The K parameter

Intent

• Compute $K$ for a converter and $K_\mathrm{crit}(D)$ for its topology.
• Decide the operating mode, and find the inductance that keeps a converter in CCM down to a given load.
• Identify the operating point (input voltage and load) that is the worst case for the mode boundary.

Theory

The boundary condition

A converter with a diode leaves CCM when the inductor current ripple reaches the dc inductor current: at the boundary the half-ripple $\Delta i_L$ equals the dc current [Erickson & Maksimović 2020, Ch. 5]. When both sides are written in terms of circuit values, the inductance, the load resistance and the switching period collect into one dimensionless parameter,

$$K = \frac{2 L}{R T_s}$$

$K$

inductor parameter (CCM vs DCM)

$L$

inductance

$R$

load resistance

$T_s$

switching period

Dimensionless inductor parameter K✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• definition

Convention

the converter operates in CCM when $K > K_\mathrm{crit}(D)$

Source

Erickson & Maksimović 2020, Ch. 5 (The Discontinuous Conduction Mode)

Equation id

K.def · equations.yaml entry

and the boundary becomes a function of the duty ratio alone, the critical value $K_\mathrm{crit}(D)$. The converter operates in CCM for $K > K_\mathrm{crit}$ and in DCM for $K < K_\mathrm{crit}$ [Erickson & Maksimović 2020, Ch. 5].

$K$ compares the time constant $L/R$ with the switching period. A larger inductance, a heavier load (smaller $R$) or a higher switching frequency all raise $K$ and push the converter towards CCM.

Critical values of the basic converters

Each topology has its own $K_\mathrm{crit}(D)$, because its dc inductor current and its ripple depend on $D$ in different ways:

$$K_\mathrm{crit} = 1 - D$$

$K_\mathrm{crit}$

K at the CCM/DCM boundary

$D$

duty ratio

Critical K of the buck converter✓ verified by 8 tests

Assumptions, convention and source

Assumptions

• ideal (lossless) switches and passive components
• at the CCM/DCM boundary
• periodic steady state
• small-ripple approximation
• resistive load R

Source

Erickson & Maksimović 2020, Ch. 5 (The Discontinuous Conduction Mode)

Derivation

step-by-step sympy derivation

Equation id

Kcrit.buck · equations.yaml entry

$$K_\mathrm{crit} = D \left(1 - D\right)^{2}$$

$K_\mathrm{crit}$

K at the CCM/DCM boundary

$D$

duty ratio

Critical K of the boost converter✓ verified by 8 tests

Assumptions, convention and source

Assumptions

• ideal (lossless) switches and passive components
• at the CCM/DCM boundary
• periodic steady state
• small-ripple approximation
• resistive load R

Notes

maximum $4/27$ at $D = 1/3$

Source

Erickson & Maksimović 2020, Ch. 5 (The Discontinuous Conduction Mode)

Derivation

step-by-step sympy derivation

Equation id

Kcrit.boost · equations.yaml entry

$$K_\mathrm{crit} = \left(1 - D\right)^{2}$$

$K_\mathrm{crit}$

K at the CCM/DCM boundary

$D$

duty ratio

Critical K of the buck-boost converter✓ verified by 8 tests

Assumptions, convention and source

Assumptions

• ideal (lossless) switches and passive components
• at the CCM/DCM boundary
• periodic steady state
• small-ripple approximation
• resistive load R

Source

Erickson & Maksimović 2020, Ch. 5 (The Discontinuous Conduction Mode)

Derivation

step-by-step sympy derivation

Equation id

Kcrit.buckboost · equations.yaml entry

For the flyback, $K$ is formed with the magnetizing inductance referred to the primary and with the actual load resistance; the turns ratio then appears in the critical value:

$$K_\mathrm{crit} = \frac{\left(1 - D\right)^{2}}{n^{2}}$$

$K_\mathrm{crit}$

K at the CCM/DCM boundary

$D$

duty ratio

$n$

turns ratio, secondary/primary

Critical K of the flyback converter✓ verified by 8 tests

Assumptions, convention and source

Assumptions

• ideal (lossless) switches and passive components
• at the CCM/DCM boundary
• periodic steady state
• small-ripple approximation
• resistive load R

Convention

$K = 2L_M/(R T_s)$ with $L_M$ referred to the primary and the actual (secondary-side) load $R$; 1:n, $n = N_s/N_p$

Source

Erickson & Maksimović 2020, Ch. 5–6 (DCM; flyback as an isolated buck-boost)

Derivation

step-by-step sympy derivation

Equation id

Kcrit.flyback · equations.yaml entry

The buck’s critical value is largest at small $D$, approaching 1. The boost’s value is zero at both ends of the duty-ratio range; setting its derivative to zero shows that it peaks at $D = 1/3$, where it is $4/27$. The buck-boost’s value falls from 1 at small $D$ to zero at large $D$.

