When designing power supplies, filters, or energy-storage circuits, guessing component values leads to melted cores, excessive ripple, or brownouts. The fundamental capacitor and inductor equations dictate exactly how these passive components store energy and respond to changing voltages and currents. This guide strips away the abstract calculus, providing the discrete engineering forms of these equations, fully tracked worked examples, and a concrete decision path to select physical parts.
The Core Capacitor and Inductor Equations Defined
In practical circuit design, we rarely use the instantaneous derivative forms ($v = L \frac{di}{dt}$). Instead, we use the discrete delta ($\Delta$) forms, which calculate average changes over a specific switching period or time interval.
Inductor Equation (Voltage-Time Relationship)
$$V_L = L \frac{\Delta I}{\Delta t}$$
Capacitor Equation (Current-Time Relationship)
$$I_C = C \frac{\Delta V}{\Delta t}$$
| Symbol | Parameter | SI Unit | Engineering Prefix Common Range |
|---|---|---|---|
| $V_L$ | Voltage across the inductor | Volts (V) | 1V to 800V |
| $I_C$ | Current through the capacitor | Amperes (A) | 1mA to 50A |
| $L$ | Inductance | Henries (H) | $\mu H$ ($10^{-6}$) to $mH$ ($10^{-3}$) |
| $C$ | Capacitance | Farads (F) | $pF$ ($10^{-12}$) to $mF$ ($10^{-3}$) |
| $\Delta I$ | Change in current (ripple) | Amperes (A) | mA to A |
| $\Delta V$ | Change in voltage (ripple/droop) | Volts (V) | mV to V |
| $\Delta t$ | Time interval (on-time or off-time) | Seconds (s) | $ns$ ($10^{-9}$) to $ms$ ($10^{-3}$) |
Rearranged Forms for Component Sizing
To solve for the physical component value or the resulting ripple, use these algebraic rearrangements:
- Solve for Inductance: $L = \frac{V_L \cdot \Delta t}{\Delta I}$
- Solve for Inductor Ripple Current: $\Delta I = \frac{V_L \cdot \Delta t}{L}$
- Solve for Capacitance: $C = \frac{I_C \cdot \Delta t}{\Delta V}$
- Solve for Capacitor Voltage Droop: $\Delta V = \frac{I_C \cdot \Delta t}{C}$
- Solve for Time (Hold-up or Charge Time): $\Delta t = \frac{C \cdot \Delta V}{I_C}$ or $\Delta t = \frac{L \cdot \Delta I}{V_L}$
Assumptions, Unit Traps, and Realistic Magnitudes
When the Formula Applies (and Its Assumptions)
These discrete equations assume ideal components and a linear, constant slope for the voltage or current change. For an inductor in a switching converter, this assumes the core has not saturated (inductance remains constant as current rises) and that the voltage drop across the switching MOSFET and diode is negligible compared to the rail voltage. For a capacitor, it assumes the Equivalent Series Resistance (ESR) voltage drop is either accounted for separately or is negligible compared to the capacitive droop.
Unit Mistakes That Break the Math
Another critical trap is misidentifying $\Delta t$. In a PWM switching circuit, $\Delta t$ is not the total switching period ($T$). It is strictly the on-time ($t_{on}$) or off-time ($t_{off}$) during which the specific voltage polarity is applied to the component.
Realistic Answer Magnitudes
If your math yields an inductor value of 500 H or a capacitor of 0.05 F for a standard DC-DC converter, you have a unit error. Realistic magnitudes for switch-mode power supplies (SMPS) are:
- Inductors: $1 \mu H$ to $100 \mu H$ (for 100kHz to 2MHz switching frequencies).
- Capacitors: $10 \mu F$ to $470 \mu F$ (for output filtering and DC-link hold-up).
Worked Example 1: Sizing a Buck Converter Inductor
Scenario: You are designing a buck converter stepping 12V down to 3.3V. The switching frequency ($f_{sw}$) is 500 kHz. The maximum load current is 2.0A. You want to limit the inductor ripple current ($\Delta I$) to 30% of the maximum load current.
- Calculate Target Ripple Current ($\Delta I$):
$\Delta I = 2.0\text{ A} \times 0.30 = 0.6\text{ A}$ - Calculate the Duty Cycle ($D$):
$D = \frac{V_{out}}{V_{in}} = \frac{3.3\text{ V}}{12\text{ V}} = 0.275$ - Calculate the Inductor On-Time ($\Delta t$):
The period $T = \frac{1}{500,000\text{ Hz}} = 2 \times 10^{-6}\text{ s}$ ($2 \mu s$).
$\Delta t (t_{on}) = D \times T = 0.275 \times 2 \times 10^{-6}\text{ s} = 5.5 \times 10^{-7}\text{ s}$ ($550\text{ ns}$). - Determine Voltage Across the Inductor ($V_L$):
During the on-time, the left side of the inductor is at 12V and the right side is at 3.3V.
