An RL circuit experiment measures how current and voltage change with time when a power supply is connected to or disconnected from a series circuit containing a resistor and an inductor, allowing self-induction and transient behavior to be examined.
An inductor generates an induced electromotive force in a direction that opposes changes in the current flowing through it. Therefore, even when connected to a DC power supply, the current does not instantaneously reach its final value but increases with time. Conversely, after the power supply is disconnected, the current does not instantly fall to zero; it decreases while the magnetic energy stored in the inductor is released.
The rate of change in an RL circuit is determined by the self-inductance L and circuit resistance R, and is expressed by the time constant τ = L/R.
This article assumes an RL series circuit using a DC power supply, resistor, inductor, switch, oscilloscope, and related equipment, and covers the introduction, precautions, items to record, sample experimental values, calculations, sources of error, example results, and discussion examples.
Note:
This article is a study reference for discussing results obtained in physics experiments at schools, universities, and similar institutions. The experimental conditions and measured values shown here are illustrative examples and are not actual measurement results.
When the current through an inductor is suddenly interrupted, a large back EMF may be generated. To prevent electric shock or damage to measuring instruments, check the supply voltage, current, inductor ratings, and whether protective diodes or resistors are present. Always turn off the power before changing wiring, and follow the instructions in your institution’s laboratory manual and those of your instructor or TA.
- What Is an RL Circuit?
- Main Items to Record in the Results
- Sample Experimental Conditions
- Sample Experimental Values
- Calculation Method
- Calculating the Total Resistance
- Calculating the Theoretical Time Constant
- Calculating the Final Current
- Rising Current After 1 ms
- Inductor Voltage After 1 ms
- Obtaining the Experimental Time Constant from 63.2%
- Obtaining the Experimental Time Constant from 36.8%
- Linearizing the Decay Curve
- Calculating Inductance from the Experimental Time Constant
- Relative Error
- Magnetic Energy Stored in the Inductor
- Main Sources of Error
- Example of How to Write the Results
- Points to Connect to the Discussion
- Discussion Example
- Discussion Example When the Experimental Time Constant Was Smaller Than the Theoretical Value
- Discussion Example When the Experimental Time Constant Was Larger Than the Theoretical Value
- Discussion Example When Oscillation Was Observed in the Waveform
- Discussion Example When the Final Current Was Smaller Than the Theoretical Value
- Summary
What Is an RL Circuit?
An RL circuit contains a resistor and an inductor. Immediately after connection to a DC power supply or switching the supply, self-induction in the inductor causes current and voltage to change with time.
Induced EMF of an Inductor
When the current through an inductor with self-inductance L changes, the following induced electromotive force is generated.
VL = −L(dI/dt)
The negative sign indicates that the induced EMF is produced in a direction that opposes the change in current. This corresponds to Lenz’s law.
Time Constant of an RL Circuit
The time constant of an ideal RL series circuit is expressed as follows.
τ = L/R
In an actual circuit, the total resistance includes not only the external resistor but also the inductor winding resistance, power-supply internal resistance, signal-generator output resistance, and other resistance components.
τ = L/Rtotal
Current When the Power Supply Is Connected
If the supply voltage is V0 and the total resistance is R, the current after the supply is connected is expressed as follows.
I(t) = (V0/R){1 − exp(−tR/L)}
The final current is the DC current after sufficient time has elapsed.
I∞ = V0/R
Inductor Voltage When the Power Supply Is Connected
The voltage across an ideal inductor is maximum immediately after connection and approaches zero with time.
VL(t) = V0exp(−tR/L)
The voltage across the resistor approaches the supply voltage as the current increases.
VR(t) = V0{1 − exp(−tR/L)}
Current Decay
When the power supply is disconnected and a path is provided for the inductor current to decay through a resistance, the current is expressed as follows.
I(t) = I0exp(−tR/L)
Values After One Time Constant
One time constant after the power supply is connected, the current reaches approximately 63.2% of its final value.
I(τ) = I∞(1 − e−1) ≈ 0.632I∞
One time constant after decay begins, the current falls to approximately 36.8% of its initial value.
I(τ) = I0e−1 ≈ 0.368I0
After about five time constants, the rising current reaches about 99.3% of its final value, while the decaying current falls to about 0.67% of its initial value. For practical purposes, the change can then be regarded as nearly complete.