Designing for CCM: the critical inductance

Setting $K$ equal to $K_\mathrm{crit}$ and solving the definition of $K$ for $L$ gives the smallest inductance that keeps the converter in CCM at a given load:

$$L_\mathrm{crit} = \frac{K_\mathrm{crit} R T_s}{2}$$

$L_\mathrm{crit}$

critical inductance (CCM/DCM boundary)

$K_\mathrm{crit}$

K at the CCM/DCM boundary

$R$

load resistance

$T_s$

switching period

Critical inductance at the CCM/DCM boundary✓ verified by 8 tests

Assumptions, convention and source

Assumptions

• ideal (lossless) switches and passive components
• at the CCM/DCM boundary
• periodic steady state
• small-ripple approximation
• resistive load R

Notes

the converter stays in CCM for $L > L_\mathrm{crit}$; check the largest $R$ (lightest load) and the duty ratio that maximises $K_\mathrm{crit}$ over the input range

Source

Erickson & Maksimović 2020, Ch. 5 (The Discontinuous Conduction Mode)

Derivation

step-by-step sympy derivation

Equation id

L.crit · equations.yaml entry

Because $K_\mathrm{crit}$ depends on $D$, and $D$ on the input voltage, the worst case is the lightest load (largest $R$) at the duty ratio that gives the largest $K_\mathrm{crit}$ over the input-voltage range. For a buck with a regulated output this is the highest input voltage, where $D$ is smallest.

Worked example

The buck example at its nominal load, and the same converter with a ten-times lighter load:

Buck example

Symbol	Value	How it is obtained
$L$ inductance	100 µH	given
$R$ load resistance	10 Ω	given
$T_s$ switching period	10 µs	Switching period
$D$ duty ratio	0.5	given
$K$ inductor parameter (CCM vs DCM)	2	Dimensionless inductor parameter K
$K_\mathrm{crit}$ K at the CCM/DCM boundary	0.5	Critical K of the buck converter
$L_\mathrm{crit}$ critical inductance (CCM/DCM boundary)	25 µH	Critical inductance at the CCM/DCM boundary

Operating mode: CCM ($K > K_\mathrm{crit}$: holds)

Buck example at light load

Symbol	Value	How it is obtained
$L$ inductance	100 µH	given
$R$ load resistance	100 Ω	given
$T_s$ switching period	10 µs	Switching period
$D$ duty ratio	0.5	given
$K$ inductor parameter (CCM vs DCM)	0.2	Dimensionless inductor parameter K
$K_\mathrm{crit}$ K at the CCM/DCM boundary	0.5	Critical K of the buck converter
$L_\mathrm{crit}$ critical inductance (CCM/DCM boundary)	250 µH	Critical inductance at the CCM/DCM boundary

Operating mode: DCM ($K > K_\mathrm{crit}$: does not hold)

At the nominal load, $K$ = 2 is well above $K_\mathrm{crit}$ = 0.5: CCM, with margin. Equivalently, the inductance of 100 µH exceeds the critical 25 µH. At the light load, $K$ falls to 0.2, below the same critical value, and the critical inductance rises to 250 µH: the converter is in DCM.

Try it

▶ Open in the equation explorer: Critical K of the boost converter
Preset from the example boost-basic

▶ Open in the simulator: Boost
Preset from the example boost-basic

Sweep $D$ from 0 to 1 and find the peak of the boost’s critical value. A boost whose $K$ lies above that peak stays in CCM at every duty ratio.

Bench exercise

On a converter with a diode rectifier and an adjustable resistive load, watch the inductor current with a current probe and increase the load resistance until the current just touches zero at the end of each period. Measure the duty ratio and the load resistance at that point, compute $K$ from the inductance and the switching period, and compare it with $K_\mathrm{crit}(D)$ for the topology.

