Inductor Voltage Equation: Formula, Derivation, and Practical Circuit Applications
The inductor voltage equation describes how voltage across an inductor relates to the rate of change of current through it. For an ideal inductor with constant inductance, the fundamental relationship is v L (t)=L[di L (t)/dt], based on the passive sign convention. Understanding this equation helps engineers analyze transient responses, switching converters, filters, and energy-storage circuits. It also explains why an inductor resists sudden changes in current and how inductance, current slope, and voltage interact in practical electronic designs.
1. What Is the Inductor Voltage Equation?
The inductor voltage equation is a fundamental relationship in circuit theory that connects the voltage across an inductor to the rate at which its current changes.
For an ideal inductor with constant inductance, the equation is:
v L (t)=L[di L (t)/dt]
Where:
| Symbol | Definition | SI unit |
|---|---|---|
| v L (t) | Instantaneous voltage across the inductor | Volt (V) |
| L | Inductance | Henry (H) |
| i L (t) | Instantaneous current through the inductor | Ampere (A) |
| di L (t)/dt | Rate of change of current with time | A/s |
| t | Time | Second (s) |
The equation assumes a constant inductance and uses the passive sign convention, meaning the reference current enters the terminal designated as positive for voltage.
The key principle is that inductor voltage depends on how quickly current changes, not simply on the amount of current flowing through the component.
- If current increases, the inductor develops a voltage that opposes the increase.
- If current decreases, the voltage polarity reverses relative to the increasing-current case.
- If current remains constant, the ideal inductor voltage is zero.
These relationships are essential for understanding transient behavior and power conversion in electronic circuits.
2. How Is the Inductor Voltage Equation Derived?
The equation originates from Faraday's law of electromagnetic induction, which relates induced voltage to the rate of change of magnetic flux linkage.
For a coil with N turns, the flux linkage is:
λ=NΦ
Where N is the number of turns and Φ is the magnetic flux through each turn, assuming the same flux links all turns.
The induced voltage, using the passive sign convention for an ideal inductor, is:
v L =dλ/dt
For a linear magnetic system, flux linkage is proportional to current:
λ=Li L
Substituting this relationship gives:
v L =d(Li L )/dt
If inductance is constant, it can be taken outside the derivative:
v L =Ldi L /dt
This derivation shows that the inductor voltage equation is a direct consequence of electromagnetic induction.
Important qualification: If inductance varies with current or time, the more general expression is v L =dλ/dt. When flux linkage is represented as λ=L(t)i L (t), the equation becomes v L =Ldi L /dt+i L dL/dt. Nonlinear magnetic materials may require a more detailed flux-linkage model.
3. How to Calculate Inductor Voltage
The basic equation can be rearranged to calculate voltage, inductance, or the rate of current change, depending on which quantities are known.
3.1 Calculate Voltage from Inductance and Current Change
When current changes approximately linearly over a known time interval, use:
V L =L△I/△t
Where:
- VL is the voltage across the inductor in volts.
- L is the inductance in henries.
- △I is the change in current in amperes.
- △t is the time interval in seconds.
Example: An ideal \(100\,\mu H\) inductor carries a current that increases from 0.2 A to 1.2 A in 100 μs. Calculate the voltage across it.
First, determine the current change:
△I=1.2-0.2=1.0A
Convert the inductance and time interval into SI units:
L=100×10 -6 H
△t=100×10 -6 s
Substitute the values:
V L =100×10 -6 ×1.0/(100×10 -6 )
V L =1V
The ideal inductor requires a 1 V terminal voltage under the stated current-ramp conditions, with the chosen passive sign convention.
3.2 Calculate the Rate of Current Change
When the voltage and inductance are known, rearrange the equation:
di L /dt=v L /L
For example, if a constant 2 V is applied across a 200μH ideal inductor:
di L /dt=2/(200×10 -6 ) =10000A/s
Converting to amperes per millisecond:
di L /dt=10A/ms
The current increases by 10 A for every millisecond that the 2 V voltage is maintained, provided the ideal constant-inductance model remains valid.
3.3 Calculate Inductance
If voltage and the current slope are known, the inductance is:
L=v L /(di L /dt)
For example, an inductor has a measured voltage of 0.5 V while its current increases at 2 A/ms:
L=0.5/2000=250×10 -6 H
Therefore:
L=250μH
This calculation is useful for checking inductance under operating conditions, although winding resistance, measurement bandwidth, parasitic effects, and magnetic nonlinearity must be considered when interpreting real measurements.