Main Items to Record in the Results
- Nominal self-inductance of the inductor
- Winding resistance of the inductor
- Nominal and measured values of the external resistor
- Total resistance of the circuit
- Supply voltage
- Theoretical time constant L/R
- Current rise waveform
- Current decay waveform
- Voltage waveform across the resistor
- Voltage waveform across the inductor
- Final current
- Current after one time constant
- Experimental time constant obtained from 63.2% or 36.8%
- Time constant obtained from the slope of a semilogarithmic graph
- Self-inductance calculated from the experimental time constant
- Relative error between theoretical and experimental values
- Oscilloscope time scale, voltage scale, and probe attenuation
- Signal-generator output resistance
- Changes in waveform due to the presence of a core or the current level
Sample Experimental Conditions
| Item | Sample Condition |
|---|---|
| Nominal inductance L | 100 mH |
| Inductor winding resistance r | 20 Ω |
| External resistance R | 80 Ω |
| Total resistance Rtotal | 100 Ω |
| Supply voltage V0 | 5.00 V |
| Theoretical time constant τ | 1.00 ms |
| Final current I∞ | 50.0 mA |
| Measurement method | Oscilloscope |
| Input waveform | Square wave |
| Measurement time | 0–5.0 ms |
Sample Experimental Values
Current Rise After Power Is Applied
| Time t (ms) | Theoretical Current (mA) | Measured Current (mA) | Percentage of Final Value |
|---|---|---|---|
| 0.0 | 0.00 | 0.3 | 0.6% |
| 0.5 | 19.67 | 19.4 | 38.8% |
| 1.0 | 31.61 | 31.3 | 62.6% |
| 1.5 | 38.84 | 38.5 | 77.0% |
| 2.0 | 43.23 | 42.9 | 85.8% |
| 2.5 | 45.90 | 45.6 | 91.2% |
| 3.0 | 47.51 | 47.2 | 94.4% |
| 4.0 | 49.08 | 48.8 | 97.6% |
| 5.0 | 49.66 | 49.4 | 98.8% |
Inductor Voltage After Power Is Applied
| Time t (ms) | Theoretical Inductor Voltage (V) | Measured Inductor Voltage (V) |
|---|---|---|
| 0.0 | 5.000 | 4.92 |
| 0.5 | 3.033 | 3.01 |
| 1.0 | 1.839 | 1.86 |
| 1.5 | 1.116 | 1.14 |
| 2.0 | 0.677 | 0.70 |
| 3.0 | 0.249 | 0.27 |
| 4.0 | 0.092 | 0.10 |
| 5.0 | 0.034 | 0.04 |
Current Decay
| Time t (ms) | Theoretical Current (mA) | Measured Current (mA) | Percentage of Initial Value |
|---|---|---|---|
| 0.0 | 50.00 | 49.7 | 99.4% |
| 0.5 | 30.33 | 30.5 | 61.4% |
| 1.0 | 18.39 | 18.6 | 37.4% |
| 1.5 | 11.16 | 11.4 | 22.9% |
| 2.0 | 6.77 | 7.0 | 14.1% |
| 2.5 | 4.10 | 4.3 | 8.7% |
| 3.0 | 2.49 | 2.7 | 5.4% |
| 4.0 | 0.92 | 1.0 | 2.0% |
| 5.0 | 0.34 | 0.4 | 0.8% |
Time Constant with Different External Resistances
| External Resistance | Coil Resistance | Total Resistance | Theoretical Time Constant | Experimental Time Constant |
|---|---|---|---|---|
| 30 Ω | 20 Ω | 50 Ω | 2.00 ms | 1.96 ms |
| 80 Ω | 20 Ω | 100 Ω | 1.00 ms | 0.99 ms |
| 180 Ω | 20 Ω | 200 Ω | 0.50 ms | 0.49 ms |
Time Constant with Different Inductances
| Inductance | Total Resistance | Theoretical Time Constant | Experimental Time Constant |
|---|---|---|---|
| 47 mH | 100 Ω | 0.47 ms | 0.46 ms |
| 100 mH | 100 Ω | 1.00 ms | 0.99 ms |
| 220 mH | 100 Ω | 2.20 ms | 2.14 ms |
Calculation Method
Calculating the Total Resistance
For an external resistance of 80 Ω and an inductor winding resistance of 20 Ω:
Rtotal = 80 + 20
= 100 Ω
Calculating the Theoretical Time Constant
For an inductance of 100 mH and total resistance of 100 Ω, first convert the inductance to henries.
