidal voltage $V(t) = V_0 \sin(\omega t)$ is applied to a series RL circuit, the inductor offers a frequency-dependent opposition called Inductive Reactance, given by:

\[Z_L = 2\pi f L\]

Unlike a resistor, an ideal inductor does not dissipate energy — it stores energy in its magnetic field. However, a real inductor also has a small DC winding resistance $r$ that must be included. The effective series resistance is therefore $R_{eff} = R + r$.

The total impedance of the series RL circuit is:

\[Z = \sqrt{R_{eff}^2 + Z_L^2}\]

The voltage across the inductor leads the current by $90°$ (i.e., $\frac{\pi}{2}$ radians). The phase angle between the applied voltage and the current (which is in phase with $V_R$) is:

\[\phi = \tan^{-1}\!\left(\frac{Z_L}{R_{eff}}\right)\]

Since the voltages across $R$ and $L$ are $90°$ out of phase, the total applied voltage is obtained by phasor (vector) addition:

\[V_{applied} = \sqrt{V_R^2 + V_L^2}\]

Note: For $R = 1\text{ k}\Omega$ (+ $20\text{ }\Omega$ winding resistance), $L = 10\text{ mH}$, and $f = 5000\text{ Hz}$, the inductive reactance $Z_L = 2\pi \times 5000 \times 0.01 \approx 314\text{ }\Omega$. The calculated phase angle is $\phi = 17.12°$ and the measured value is approximately $18°$. The small discrepancy is due to component tolerances and the winding resistance of the inductor.

4. Circuit Diagram / Setup

  1. Series Connection: Connect the Resistor ($R$) and Inductor ($L$) in series.
  2. AC Source: Connect the free end of the inductor to WG (Wave Generator output of SEELab3/ExpEYES-17).
  3. Ground: Connect the free end of the resistor to GND.
  4. Measurement (Applied Voltage): Connect the WG end also to A1 to monitor the applied sinusoidal voltage.
  5. Measurement (Resistor Voltage): Connect the junction between $L$ and $R$ to A2 to monitor $V_R$ (which is in phase with the current).

5. Procedure

  1. Open the SEELab3 / ExpEYES app and select the “RL Steady State” experiment.
  2. Set the wave generator (WG) to output a sinusoidal signal at $f = 5000\text{ Hz}$.
  3. The oscilloscope screen will display three traces:
    • The applied voltage (from WG / A1)
    • The voltage across R ($V_R$, in phase with the current. A2.)
    • The voltage across L ($V_L$, leading the current by $90°$ . calculated as A1-A2 )
  4. Note the peak amplitudes of each trace from the display.
  5. Observe the phase difference ($\phi$) between the applied voltage and $V_R$ as reported by the software.
  6. Vary the frequency (e.g., 1000 Hz, 2000 Hz, 5000 Hz, 10000 Hz) and repeat Steps 3–5 to study how $Z_L$ and $\phi$ change with frequency.
  7. Compare the measured phase angle and voltages with theoretically calculated values, remembering to include the winding resistance $r$ of the inductor.
Distance Plot

Steady State Response (Phone App)

Falling Data

Steady State Setup For ExpEYES17

6. Observation Table

Resistor ($R$): ____ $\Omega$     Inductor ($L$): ____ mH     Winding Resistance ($r$): ____ $\Omega$

Effective Resistance $R_{eff} = R + r$: ____ $\Omega$

Frequency $f$ (Hz) $Z_L = 2\pi fL$ ($\Omega$) Theoretical $\phi$ (°) Measured $V_R$ (V) Measured $V_L$ (V) Measured $\phi$ (°) $\sqrt{V_R^2 + V_L^2}$ (V) Applied $V$ (V)
1000              
2000              
5000              
10000              
20000              

7. Results and Discussion

8. Precautions

  1. Include Winding Resistance: A real inductor has a DC resistance $r$ in its winding. Always measure this with a multimeter and add it to $R$ when calculating the theoretical phase angle; ignoring it leads to a systematic error.
  2. Frequency Selection: Choose a frequency where $Z_L$ is comparable to $R_{eff}$ to get a clearly observable phase difference. For $L = 10\text{ mH}$ and $R = 1\text{ k}\Omega$, this is around $f \approx \frac{R}{2\pi L} \approx 15.9\text{ kHz}$. Lower frequencies like $5000\text{ Hz}$ give a small but measurable phase angle.
  3. Core Saturation: Do not use very large signal amplitudes with iron-core inductors, as the core may saturate, making the inductance non-linear and waveforms distorted.
  4. Stable Waveform: Allow the waveform display to stabilize for a few seconds before recording amplitudes and phase differences.

9. Troubleshooting

Symptom Possible Cause Corrective Action
No phase shift visible; traces overlap Frequency too low — $Z_L \ll R$, so inductor is nearly a short circuit at DC-like frequencies. Increase frequency significantly (try 5000 Hz or higher).
Measured $\phi$ much larger than theoretical Winding resistance $r$ not accounted for in calculation. Measure $r$ with a multimeter and use $R_{eff} = R + r$ in the formula.
Waveforms are distorted / non-sinusoidal Iron-core inductor saturating at high signal levels. Reduce the WG output amplitude, or use an air-core inductor.
$V_L$ trace is flat or missing Loose connection at the inductor or wrong channel assignment. Check all connections; verify channel assignments in the software.

10. Viva-Voce Questions

Q1. What is inductive reactance and how does it differ from resistance?

Ans: Inductive reactance ($Z_L = 2\pi fL$) is the opposition offered by an inductor to alternating current. Unlike resistance, it is directly proportional to frequency — it increases as frequency increases. A pure inductor stores energy in a magnetic field and releases it without dissipation, whereas a resistor permanently converts electrical energy into heat.

Q2. Why does the voltage across an inductor lead the current by 90°?

Ans: The voltage across an inductor is proportional to the rate of change of current: $V_L = L\frac{dI}{dt}$. A sinusoidal current $I = I_0 \sin(\omega t)$ gives $V_L = LI_0\omega\cos(\omega t) = LI_0\omega\sin(\omega t + 90°)$. Therefore the voltage always leads the current by exactly $90°$ in a pure inductor.

Q3. How does the behavior of an RL circuit differ from an RC circuit as frequency increases?

Ans: In an RL circuit, inductive reactance $Z_L = 2\pi fL$ increases with frequency, so $V_L$ grows and the phase angle $\phi$ increases toward $90°$ at very high frequencies (the circuit becomes predominantly inductive). In an RC circuit, capacitive reactance $Z_C = \frac{1}{2\pi fC}$ decreases with frequency, so $V_C$ shrinks and the phase angle decreases toward $0°$ — the two circuits are duals of each other.

Q4. Why must the winding resistance of the inductor be included in calculations?

Ans: A real inductor is wound from a long length of thin wire, which has a measurable DC resistance $r$. This resistance is always in series with the inductive reactance. Ignoring it means using an incorrect value of effective resistance $R_{eff}$, which causes the theoretical phase angle to differ from the measured value, even when the component values are otherwise accurate.

Q5. At what frequency will the voltages across R and L be equal in a series RL circuit?

Ans: The voltages are equal when $V_R = V_L$, which occurs when $R_{eff} = Z_L$, i.e., $R_{eff} = 2\pi fL$. Solving: $f = \frac{R_{eff}}{2\pi L}$. For $R_{eff} = 1020\text{ }\Omega$ and $L = 10\text{ mH}$, this gives $f \approx \frac{1020}{2\pi \times 0.01} \approx 16.2\text{ kHz}$. At this frequency, the phase angle $\phi = 45°$.