4. Inductor Voltage Equation for Different Current Waveforms
The voltage across an inductor depends on the shape of its current waveform. The same inductance can produce different voltages depending on how rapidly the current changes.
| Current waveform | Current behavior | Ideal inductor voltage |
|---|---|---|
| Constant current | Current does not change | v L =0 |
| Linear rising current | Current increases at a constant rate | Constant positive voltage |
| Linear falling current | Current decreases at a constant rate | Constant negative voltage |
| Sinusoidal current | Current changes periodically | Sinusoidal voltage with a phase lead |
| Exponential current | Current changes at a continuously varying rate | Voltage varies with the current derivative |
4.1 Inductor Voltage with Constant Current
When current is constant:
di L /dt=0
Therefore:
v L =0
This result applies to an ideal inductor in steady-state DC operation. A practical inductor can still exhibit a voltage drop due to winding resistance and other parasitic effects.
4.2 Inductor Voltage with a Linear Current Ramp
For a linear current ramp:
i L (t)=I 0 +kt
Where I 0 is the initial current and k is the current slope.
Differentiating gives:
di L /dt=k
Thus:
v L =Lk
The voltage remains constant while the current slope remains constant. A steeper current ramp requires a proportionally larger voltage for the same inductance.
4.3 Inductor Voltage with Sinusoidal Current
Suppose the current is:
i L (t)=I m sin(ωt)
Differentiating:
v L (t)=ωLI m cos(ωt)
Since cos(ωt)=sin(ωt+90°):
v L (t)=ωLI m sin(ωt+90°)
The voltage leads the current by 90° in an ideal inductor under sinusoidal steady-state conditions.
Because ω=2πf, the voltage amplitude is:
V m =2πfLI m
This relationship connects the time-domain inductor voltage equation to inductive reactance in AC circuit analysis.
5. Inductor Voltage Equation in the Time and Frequency Domains
The inductor voltage equation can be analyzed in either the time domain or the frequency domain. The appropriate method depends on whether the circuit is being studied during a transient or under sinusoidal steady-state conditions.
5.1 Time-Domain Equation
The time-domain expression is:
v L (t)=Ldi L (t)/dt
It is particularly useful for analyzing switching events, startup behavior, current ramps, and transient responses.
If the voltage is known, integrating it gives the change in current:
i L (t)=i L (t 0 )+(1/L)∫ t t 0 v L (τ)dτ
This equation assumes constant inductance. It also demonstrates that the current change depends on the accumulated voltage over time, often called the inductor's volt-second relationship.
5.2 Frequency-Domain Equation
For sinusoidal steady-state analysis, the inductor is represented by its complex impedance:
Z L =jωL=j2πfL
Therefore:
V L =jωL I L
Here, jrepresents a 90-degree phase rotation, while V L and I L are voltage and current phasors.
The magnitude of the inductive reactance is:
X L =2πfL
Inductive reactance increases with both frequency and inductance. This is why an inductor may provide a low-impedance path for slowly varying current but oppose higher-frequency current variations more strongly, within the component's usable frequency range.
The time-domain and frequency-domain equations describe the same ideal component. However, the frequency-domain representation assumes sinusoidal steady-state analysis and does not, by itself, describe arbitrary switching transients.
6. Inductor Voltage Equation in Switching Power Converters
The inductor voltage equation is especially important in DC-DC converter s, where switching changes the voltage applied to the inductor and controls the rate of current increase or decrease.
6.1 Buck Converter
In an ideal buck converter operating in continuous conduction mode, the inductor alternates between two main switching states.
During the switch-on interval:
v L =V in -V out
The inductor current slope is:
di L /dt=(V in -V out )/L
During the switch-off interval, assuming an ideal freewheeling path:
v L =-V out
The current slope becomes:
di L /dt=-V out /L
The current rises during the on-time and falls during the off-time. In steady-state continuous conduction, the average inductor voltage over one complete switching period is zero.
For switching period T s and duty cycle D:
D(V in -V out )T s +(1-D)-V out )T s =0
Simplifying yields the ideal buck-converter voltage relationship:
V out =DV in
This derivation illustrates how the inductor voltage equation links switching behavior to converter output voltage.
6.2 Calculating Buck Converter Inductor Current Ripple
The peak-to-peak current ripple during the switch-on interval is:
△I L =(V in -V out )D/(Lf s )
Where f s is the switching frequency and D is the duty cycle.
Consider an ideal buck converter with the following operating parameters:
| Parameter | Value |
|---|---|
| Input voltage | 12 V |
| Output voltage | 5 V |
| Switching frequency | 100 kHz |
| Duty cycle | 0.417 (approximately) |
| Inductance | 100 μH |
The approximate peak-to-peak ripple is:
△I L =(12-5)×0.417/(100×10 -6 ×100000
△I L ≈0.292A
This is an idealized calculation. Actual ripple can differ due to component tolerances, parasitic resistance, switching delays, and operating-mode changes.