L = 100 mH = 0.100 H
τ = L/R
= 0.100 / 100
= 1.00 × 10−3 s
= 1.00 ms
Calculating the Final Current
I∞ = V0/R
= 5.00 / 100
= 0.0500 A
= 50.0 mA
Rising Current After 1 ms
I(t) = I∞{1 − exp(−t/τ)}
= 50.0{1 − exp(−1.00/1.00)}
= 50.0(1 − 0.3679)
= 31.6 mA
Inductor Voltage After 1 ms
VL(t) = V0exp(−t/τ)
= 5.00exp(−1.00/1.00)
= 1.84 V
Obtaining the Experimental Time Constant from 63.2%
If the final current is 49.7 mA, 63.2% of it is:
0.632 × 49.7 = 31.4 mA
Read the time at which the current-rise curve reaches 31.4 mA. In this example it is about 0.99 ms, so the experimental time constant is 0.99 ms.
Obtaining the Experimental Time Constant from 36.8%
If the current at the start of decay is 49.7 mA, 36.8% is:
0.368 × 49.7 = 18.3 mA
Read the time at which the decay curve falls to 18.3 mA and use it as the time constant.
Linearizing the Decay Curve
Taking the natural logarithm of both sides of the current-decay equation gives:
lnI = lnI0 − (R/L)t
On a graph with time t on the horizontal axis and lnI on the vertical axis, the slope is −R/L, or −1/τ.
Slope a = −1/τ
τ = −1/a
For example, if the slope is −1.01 × 103 s−1:
τ = −1/(−1.01 × 103)
= 9.90 × 10−4 s
= 0.990 ms
Calculating Inductance from the Experimental Time Constant
For an experimental time constant of 0.990 ms and total resistance of 101 Ω:
L = τR
= 0.990 × 10−3 × 101
= 9.999 × 10−2 H
≈ 100 mH
Relative Error
For a theoretical time constant of 1.00 ms and experimental value of 0.99 ms:
Relative error (%) = |Experimental value − Theoretical value| / Theoretical value × 100
= |0.99 − 1.00| / 1.00 × 100
= 1.0%
Magnetic Energy Stored in the Inductor
For a final current of 50.0 mA and inductance of 0.100 H:
U = (1/2)LI2
= (1/2) × 0.100 × (0.0500)2
= 1.25 × 10−4 J
Main Sources of Error
| Source of Error | Effect on Measurement | Likely Trend in Results | Improvement |
|---|---|---|---|
| Inductor winding resistance | Adds resistance beyond the external resistor | Time constant is smaller than calculated using only the external resistor | Measure winding resistance and include it in total resistance |
| Resistor tolerance | Actual total resistance differs from nominal value | Time constant and final current shift | Use measured resistance values |
| Inductance tolerance | Actual L differs from nominal value | Rise and decay times change | Calculate L from experimental results and compare |
| Core nonlinearity | L changes with current or magnetic flux density | Waveform departs from a simple exponential | Measure at low current and avoid saturation |
| Core hysteresis and eddy currents | Magnetic losses occur | Deviation from ideal equations and waveform delay | Use a high-frequency core or air-core inductor |
| Signal-generator output resistance | Typically about 50 Ω is added in series | Time constant and final current become smaller | Include output resistance in total resistance |
| Power-supply internal resistance | Terminal voltage and total resistance change under load | Final current becomes smaller than theoretical value | Measure terminal voltage while current flows |
| Oscilloscope input resistance | Adds a parallel load at the measurement point | Measured voltage and time constant change | Use a 10× probe or similar high-impedance probe |
| Probe input capacitance | Slows high-frequency components | Initial rise becomes rounded | Use a compensated low-capacitance probe |
| Inductor parasitic capacitance | Inductor no longer behaves as a pure L | Oscillation or ringing occurs during fast switching | Measure at a sufficiently low frequency |
| Parasitic inductance of switches and wiring | Adds inductive components | Voltage spikes appear at interruption | Keep wiring short and loop area small |
| Protective diode | Changes the discharge path and voltage | Decay time constant differs from the theoretical circuit | Analyze using an equivalent circuit including the diode |
| Reading time zero | Switching time and measurement start are offset | Systematic error in time-constant reading | Use the trigger function |
| Noise at low current | Relative error increases near the end of decay | End of semilog plot departs from a straight line | Exclude points near the noise floor from fitting |
| Inductor heating | Winding resistance increases with time | Final current decreases and time constant changes | Measure at low current for a short time |
Why the Current Cannot Change Instantaneously
The inductor voltage is expressed as VL = LdI/dt. Changing the current instantaneously would require dI/dt to become infinite, which would require an infinite voltage.
