1 Introduction
Boost topologies are crucial in power electronics, but choosing the right inductor value is not always as simple as commonly assumed. In DC-DC boost converters, the selected inductor value affects input current ripple, output capacitance, and transient response. Choosing the correct inductor value helps optimize converter size and cost and ensures operation in the desired conduction mode. This article describes a method for calculating inductor values to maintain the required ripple current and selected conduction mode over a range of input voltages, introduces a mathematical method for calculating the upper and lower bounds of the input voltage mode, and explores how to accelerate these design steps using ON Semiconductor's WebDesigner™ online design tool.
2 conduction modes
The conduction mode of a boost converter is determined by the magnitude of the peak-to-peak inductor ripple current (ΔIL) relative to the DC input current (IIN). This ratio can be defined as the inductor ripple factor (KRF). The higher the inductance, the lower the ripple current and KRF.
In Continuous On-Mode (CCM), the instantaneous inductor current does not reach zero during a normal switching cycle (Figure 1). Therefore, CCM remains constant when ΔIL is less than twice IIN or KRF < 2. The MOSFET or diode must be turned on in CCM. This mode is typically used in medium- and high-power converters to minimize the peak and RMS values of current in the components. Discontinuous On-Mode (DCM) occurs when KRF > 2 and the inductor current is allowed to decay to zero in each switching cycle (Figure 2). The inductor current remains zero until the start of the next switching cycle, and neither the diode nor the MOSFET is turned on. This non-conducting time is called the tie. DCM can provide a lower inductance value and avoids reverse recovery losses in the output diode.
Figure 1 – CCM Operation Figure 2 – DCM Operation
When KRF=2, the converter is considered to be in critical conduction mode (CrCM) or boundary conduction mode (BCM). In this mode, the inductor current reaches zero at the end of the cycle, just as a MOSFET turns on at the beginning of the next cycle. For applications requiring a certain range of input voltage (VIN), fixed-frequency converters are typically designed to operate in the desired single conduction mode (CCM or DCM) within a specified VIN range under maximum load. As the load decreases, a CCM converter will eventually enter DCM operation. The load that causes a change in conduction mode at a given VIN is called the critical load (ICRIT). The inductance value that triggers CrCM/BCM at a given VIN is called the critical inductance (LCRIT), which typically occurs under maximum load conditions.
3. Ripple current and VIN
As is well known, when the input voltage is half the output voltage (VOUT), i.e., the duty cycle (D) is 50% (Figure 3), the maximum inductor ripple current of a DC-DC boost converter operating at a fixed output voltage in continuous conduction mode will occur. This can be expressed mathematically by setting the derivative of the ripple current with respect to D (the slope of the tangent) to zero and solving for D. For simplicity, assume the converter efficiency is 100%.
according to
Figure 3 – Inductor ripple current in CCM
4CCM work
To select the inductor value (L) for the CCM boost converter, the highest possible KRF value needs to be chosen to ensure CCM operation across the entire input voltage range and to avoid peak current being affected by the MOSFETs, diodes, and output capacitors. The minimum inductor value is then calculated. The maximum KRF value is typically chosen between 0.3 and 0.6, but can be as high as 2.0 for CCM. As mentioned earlier, the maximum ripple current ΔIL occurs when D=0.5. So, at what duty cycle will the maximum KRF occur? We can determine this using a derived method.
Assuming η = 100%, then
The pseudo-solution D=1 can be ignored because it is practically impossible to occur in steady state (for a boost converter, the duty cycle must be less than 1.0). Therefore, the ripple factor KRF is highest when D=⅓ or VIN=⅔VOUT, as shown in Figure 4. The same method can also be used to derive the maximum values LMIN, LCRIT, and ICRIT at the same point.
Figure 4 – Maximum value of CCM ripple coefficient KRF when D=⅓
For CCM operation, the minimum inductance value (LMIN) should be calculated at the actual operating input voltage (VIN(CCM)) closest to V⅔VOUT. Depending on the specific input voltage range of the application, VIN(CCM) may appear at the minimum VIN, maximum VIN, or somewhere in between. Solve equation (5) to find L, and recalculate it based on KRF under VIN(CCM) to obtain the following result.
VIN(CCM) is the actual working VIN that is closest to ⅔VOUT.