Design implication: Inductance, switching frequency, and duty cycle determine the inductor current ripple. That ripple affects peak current, output filtering, conduction losses, and the selection of associated power components.
7. How Does the Inductor Voltage Equation Relate to Stored Energy?
An inductor stores energy in its magnetic field when current flows through it. For an ideal linear inductor, the stored energy is:
E L =(1/2)Li L 2
Where E L is the stored energy in joules.
The instantaneous power entering the inductor is:
p L =v L i L
Substituting the voltage equation gives:
p L =Li L di L /dt
Since L is constant:
p L =(d/dt)(1/2)Li L 2
This establishes the relationship between inductor voltage, current, power, and magnetic energy.
- Positive power: The inductor absorbs energy under the passive sign convention.
- Negative power: The inductor releases energy back into the circuit.
- Zero voltage: An ideal inductor with constant current has no instantaneous power transfer, even though it may retain stored magnetic energy.
In switching converters, this energy-storage behavior allows inductors to transfer energy between switching intervals and help maintain continuous output current.
8. Practical Factors That Affect Inductor Voltage
The ideal equation is a useful starting point, but real inductors have electrical and magnetic limitations that influence their voltage and current behavior.
| Practical factor | Effect on inductor behavior | Design consideration |
|---|---|---|
| Winding resistance (DCR) | Produces a resistive voltage drop and power loss | Evaluate copper losses and temperature rise |
| Core saturation | Reduces incremental inductance as current increases | Verify inductance at peak operating current |
| Parasitic capacitance | Creates self-resonance and changes high-frequency behavior | Check self-resonant frequency and impedance |
| Core losses | Generate heat under AC or switching excitation | Evaluate core material and operating frequency |
| Inductance tolerance | Changes current slope and ripple | Include worst-case component variation |
| Temperature variation | Can change inductance, resistance, and magnetic properties | Check operation across the expected temperature range |
8.1 Winding Resistance and the Terminal Voltage
For a practical inductor modeled as an ideal inductance in series with winding resistance R s , the terminal voltage is approximately:
v terminal =Ldi L /dt}+i L R s
This expression assumes a constant inductance and neglects additional parasitic effects.
The resistive term matters particularly in power circuits because it contributes to voltage drop, efficiency loss, and self-heating.
8.2 Core Saturation and Nonlinear Inductance
The basic equation assumes that inductance remains constant. In a real magnetic component, inductance can decrease as the magnetic core approaches saturation.
If the flux linkage is a nonlinear function of current, the more general relationship is:
v L =dλ/dt
For a static nonlinear flux-linkage curve, the incremental inductance is:
L inc =dλ/di L
Consequently:
v L =L inc di L /dt
This formulation is useful for small variations around an operating point. For large excursions, hysteresis, core losses, and dynamic magnetic effects may require a more complete model.
8.3 PCB Layout and Switching-Inductor Voltage
In a switching power supply, the inductor does not operate independently of the PCB layout. The placement of the inductor, switching devices, input capacitors, and output capacitors affects parasitic inductance, electromagnetic interference (EMI), and voltage overshoot.
Practical layout considerations include:
- Minimize high-di/dt current-loop areas to reduce parasitic inductance and radiated noise.
- Keep switching-node copper appropriately sized to balance electrical, thermal, and EMI requirements.
- Provide adequate clearance and creepage where required by the circuit voltage and applicable safety standards.
- Use suitable copper thickness and thermal management for the expected current.
- Evaluate voltage overshoot and ringing with appropriate measurement techniques rather than relying solely on ideal calculations.
These considerations are particularly important in compact power-conversion circuits, where fast switching edges and high current can make parasitic effects significant.
9. Common Mistakes When Using the Inductor Voltage Equation
Correct application of the equation requires attention to units, sign conventions, and the component's operating conditions.
| Common mistake | Why it is incorrect | Correct approach |
|---|---|---|
| Using current instead of current change | Voltage depends on the current derivative, not current magnitude alone | Calculate di L /dt |
| Mixing microhenries and seconds incorrectly | Unit errors can produce results off by orders of magnitude | Convert values to SI units before calculation |
| Ignoring voltage polarity | A decreasing current produces the opposite voltage sign under a fixed reference convention | Define current direction and voltage polarity first |
| Assuming an inductor always has zero DC voltage | This is true only for an ideal inductor at steady-state DC | Include DCR and other losses in practical models |
| Ignoring saturation | Effective inductance can decrease at high current | Check inductance at the intended bias current |
| Applying v=Ldi/dtwithout checking assumptions | The equation in this form assumes constant inductance | Use the flux-linkage relationship when inductance varies |
A useful verification method is to check the units:
H×A/s=V
Because one henry is equivalent to one volt-second per ampere, the equation is dimensionally consistent.
