Measurement of Resistance and Impedance of an Inductor
Introduction
This worksheet guides you through an experimental procedure to measure both the DC resistance and AC impedance of an inductor. You will analyze how these parameters differ when using an inductor with and without an iron core, and observe the frequency dependence of impedance.
Theoretical Background
An inductor is a passive electrical component that stores energy in its magnetic field when electric current flows through it. The two key electrical parameters of an inductor are:
DC Resistance (R): This is the ohmic resistance of the wire used to wind the inductor coil. It remains constant regardless of the frequency.
$R = \frac{V_{DC}}{I_{DC}}$
Inductive Reactance (XL): This is the opposition to change in current due to inductance. It depends on the frequency of the AC signal.
$X_L = 2\pi f L$
Where f is the frequency in Hz and L is the inductance in henries.
Impedance (Z): The total opposition to current flow in an AC circuit containing resistance and reactance.
$Z = \sqrt{R^2 + X_L^2}$
The phase angle between voltage and current is:
$\phi = \tan^{-1}\left(\frac{X_L}{R}\right)$
Iron Core Effect: Adding an iron core increases the inductance value significantly compared to an air core because:
- Iron has much higher permeability (μ) than air
- The relationship between inductance and permeability is: $L \propto \mu$
- This increases the inductive reactance and therefore the impedance
Objectives
- Measure the DC resistance of an inductor with and without an iron core
- Determine the AC impedance of the inductor at different frequencies
- Calculate the inductance value from the measured impedance
- Analyze the effect of an iron core on impedance and inductance
- Verify the relationship between impedance and frequency
Materials Required
- Inductor coil (solenoid)
- Removable iron core
- DC power supply (0-12V)
- AC signal generator (variable frequency)
- Digital multimeter (DMM)
- Oscilloscope (dual channel)
- Connecting wires
- Ammeter (DC and AC)
- Voltmeter (DC and AC)
- Resistor (known value, approx. 100Ω)
Circuit Diagrams
DC Resistance Measurement Circuit
AC Impedance Measurement Circuit
Experimental Procedure
Part A: DC Resistance Measurement
Set up the circuit as shown in the DC resistance measurement diagram.
Ensure the power supply is turned off before making connections.
First, measure the resistance directly using a multimeter in resistance mode.
Record this value as Rdirect in Table 1.
Connect the DC power supply, ammeter, and voltmeter as shown.
Set the power supply to approximately 5V DC.
Turn on the power supply and record the voltage (V) and current (I) readings.
Calculate the resistance using Ohm's Law: R = V/I.
Repeat the measurement three times with different voltage settings.
Perform these measurements both with and without the iron core inserted.
Part B: AC Impedance Measurement
Set up the circuit as shown in the AC impedance measurement diagram.
Connect the function generator to supply an AC sinusoidal voltage.
Set the oscilloscope to display both voltage across and current through the inductor.
For current measurement, use a small known resistor (Rs) in series and measure the voltage across it.
Start with a frequency of 50 Hz and set the voltage amplitude to approximately 5V peak-to-peak.
Record the voltage across the inductor (VL) and the voltage across the series resistor (VR).
Calculate the current using: I = VR/Rs.
Calculate the impedance: Z = VL/I.
Measure the phase angle (φ) between voltage and current using the oscilloscope.
Repeat the measurements for frequencies: 100 Hz, 200 Hz, 500 Hz, 1 kHz, 2 kHz, and 5 kHz.
Perform these measurements both with and without the iron core inserted.
