This experiment delves into the concept of inductive reactance and its direct proportionality to the frequency of an alternating current (AC) signal. By utilizing a function generator, we will apply varying frequencies to an inductor and measure the resulting voltage and current to calculate its inductive reactance.
Objective
To experimentally determine the inductive reactance of a given inductor at various input signal frequencies and to verify the linear relationship between inductive reactance and frequency.
Theoretical Background
Inductive reactance ($X_L$) is the opposition offered by an inductor to the flow of alternating current. It is analogous to resistance in a DC circuit but is dependent on the frequency of the AC signal and the inductance of the coil. The formula for inductive reactance is given by:
$X_L = 2\pi fL$
Where:
$X_L$ is the inductive reactance in Ohms ($\Omega$)
$f$ is the frequency of the AC signal in Hertz (Hz)
$L$ is the inductance of the coil in Henries (H)
From this equation, it is evident that inductive reactance is directly proportional to the frequency. Therefore, as the frequency of the AC signal increases, the inductive reactance should also increase linearly.
Experimentally, inductive reactance can be determined by measuring the RMS voltage ($V_L$) across the inductor and the RMS current ($I_L$) flowing through it, using the following relationship, which is a form of Ohm's Law for AC circuits:
$X_L = \frac{V_L}{I_L}$
Required Apparatus
Function Generator
Inductor of a known value (e.g., 10 mH, 100 mH)
Digital Multimeter (DMM) or Oscilloscope with probes
A resistor of known value (e.g., 1 k$\Omega$, to act as a current sensing resistor)
Connecting wires and a breadboard
Circuit Diagram
The circuit consists of the function generator connected in series with the inductor and a known resistor. The resistor is used to measure the current flowing through the circuit.
Procedure
Measure Component Values: Before setting up the circuit, use the DMM to measure the actual resistance of the resistor and the DC resistance of the inductor. Record these values.
Circuit Setup: Construct the series circuit as shown in the diagram above on a breadboard.
Initial Settings:
Set the function generator to produce a sine wave.
Set the initial frequency to a low value, for example, 100 Hz.
Set the output voltage of the function generator to a suitable value (e.g., 5V peak-to-peak).
Voltage and Current Measurement:
Connect one channel of the oscilloscope across the function generator to measure the total applied voltage ($V_{in}$).
Connect another channel of the oscilloscope across the known resistor ($R$) to measure the voltage across it ($V_R$).
Alternatively, use the DMM set to AC voltage to measure the RMS voltage across the resistor ($V_R$) and the inductor ($V_L$).
Calculate Current: The current flowing through the series circuit (IL) can be calculated using Ohm's law on the known resistor:
IL=RVR
Calculate Experimental Inductive Reactance: Measure the voltage across the inductor (VL) using the oscilloscope or DMM. Calculate the experimental inductive reactance using:
XL(exp)=ILVL
Vary the Frequency: Increase the frequency of the function generator in steps (e.g., 200 Hz, 300 Hz, 400 Hz, up to 1 kHz or higher).
Record Data: For each frequency step, repeat the measurements of $V_R$ and $V_L$ and calculate the corresponding current ($I_L$) and experimental inductive reactance ($X_{L(exp)}$). Record all values in a tabular format.
Theoretical Calculation: For each frequency, calculate the theoretical inductive reactance using the formula $X_{L(th)} = 2\pi fL$, where L is the known inductance of the coil.
Observations and Calculations
| Frequency (f) in Hz | Voltage across Resistor (VR) in Volts | Current (IL=VR/R) in Amperes | Voltage across Inductor (VL) in Volts | Experimental XL=VL/IL in Ω | Theoretical XL=2πfL in Ω | Percentage Error (%) |
Percentage Error Calculation:
%Error=XL(th)∣XL(th)−XL(exp)∣×100
Graph
Plot a graph of Experimental Inductive Reactance ($X_{L(exp)}$) on the y-axis against Frequency ($f$) on the x-axis. The graph should be a straight line passing through the origin, which visually confirms the linear relationship between inductive reactance and frequency.
Precautions
Ensure all connections are tight and correct before switching on the power supply.
Start with the lowest voltage and frequency settings on the function generator.
The internal resistance of the function generator and the DC resistance of the inductor can introduce errors in the measurements. These should be considered for more accurate results.
Use appropriate ranges on the multimeter or oscilloscope to get accurate readings.
Handle the electronic components with care.
This video demonstrates a practical experiment on inductive reactance. https://youtu.be/z9oAIZSofQE?si=xwYJywGZN0SkRx-d
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