$V_L = 12\text{ V} - 3.3\text{ V} = 8.7\text{ V}$ - Solve for Inductance ($L$):
$L = \frac{V_L \cdot \Delta t}{\Delta I} = \frac{8.7\text{ V} \cdot 5.5 \times 10^{-7}\text{ s}}{0.6\text{ A}}$
$L = \frac{4.785 \times 10^{-6}}{0.6} = 7.975 \times 10^{-6}\text{ H}$
Result: $7.975 \mu H$. Round up to the nearest standard value: $8.2 \mu H$.
Worked Example 2: Sizing a Microcontroller Hold-Up Capacitor
Scenario: An IoT sensor node draws a steady 150 mA ($0.15\text{ A}$) at 5.0V. Due to a long cable run, the power supply occasionally drops out for 2 milliseconds ($2\text{ ms}$). The microcontroller's brownout reset triggers at 4.2V. We need a capacitor to bridge this 2 ms gap.
- Identify Knowns in Base SI Units:
$I_C = 0.15\text{ A}$
$\Delta t = 0.002\text{ s}$
$\Delta V = 5.0\text{ V} - 4.2\text{ V} = 0.8\text{ V}$ - Solve for Capacitance ($C$):
$C = \frac{I_C \cdot \Delta t}{\Delta V} = \frac{0.15\text{ A} \cdot 0.002\text{ s}}{0.8\text{ V}}$
$C = \frac{0.0003}{0.8} = 0.000375\text{ F}$ - Convert to Engineering Prefix:
$0.000375\text{ F} \times 10^6 = 375 \mu F$.
Result: Select a standard value of $390 \mu F$ or $470 \mu F$.
Decision Path: Selecting Your Physical Component
Math gives you a theoretical value; physics dictates the physical part. Use this decision tree to terminate your design with a concrete component selection.
| Condition / Application | Required Characteristic | Concrete Part Pick (Example) |
|---|---|---|
| Inductor: SMPS Buck/Boost ($1 \mu H - 20 \mu H$), $I_{peak} > 2\text{A}$ | Shielded ferrite drum core to prevent EMI; high saturation current ($I_{sat}$) rating. | Würth Elektronik 7447742082 ($8.2 \mu H$, 3.0A $I_{sat}$, shielded SMD) |
| Inductor: Low-frequency filtering / Choke ($> 100 \mu H$), $I < 1\text{A}$ | High inductance, unshielded or toroidal is acceptable if EMI is managed. | Bourns SRP1265A-101M ($100 \mu H$, 2.5A, shielded high-current) |
| Capacitor: SMPS Output Filter (High ripple current, $< 50\text{V}$) | Ultra-low ESR to minimize $I^2R$ heating and output voltage ripple. | Panasonic EEH-ZA1V391XP ($390 \mu F$, 35V, Polymer Hybrid, low ESR) |
| Capacitor: DC-Link / Hold-up (High energy density, $> 50\text{V}$) | High CV product (Capacitance $\times$ Voltage), long life at high temp. | Rubycon ZLH Series (e.g., $470 \mu F$, 63V, Low-ESR Aluminum Electrolytic) |
Real-World Parasitics: When the Ideal Math Fails
The equations above assume ideal behavior. On the bench, three parasitics will alter your results:
- Inductor Core Saturation: If your peak current ($I_{out} + \frac{\Delta I}{2}$) exceeds the component's $I_{sat}$ rating, the inductance $L$ drops precipitously. The $\Delta I$ will spike, potentially destroying your switching MOSFET. Always check the datasheet's $I_{sat}$ curve, not just the RMS current rating.
- Capacitor ESR (Equivalent Series Resistance): In Example 2, we calculated a 375 $\mu F$ requirement based purely on capacitive droop. However, when the 150 mA load steps on instantly, the ESR causes an immediate voltage drop ($V_{drop} = I \times ESR$). If your capacitor has 0.5$\Omega$ ESR, you lose an immediate 75 mV before the capacitive droop even begins. For high-step loads, select polymer or multi-layer ceramic capacitors (MLCC) with ESR in the milliohm range.
- DC Bias Effect in MLCCs: If you substitute the $390 \mu F$ electrolytic in Example 2 with a bank of X5R/X7R ceramic capacitors, be aware that ceramic capacitance drops significantly under DC bias. A "100 $\mu F$, 10V" MLCC might only provide 40 $\mu F$ of actual capacitance when biased at 5V. Always consult the manufacturer's DC bias derating curve.
Default Recommendation: For general-purpose 5V to 12V microcontroller and IoT prototyping where exact ripple math is secondary to reliability, default to Panasonic OS-CON or Polymer Hybrid (EEH-ZA series) capacitors for bulk storage, and Würth WE-PD or WE-HCI series shielded inductors for local DC-DC conversion. They offer the best balance of low parasitics, predictable saturation curves, and widespread availability without requiring complex thermal derating calculations.
