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

SEELab3 or ExpEYES-17 unit
One Inductor ($L = 10\text{ mH}$, with internal DC resistance $r \approx 20\text{ }\Omega$)
One Resistor ($R = 1\text{ k}\Omega$)
Connecting wires
PC or Smartphone with SEELab3 / ExpEYES software

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

Series Connection: Connect the Resistor ($R$) and Inductor ($L$) in series.
AC Source: Connect the free end of the inductor to WG (Wave Generator output of SEELab3/ExpEYES-17).
Ground: Connect the free end of the resistor to GND.
Measurement (Applied Voltage): Connect the WG end also to A1 to monitor the applied sinusoidal voltage.
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

Open the SEELab3 / ExpEYES app and select the “RL Steady State” experiment.
Set the wave generator (WG) to output a sinusoidal signal at $f = 5000\text{ Hz}$.
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 )
Note the peak amplitudes of each trace from the display.
Observe the phase difference ($\phi$) between the applied voltage and $V_R$ as reported by the software.
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.
Compare the measured phase angle and voltages with theoretically calculated values, remembering to include the winding resistance $r$ of the inductor.

Steady State Response (Phone App)

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

The voltage across the inductor was observed to lead the current (and $V_R$) by approximately $90°$, which is opposite in sense to the RC circuit where $V_C$ lags.
At $f = 5000\text{ Hz}$, the measured phase angle was found to be ____ °, against a theoretical value of $17.12°$.
The vector sum $\sqrt{V_R^2 + V_L^2}$ was found to be ____ V, agreeing closely with the directly measured applied voltage of ____ V, verifying Kirchhoff’s Voltage Law in phasor form.
As frequency increases, $Z_L$ increases, causing the phase angle $\phi$ to increase and $V_L$ to dominate over $V_R$. This is the dual behavior of an RC circuit, where $Z_C$ decreases with frequency.

8. Precautions

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.
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.
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.
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 $\phi = 45°$.

Steady-State Response of a Series RL Circuit