Real passive components

Intent

• Estimate the conduction loss in a winding or in a capacitor’s series resistance from the rms current.
• Explain where an inductor’s self-resonance comes from, and why an inductance measured near it is wrong.
• Estimate the output ripple added by a capacitor’s series resistance.

Theory

Series resistance and conduction loss

A real inductor has the resistance of its winding [Erickson & Maksimović 2020, Ch. 3], and a real capacitor an equivalent series resistance (ESR) [Singh 2014]. A current through a resistance dissipates power in proportion to the square of its rms value [Erickson & Maksimović 2020, Ch. 3]:

$$P = I_\mathrm{rms}^{2} R_x$$

$P$

power

$I_\mathrm{rms}$

rms current

$R_x$

resistance carrying the current

Conduction loss in a resistance✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• periodic steady state

Source

Erickson & Maksimović 2020, Ch. 3 (Steady-State Equivalent Circuit Modeling, Losses, and Efficiency)

Derivation

step-by-step sympy derivation

Equation id

loss.cond · equations.yaml entry

The same equation gives the conduction loss of a transistor’s on-resistance (see the semiconductors page). It is the rms value that counts, not the average: a pulsed current loses more than a steady one with the same average (rms of a pulse train).

Self-resonance

Neighbouring turns of a winding are separated by insulation, so a small capacitance appears across each turn [Texas Instruments 2021]. Together these act as a capacitance in parallel with the inductance, and the two resonate:

$$f_\mathrm{srf} = \frac{1}{2 \pi \sqrt{L C_p}}$$

$f_\mathrm{srf}$

self-resonant frequency

$C_p$

parasitic winding capacitance

$L$

inductance

Self-resonant frequency of an inductance with parallel parasitic capacitance✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• undamped L-C resonance

Notes

above $f_\mathrm{srf}$ the component behaves as a capacitor; near it, an impedance measurement reads neither $L$ nor $C_p$

Source

Alexander & Sadiku 2017, Ch. 14 (Frequency Response)

Derivation

step-by-step sympy derivation

Equation id

passive.f_srf · equations.yaml entry

Below this frequency the component is inductive; above it the parallel capacitance dominates and the component behaves as a capacitor [Alexander & Sadiku 2017, Ch. 14]. Wire-wound inductors with many turns can have a high inductance but often a low self-resonant frequency [Texas Instruments 2021]. The self-resonant frequency can be measured with an impedance analyzer: it is the frequency at which the phase of the impedance crosses zero [Texas Instruments 2021].

Capacitor series resistance and output ripple

A converter’s output capacitor carries the ripple current through its series resistance as well as through its capacitance, and the series resistance adds a ripple voltage in phase with that current [Singh 2014]:

$$\Delta v_\mathrm{ESR} = R_\mathrm{ESR} \Delta i_{L,\mathrm{pp}}$$

$\Delta v_\mathrm{ESR}$

ESR ripple, peak-to-peak

$\Delta i_{L,\mathrm{pp}}$

inductor ripple, peak-to-peak

$R_\mathrm{ESR}$

equivalent series resistance

Ripple voltage across a capacitor's series resistance✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• the whole inductor ripple current flows into the output capacitor; the load draws only dc

Notes

peak-to-peak and in phase with the ripple current; it adds to the capacitive ripple

Source

Singh 2014

Derivation

step-by-step sympy derivation

Equation id

cap.esr.ripple · equations.yaml entry

This term adds to the capacitive ripple (buck output ripple), and the capacitor’s series inductance (ESL) adds steps at the switching edges [Singh 2014]. The two terms peak at different instants, so the sum of their peak-to-peak values is an upper bound on the total.

Worked example

An example winding, inductor and output capacitor:

Non-ideal passives example

Symbol	Value	How it is obtained
$I_\mathrm{rms}$ rms current	2 A	given
$R_x$ resistance carrying the current	50 mΩ	given
$L$ inductance	1 mH	given
$C_p$ parasitic winding capacitance	10 pF	given
$R_\mathrm{ESR}$ equivalent series resistance	50 mΩ	given
$\Delta i_{L,\mathrm{pp}}$ inductor ripple, peak-to-peak	600 mA	given
$P$ power	200 mW	Conduction loss in a resistance
$f_\mathrm{srf}$ self-resonant frequency	1.59 MHz	Self-resonant frequency of an inductance with parallel parasitic capacitance
$\Delta v_\mathrm{ESR}$ ESR ripple, peak-to-peak	30 mV	Ripple voltage across a capacitor's series resistance

The winding loses 200 mW at the given rms current. The inductance and its parasitic capacitance resonate at 1.59 MHz. The capacitor’s series resistance adds 30 mV peak-to-peak to the output ripple; compare it with the capacitive ripple on the buck page.