For the critical inductance and the changes in VIN and IOUT, KRF=2, we can obtain...
5DCM work
As shown in Figure 5, DCM mode operation remains unchanged when the inductance value is less than LCRIT at a given operating VIN and output current (IOUT). For DCM converters, the shortest idle time can be selected to ensure DCM operation across the entire input voltage range. The minimum tiele is typically 3%-5% of the switching cycle, but may be longer at the cost of increased device peak current. The minimum tiele is then used to calculate the maximum inductance value (LMAX). LMAX must be lower than the lowest LCRIT within the VIN range. For a given VIN, CrCM is triggered when the inductance value equals LCRIT (tidle=0).
Figure 5 – Changes in LCRIT and Standardized VIN
To calculate LMAX for the selected minimum idle time (tidle(min)), the DCM volt-second balance equation is first used to find tON(max) (the maximum allowable MOSFET on-time) as a function of VIN, where tdis is the inductor discharge time.
LMAX follows a curve similar to LCRIT, peaking at VIN = ⅔VOUT. To ensure minimum latency, calculate the lowest LMAX value at the actual operating input voltage (VIN(DCM)) opposite to this operating point. Depending on the actual input voltage range of the application, VIN(DCM) will be equal to the minimum or maximum operating VIN. If the overall input voltage range is above or below ⅔VOUT (inclusive), then VIN(DCM) is the input voltage furthest from ⅔VOUT. If the input voltage range covers ⅔VOUT, calculate the inductance at both the minimum and maximum VIN, and choose the lower (worst-case) inductance value. Alternatively, evaluate VIN graphically to determine the worst-case scenario.
6 Input Voltage Mode Boundary
When the output current of the boost converter is less than the maximum value of ICRIT and VIN, CCM operation will be triggered if the input voltage increases above the upper mode boundary or decreases below the lower mode boundary, i.e., IOUT is greater than ICRIT. DCM operation occurs between the two VIN mode boundaries, i.e., when IOUT is less than ICRIT. To graphically represent these conduction mode boundaries under VIN, plot the critical load (using the selected inductor) versus input voltage and associated output current on the same graph. Then find the two VIN values that intersect the two curves on the X-axis (Figure 6).
Figure 6 – Input Voltage Mode Boundary
To represent the mode boundary of VIN algebraically, first set the expression for the critical load to be equal to the relevant output current, in order to find the intersection point:
This can be rewritten as a cubic equation, and KCM can be calculated using constants.
Here, the three solutions to the cubic equation x³ + ax² + bx + c = 0 can be obtained using trigonometric functions [1][2]. In this case, the coefficient of "b" in the x¹ term is zero. We define the solution as the vector VMB.
We know
Due to the physical limitations of the boost converter, any solution where VMB≤0 or VMB>VOUT is negligible. Both positive solutions are valid values of VIN at the mode boundary.
7-Pattern Boundaries – Design Examples
We assume a DCM boost converter with the following specifications:
Substituting VOUT and the calculated θ value into (29), the VIN value at the mode boundary is obtained:
Ignoring the spurious solution (-3.36V), we obtain two input voltage mode boundaries at 4.95V and 10.40V. These calculated values agree with the intersection shown in Figure 7.
Figure 7 – Calculated mode boundary
8. Design accelerated using WebDesigner™ BoostPowertrain
Manually repeating these design calculations for different boost inductor values can be tedious and time-consuming. The complex cubic equations also make calculating input voltage mode boundaries quite cumbersome and error-prone. Using online design tools like ON Semiconductor's WebDesigner™ makes the design process much easier and significantly faster. The BoostPowertrain design module (Figure 8) automatically performs all these calculations (including the impact of actual energy efficiency) and recommends the optimal inductor value based on your application requirements. You can select actual inductor component values from a wide built-in database or enter your own custom inductor specifications to instantly calculate ripple current and mode boundaries, and their impact on output capacitance, MOSFET, diode losses, and overall energy efficiency.
Figure 8-WebDesigner™BoostPowertrain
You can click here to get the WebDesignerBoostPowertrain design tool.
9 Conclusions
Inductance values affect many aspects of boost converters, and an inappropriate selection can lead to excessive cost, oversized design, or poor performance. By understanding the relationship between inductance value, ripple current, duty cycle, and conduction mode, designers can ensure the desired performance across the input voltage range.