10. How PCBMASTER's Engineering Capabilities Relate to Inductor-Based Designs
Inductor voltage calculations are one part of a broader power-electronics design process. A circuit must also account for component ratings, current paths, switching behavior, thermal performance, and PCB manufacturability.
For projects involving DC-DC converters, power-management circuits, and other inductor-based designs, PCBMASTER provides PCB fabrication, PCBA assembly, and SMT services. Its engineering support includes free document checking to identify potential manufacturability and design issues before production.
Three considerations are particularly relevant to these projects:
- Electrical requirements: Confirm that the PCB design accommodates the operating voltage, peak current, current ripple, and expected switching behavior.
- Manufacturing feasibility: Review copper thickness, trace geometry, component footprints, and thermal requirements before fabrication.
- Production quality: Establish suitable inspection and testing requirements for the assembled board and its intended application.
PCBMASTER reports certifications including ISO 9001, IATF 16949, UL, and RoHS-related compliance. These credentials and services can support manufacturing qualification, although the specific applicability of each certification or compliance claim should be verified for the product and project concerned.
The key distinction is that a correct inductor voltage calculation establishes an electrical design requirement; it does not, by itself, guarantee reliable circuit performance or PCB manufacturability. Both electrical validation and manufacturing review are necessary.
11. Frequently Asked Questions About the Inductor Voltage Equation
The following questions address common points that arise when applying the inductor voltage equation in circuit analysis and power-electronics design.
11.1 What Is the Basic Formula for Inductor Voltage?
For an ideal inductor with constant inductance, the formula is:
v L =Ldi L /dt
The voltage equals the inductance multiplied by the rate of current change.
11.2 Why Does an Inductor Oppose Changes in Current?
A changing current changes the magnetic flux linked to the inductor, inducing a voltage that opposes the change according to Lenz's law. A larger inductance requires a greater voltage to produce the same current slope.
11.3 Can the Voltage Across an Inductor Be Negative?
Yes. Voltage polarity depends on the chosen reference directions. With the passive sign convention, a negative voltage indicates that current is decreasing if inductance is positive and constant.
11.4 What Happens to Inductor Voltage When Current Is Constant?
For an ideal inductor with constant current, the voltage is zero because di L /dt=0. A practical inductor may still have a resistive voltage drop.
11.5 How Is Inductor Voltage Related to Inductive Reactance?
For sinusoidal steady-state operation, inductive reactance is X L =2πfL. It describes the magnitude of the inductor's opposition to sinusoidal current and leads to the phasor relationship V L =jωL I L .
11.6 How Do You Calculate Inductor Current Ripple?
For a constant voltage applied during a switching interval, calculate the current change using:
△I L =V L △t/L
For a buck converter's on-time, this becomes △I L =(V in -V out )D/(Lf s ), assuming ideal continuous-conduction operation.
11.7 Does Inductor Voltage Depend on Frequency?
It depends on the current waveform. For sinusoidal current with a fixed amplitude, voltage amplitude increases with frequency according to V m =2πfLI m . For other waveforms, the voltage depends on the instantaneous current derivative.
11.8 What Is the Difference Between an Ideal and a Real Inductor?
An ideal inductor stores and releases magnetic energy without resistive or core losses. A real inductor has winding resistance, parasitic capacitance, core losses, and potentially nonlinear inductance, all of which can affect circuit performance.
12. Conclusion: Applying the Inductor Voltage Equation Correctly
The fundamental inductor voltage equation,
v L =Ldi L /dt
provides the foundation for analyzing current ramps, transient responses, AC circuits, and switching power converters. Its practical value lies in predicting how inductance and applied voltage determine current behavior.
For accurate circuit design, engineers should also consider voltage polarity, unit consistency, inductance variation, winding resistance, core saturation, and PCB parasitics. Combining the ideal equation with realistic component models and sound PCB design practices leads to more reliable power-electronics implementations.
Tags: #InductorVoltageEquation #Inductor #Inductance #CircuitAnalysis #PowerElectronics #DCDCConverter #PCBDesign #PCBMASTER #ElectricalEngineering
About the Author
Carol Luo
PCB Design EngineerI'm Carol, a PCB Engineer at PCBMASTER with experience in PCB design and manufacturing engineering since 2018. I focus on translating engineering requirements into reliable PCB solutions, with expertise in stack-up design, material selection, and design-for-manufacturing (DFM). I share practical engineering insights from real-world PCB design and production experience.
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