Try it

▶ Open in the equation explorer: Self-resonant frequency of an inductance with parallel parasitic capacitance
Preset from the example passives-real

Sweep $C_p$ on its logarithmic axis: the self-resonant frequency falls with the square root of the parasitic capacitance. Then open the conduction-loss equation and sweep the rms current to see the square law.

Bench exercise

With an impedance analyzer, or an LCR meter whose test frequency can be stepped, measure the impedance magnitude and phase of a wound inductor with many turns over frequency. Find the self-resonant frequency where the phase crosses zero [Texas Instruments 2021], and estimate the parasitic capacitance from the equation above. Then note how the inductance reading at a single test frequency drifts as that frequency approaches the resonance.

Gotchas

• Measure an inductance well below its self-resonance. Near the resonance the reading is neither the inductance nor the capacitance, and for windings with many turns the resonance can be low [Texas Instruments 2021].
• The rms current sets the loss. A winding or a series resistance that carries a pulsed current loses more than its average current suggests.
• Edges add their own ripple. The capacitor’s series inductance produces steps in the output voltage at every switching edge, which neither the capacitive nor the resistive term predicts [Singh 2014].

Go deeper

• Application note Output Ripple Voltage for Buck Switching Regulator (SLVA630A) — S. P. Singh, Texas Instruments · retrieved 2026-09-23
  Buck output ripple with the capacitor's charge, series resistance (ESR) and series inductance (ESL) taken into account.
• Application note Sensor Design for Inductive Sensing Applications Using LDC (SNOA930C) — Texas Instruments · retrieved 2026-09-23
  Written for inductive sensing; its explanation of the capacitance between turns, the self-resonant frequency and how to measure it applies to any wound inductor.
• Book chapter Steady-State Equivalent Circuit Modeling, Losses, and Efficiency (Ch. 3) — R. W. Erickson, D. Maksimović · retrieved 2026-09-23
  DC transformer models and how winding and semiconductor resistances limit the conversion ratio and efficiency.

Quiz

1. A winding with $50\,\mathrm{m}\Omega$ of resistance carries $3\,\mathrm{A}$ rms. What is its conduction loss?

   1. $0.15\,\mathrm{W}$
   2. $0.3\,\mathrm{W}$
   3. $0.45\,\mathrm{W}$
   4. $1.5\,\mathrm{W}$
   Answer and explanation

   Correct answer: C The loss is the square of the rms current times the resistance: $9 \times 0.05$.

2. A $1\,\mathrm{mH}$ inductor has $10\,\mathrm{pF}$ of parasitic capacitance. What is its self-resonant frequency?

   1. $159\,\mathrm{kHz}$
   2. $1.59\,\mathrm{MHz}$
   3. $15.9\,\mathrm{MHz}$
   4. $50\,\mathrm{kHz}$
   Answer and explanation

   Correct answer: B The product of the inductance and the capacitance is $10^{-14}\,\mathrm{s^2}$; its square root is $0.1\,\mu\mathrm{s}$, and $1/(2\pi \times 0.1\,\mu\mathrm{s})$ is about $1.59\,\mathrm{MHz}$.

3. How does an inductor behave above its self-resonant frequency?

   1. As a larger inductor
   2. As a capacitor
   3. As a short circuit at every frequency
   4. As a pure resistor
   Answer and explanation

   Correct answer: B Above the resonance the admittance of the parallel capacitance exceeds that of the inductance, so the impedance is capacitive.

4. A buck output capacitor has $20\,\mathrm{m}\Omega$ of series resistance and carries a $1\,\mathrm{A}$ peak-to-peak ripple current. What ripple does the series resistance add?

   1. $2\,\mathrm{mV}$ peak-to-peak
   2. $10\,\mathrm{mV}$ peak-to-peak
   3. $20\,\mathrm{mV}$ peak-to-peak
   4. $40\,\mathrm{mV}$ peak-to-peak
   Answer and explanation

   Correct answer: C The voltage across a resistance follows its current, so its peak-to-peak value is the resistance times the peak-to-peak current.

5. Two currents have the same average: one is constant, the other a pulse train with $D = 0.25$. How does the loss in the same resistance compare?

   1. The same
   2. The pulse train loses twice as much
   3. The pulse train loses four times as much
   4. The constant current loses more
   Answer and explanation

   Correct answer: C For the same average, the pulse is four times as high, so its rms value is twice the average and the loss, proportional to its square, is four times as large.

6. How can an inductor's self-resonant frequency be found with an impedance analyzer?

   1. Where the impedance magnitude is smallest
   2. Where the phase of the impedance crosses zero
   3. Where the inductance reading is largest
   4. At the analyzer's default test frequency
   Answer and explanation

   Correct answer: B At the parallel resonance the impedance is real, so its phase passes through zero as it changes from inductive to capacitive.