Because actual supply voltages are finite, the current through an inductor changes continuously.
Why the Inductor Voltage Is Large Immediately After Connection
Immediately after connection, the rate of current increase is greatest, so the back EMF caused by self-induction is also greatest. Ideally, nearly all the supply voltage appears across the inductor immediately after connection, while the resistor voltage is nearly zero.
As time passes, the rate of current increase decreases, so the inductor voltage falls and the resistor voltage rises.
Why an Inductor Behaves Like a Wire After Sufficient Time
In DC steady state, the current is constant and dI/dt = 0. Therefore, the induced EMF of an ideal inductor is zero, and it behaves approximately like a short circuit.
However, a real inductor has winding resistance and is therefore not a perfect short circuit.
Why a Larger Resistance Produces a Smaller Time Constant
Because the time constant of an RL circuit is τ = L/R, increasing resistance decreases the time constant. This is the opposite of an RC circuit, in which increasing resistance increases the time constant.
During decay, a larger resistance converts magnetic energy into heat more quickly, causing the current to decrease faster.
Why a Larger Inductance Makes the Change Slower
An inductor with a larger inductance generates a larger back EMF for the same rate of current change. It therefore opposes current changes more strongly, requiring more time for both rise and decay.
Example of How to Write the Results
An inductor with a self-inductance of 100 mH and winding resistance of 20 Ω was connected in series with an external 80 Ω resistor, and a 5.00 V square wave was applied while the voltage across the resistor was measured.
The current increased exponentially from approximately 0 mA immediately after connection to 31.3 mA at 1.0 ms and 49.4 mA at 5.0 ms. After sufficient time, the current was approximately 49.7 mA.
The experimental time constant obtained from the time required to reach 63.2% of the final current was 0.99 ms. The relative error compared with the theoretical time constant of 1.00 ms, calculated from the total resistance of 100 Ω and inductance of 100 mH, was 1.0%.
Points to Connect to the Discussion
- Were the current rise and decay exponential?
- Was the current after one time constant close to 63.2% or 36.8%?
- Did the theoretical time constant L/R agree with the experimental time constant?
- Was the inductor winding resistance included in the total resistance?
- Did the final current agree with V/R?
- How did the inductor voltage change immediately after connection and in steady state?
- How did increasing the external resistance affect the time constant?
- How did increasing the inductance affect the time constant?
- Was the semilogarithmic graph linear?
- How did signal-generator output resistance affect the results?
- Could core nonlinearity or magnetic saturation be neglected?
- Were voltage spikes or ringing observed when the circuit was interrupted?
- Did heating of the inductor change its winding resistance?
Discussion Example
In this experiment, the time dependence of the current after connection and disconnection of the power supply was measured in an RL circuit consisting of a 100 mH inductor with a winding resistance of 20 Ω and an external 80 Ω resistor connected in series.
The current was nearly 0 immediately after the power supply was connected. It then increased rapidly, but the rate of increase became smaller with time, and the current eventually approached approximately 49.7 mA. Conversely, during decay the current fell rapidly from its initial value and then approached 0 more gradually.
The rising current approximately followed the exponential relation I(t) = I∞{1 − exp(−t/τ)}, while the decaying current approximately followed I(t) = I0exp(−t/τ).
Including the 20 Ω winding resistance of the inductor in addition to the 80 Ω external resistor gives a total resistance of 100 Ω. Therefore, the theoretical time constant is τ = L/R = 0.100/100 = 1.00 ms.
The time at which the current reached 63.2% of its final value was approximately 0.99 ms. In the decay waveform, the time required for the current to fall to 36.8% of its initial value was also approximately 1.0 ms.