Safety Precautions:
- Do not exceed the voltage ratings of the components
- Always turn off power before making circuit changes
- Ensure proper grounding of all measurement equipment
- The inductor may get warm during prolonged testing - allow cooling periods
Data Collection
Table 1: DC Resistance Measurements
| Core Type | Rdirect (Ω) | Trial | Voltage (V) | Current (I) | Calculated R=V/I (Ω) |
|---|---|---|---|---|---|
| Air Core | 1 | ||||
| 2 | |||||
| 3 | |||||
| Iron Core | 1 | ||||
| 2 | |||||
| 3 |
Table 2: AC Impedance Measurements (Air Core)
| Frequency (Hz) | VL (V) | VR (V) | Current I=VR/Rs (A) | Impedance Z=VL/I (Ω) | Phase Angle φ (°) | Calculated Inductance L (H) |
|---|---|---|---|---|---|---|
| 50 | ||||||
| 100 | ||||||
| 200 | ||||||
| 500 | ||||||
| 1000 | ||||||
| 2000 | ||||||
| 5000 |
Table 3: AC Impedance Measurements (Iron Core)
| Frequency (Hz) | VL (V) | VR (V) | Current I=VR/Rs (A) | Impedance Z=VL/I (Ω) | Phase Angle φ (°) | Calculated Inductance L (H) |
|---|---|---|---|---|---|---|
| 50 | ||||||
| 100 | ||||||
| 200 | ||||||
| 500 | ||||||
| 1000 | ||||||
| 2000 | ||||||
| 5000 |
Calculations and Analysis
Calculation Methods
1. Calculating Impedance (Z):
$Z = \frac{V_L}{I} = \frac{V_L \times R_s}{V_R}$
Where:
- VL = Voltage across the inductor
- I = Current through the circuit
- Rs = Value of the series resistor
- VR = Voltage across the series resistor
2. Calculating Inductance (L):
$Z = \sqrt{R^2 + X_L^2}$
$X_L = \sqrt{Z^2 - R^2}$
$L = \frac{X_L}{2\pi f}$
Where:
- Z = Measured impedance
- R = DC resistance (from Part A)
- XL = Inductive reactance
- f = Frequency in Hz
- L = Inductance in henries
3. Calculating Phase Angle (φ):
$\phi = \cos^{-1}\left(\frac{R}{Z}\right) = \sin^{-1}\left(\frac{X_L}{Z}\right) = \tan^{-1}\left(\frac{X_L}{R}\right)$
The phase angle can be measured directly from the oscilloscope as the time difference between voltage and current peaks:
$\phi (degrees) = \frac{\Delta t}{T} \times 360°$
Where:
- Δt = Time difference between voltage and current peaks
- T = Period of one complete cycle
Analysis Tasks
- Calculate the average DC resistance for both air core and iron core configurations.
- For each frequency, calculate the inductance using the formula: $L = \frac{\sqrt{Z^2 - R^2}}{2\pi f}$
- Plot a graph of impedance (Z) versus frequency (f) for both air core and iron core.
- Plot a graph of calculated inductance (L) versus frequency (f) for both configurations.
- Calculate the percentage increase in impedance when adding the iron core at each frequency.
Discussion Questions
- How does the DC resistance compare between the inductor with and without an iron core? Explain any differences.
- Describe how the impedance changes with frequency for both the air core and iron core inductor. Does this match theoretical expectations?
- Why does the inductance value (L) appear to change with frequency? Should the inductance of an ideal inductor change with frequency?
- Compare the phase angle measurements at different frequencies. Explain the relationship between phase angle and frequency.
- What is the effect of the iron core on the inductance value? Explain in terms of magnetic permeability.
- At very high frequencies, inductor behavior may deviate from ideal. What factors might cause this non-ideal behavior?
- If you were designing a circuit that required high inductance but low DC resistance, what approach would you take in selecting or designing an inductor?
Error Analysis
Sources of Experimental Error
Identify and quantify possible sources of error in your measurements:
1. Instrument Errors:
- Multimeter accuracy (typically ±(0.5% + 2 digits) for DC measurements)
- Oscilloscope accuracy (typically ±3% for voltage measurements)
- Signal generator frequency accuracy (typically ±0.01%)
2. Random Errors:
- Reading fluctuations
- Connection resistances
3. Systematic Errors:
- Inductor temperature effects
- Stray capacitance at higher frequencies
- Core saturation effects
4. Propagation of Errors:
For calculated values like impedance (Z) and inductance (L), the error propagation can be estimated using:
For $Z = \frac{V_L}{I}$:
$\frac{\Delta Z}{Z} = \sqrt{\left(\frac{\Delta V_L}{V_L}\right)^2 + \left(\frac{\Delta I}{I}\right)^2}$
For $L = \frac{X_L}{2\pi f}$:
$\frac{\Delta L}{L} = \sqrt{\left(\frac{\Delta X_L}{X_L}\right)^2 + \left(\frac{\Delta f}{f}\right)^2}$
Table 4: Error Analysis
| Measurement | Estimated Error (%) | Source of Error | Ways to Minimize |
|---|---|---|---|
| DC Resistance | |||
| AC Voltage | |||
| AC Current | |||
| Impedance | |||
| Phase Angle | |||
| Inductance |
Conclusion
Summarize your findings about:
- The relationship between impedance and frequency for an inductor
- The effect of an iron core on inductance and impedance
- The practical implications of your results for circuit design
- Compare your experimental results with theoretical predictions