Gotchas

• Use consistent quantities in $K$. For isolated converters, $K$ uses the magnetizing inductance referred to the primary with the actual load resistance on the secondary side; the turns ratio then sits in $K_\mathrm{crit}$. Mixing referred and unreferred values gives the wrong mode [Erickson & Maksimović 2020, Ch. 6].
• Check the whole input-voltage range. In closed loop the duty ratio moves with the input voltage, and with it $K_\mathrm{crit}$.
• Compare with the smallest inductance the inductor will have in operation, not with its nominal value, since the critical inductance is a minimum.

Go deeper

• Book chapter The Discontinuous Conduction Mode (Ch. 5) — R. W. Erickson, D. Maksimović · retrieved 2026-09-23
  Origin of DCM, the K and K_crit boundary and the DCM conversion ratios derived with charge balance.
• Course Converter Circuits — University of Colorado Boulder (Coursera) · retrieved 2026-09-23
  DCM analysis and the isolated converter circuits, following the same textbook.
• Application note Understanding Boost Power Stages in Switchmode Power Supplies (SLVA061) — E. Rogers, Texas Instruments · retrieved 2026-09-23
  The boost counterpart: steady-state and small-signal analysis in CCM and DCM.

Quiz

1. A converter has $L = 50\,\mu\mathrm{H}$, $R = 5\,\Omega$ and $T_s = 10\,\mu\mathrm{s}$. What is $K$?

   1. 0.5
   2. 1
   3. 2
   4. 4
   Answer and explanation

   Correct answer: C $K$ is $2L/(R T_s)$: twice $50\,\mu\mathrm{H}$ is $100\,\mu\mathrm{H}$, and $R T_s$ is $50\,\mu\mathrm{s}\cdot\Omega$, so the ratio is 2.

2. A buck converter runs at $D = 0.4$ with $K = 0.5$. In which mode does it operate?

   1. CCM
   2. DCM
   3. Exactly at the boundary
   4. It cannot be decided without the input voltage
   Answer and explanation

   Correct answer: B For the buck $K_\mathrm{crit}$ is $1 - D$, which is 0.6 here. Since $K$ is below it, the inductor current reaches zero: DCM.

3. Which change pushes a converter towards DCM?

   1. A larger inductance
   2. A higher switching frequency
   3. A heavier load (smaller $R$)
   4. A lighter load (larger $R$)
   Answer and explanation

   Correct answer: D A larger $R$ lowers $K$. A larger inductance, a higher switching frequency and a heavier load all raise $K$ and push the converter towards CCM.

4. At which duty ratio is the critical value $K_\mathrm{crit}$ of the boost converter largest?

   1. As $D$ approaches 0
   2. $D = 1/3$
   3. $D = 0.5$
   4. As $D$ approaches 1
   Answer and explanation

   Correct answer: B The boost value $D(1 - D)^2$ is zero at both ends. Its derivative $(1 - D)(1 - 3D)$ vanishes at $D = 1/3$, where the value is $4/27$.

5. A buck converter must stay in CCM down to a load of $R = 20\,\Omega$, where $K_\mathrm{crit}$ is 0.5, with $T_s = 10\,\mu\mathrm{s}$. What is the smallest inductance that does this?

   1. $25\,\mu\mathrm{H}$
   2. $50\,\mu\mathrm{H}$
   3. $100\,\mu\mathrm{H}$
   4. $200\,\mu\mathrm{H}$
   Answer and explanation

   Correct answer: B The critical inductance is $K_\mathrm{crit} R T_s/2$: half of 0.5 times $200\,\mu\mathrm{s}\cdot\Omega$, that is $50\,\mu\mathrm{H}$.

6. A buck converter regulates a fixed output voltage from a range of input voltages. At which input voltage is it most likely to enter DCM at light load?

   1. The lowest input voltage
   2. The highest input voltage
   3. The middle of the range
   4. The input voltage does not matter
   Answer and explanation

   Correct answer: B A higher input voltage needs a smaller duty ratio, and the buck value $1 - D$ grows as $D$ shrinks, so the boundary is reached at a larger $K$.