The experimental time constant of 0.99 ms was close to the theoretical value of 1.00 ms, with a relative error of 1.0%. This confirms that the rate of transient change in an RL circuit is determined by the ratio of inductance to total resistance.
Immediately after connection, the current was nearly zero, so the voltage drop across the resistor was also nearly zero and most of the supply voltage appeared across the inductor. This occurs because the inductor generates a back EMF that opposes the increase in current.
As time passed, the rate of current increase decreased and the induced EMF of the inductor became smaller. Consequently, the voltage across the resistor increased, and in steady state most of the supply voltage appeared across the resistive components.
After sufficient time in DC steady state, the current is constant, so dI/dt = 0 and the induced EMF of an ideal inductor becomes zero. However, because a real inductor has winding resistance, the voltage across the entire inductor is not exactly zero.
Increasing the external resistance reduced the time constant, so the current approached its final value more quickly. Since τ = L/R in an RL circuit, this is the opposite of the relationship in an RC circuit, where resistance increases the time constant.
Increasing the inductance, on the other hand, increased the time constant and slowed the current change. A larger inductance produces a larger back EMF for the same rate of current change and therefore opposes the change more strongly.
Possible causes of differences between theoretical and experimental values include tolerances in the inductance and resistance, signal-generator output resistance, power-supply internal resistance, and oscilloscope input characteristics.
In particular, a signal generator generally has a finite output resistance. Ignoring this resistance underestimates the actual total resistance and can therefore cause the theoretical time constant to be calculated larger than its actual value.
For an inductor with a magnetic core, inductance is not necessarily constant with current. As the current increases and the core approaches magnetic saturation, the inductance may decrease and the waveform may deviate from a simple exponential.
When the power supply is interrupted, the inductor attempts to maintain the current and can therefore generate a large back EMF. Depending on parasitic elements in the switch, wiring, and probe, voltage spikes or ringing may appear on the oscilloscope.
Near the end of decay, the current becomes small, so measurement noise and instrument resolution account for a larger fraction of the measured value. Therefore, when using a semilogarithmic graph, points close to the noise floor should be excluded from the fitting range.
To improve measurement accuracy, it is important to use the total resistance including the inductor winding resistance, signal-generator output resistance, and external resistance. Measurements should also be made over an appropriate current range and for a short duration to avoid inductor heating and magnetic saturation.
These results experimentally confirm that self-induction prevents the current in an RL circuit from changing instantaneously, that the rise and decay proceed exponentially, and that the rate of change is determined by the time constant τ = L/R.
Discussion Example When the Experimental Time Constant Was Smaller Than the Theoretical Value
A smaller experimental time constant may have resulted from failing to include the inductor winding resistance or signal-generator output resistance in the total resistance, from the actual inductance being smaller than its nominal value, or from magnetic saturation reducing the inductance of the core.
Discussion Example When the Experimental Time Constant Was Larger Than the Theoretical Value
A larger experimental time constant may have resulted from the actual inductance being larger than its nominal value, overestimating the total resistance, or observing a slower rise because of the oscilloscope bandwidth or probe input capacitance.
Discussion Example When Oscillation Was Observed in the Waveform
Oscillation or ringing immediately after switching may indicate that the circuit behaved as an RLC circuit rather than a pure RL circuit because of the inductor’s parasitic capacitance, wiring parasitic inductance, and oscilloscope-probe capacitance.
Discussion Example When the Final Current Was Smaller Than the Theoretical Value
A smaller final current may have resulted from omitting the inductor winding resistance, power-supply or signal-generator internal resistance, or wire resistance from the calculation, or from an increase in winding resistance caused by heating of the inductor during operation.
Summary
In an RL circuit, self-induction prevents the current from changing instantaneously, so the current rises exponentially after the power supply is connected and decays exponentially after it is disconnected.
The rate of change is determined by the time constant τ = L/R. After one time constant, the rising current reaches approximately 63.2% of its final value, while the decaying current falls to approximately 36.8% of its initial value.
In the discussion, it is important to relate the total resistance including the inductor winding resistance, back EMF, final current, semilogarithmic graph, signal-generator output resistance, magnetic saturation, parasitic components, and voltage spikes at interruption to the experimental results.
