How to choose the correct inductor value in the dc-dc boost converter
Conduction mode
The conduction mode of the boost converter is determined by the magnitude of the peak-to-peak (ΔIL) of the inductor ripple current with respect 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.
(1) , where (2)
In continuous conduction mode (CCM), the instantaneous inductor current does not reach zero during the normal switching cycle (Figure 1). Therefore, when ΔIL is less than 2 times of IIN or KRF < 2, the CCM remains unchanged. The MOSFET or diode must be turned on with CCM. This mode is typically applied to medium power and high power converters to minimize peak and rms currents in the component. The discontinuous conduction mode (DCM) occurs when KRF > 2 and the inductor current is allowed to decay to zero during each switching cycle (Figure 2). Until the start of the next switching cycle, the inductor current remains at zero and neither the diode nor the MOSFET conducts. This non-conduction time is called tidle. DCM provides lower inductance values ​​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 the MOSFET turns on at the beginning of the next cycle. For applications that require a range of input voltages (VIN), fixed-frequency converters are typically designed to operate in the specified single-conduction mode (CCM or DCM) within the specified VIN range at maximum load. As the load decreases, the CCM converter will eventually enter the DCM operation. At a given VIN, the load that causes the conduction mode to change is the critical load (ICRIT). At a given VIN, the inductance of the CrCM / BCM is called the critical inductance (LCRIT), which usually occurs at the maximum load.
Ripple current and VIN
It is well known that when the input voltage is half of the output voltage (VOUT), that is, the duty cycle (D) is 50% (Fig. 3), the DC-DC boost converter operates at a fixed output voltage in continuous conduction mode. The maximum inductor ripple current will appear. This can be mathematically represented by setting the derivative of the ripple current with respect to D (the slope of the tangent) equal to zero and solving for D. For simplicity, assume that the converter is 100% efficient.
according to (3), (4) and (5),
And pass the volt-second balance of the inductor of CCM or CrCM (6),
then (7).
Set the derivative to zero, (8)
We can draw (9).
Figure 3 – Inductor Ripple Current in CCM
CCM work
To select the inductor value (L) of the CCM boost converter, the highest KRF value needs to be selected to ensure CCM operation over the entire input voltage range and to avoid peak currents being affected by MOSFETs, diodes, and output capacitors. Then calculate the minimum inductance value. The highest KRF value is usually chosen between 0.3 and 0.6, but can be as high as 2.0 for CCM. As mentioned earlier, when D = 0.5, the maximum ripple current ΔIL appears. So, how many duty cycles will there be a KRF maximum? We can find it by derivation.
Assuming η = 100%, then (10),
Substituting (2), (6), (7), and (10) into (1) yields:
(11)
(12).
Solving D, you can get (13).
The pseudo-solution of D = 1 can be ignored because it is virtually impossible to achieve at steady state (for boost converters, the duty cycle must be less than 1.0). Therefore, the ripple factor KRF is highest when D = 1â„3 or VIN = 2â„3VOUT, as shown in Figure 4. Using the same method, the maximum values ​​LMIN, LCRIT and ICRIT at the same point can be derived.
Figure 4 – Maximum CCM ripple factor KRF when D = 1â„3
For CCM operation, the minimum inductance value (LMIN) should be calculated at the actual operating input voltage (VIN(CCM)) closest to 2â„3 VOUT. Depending on the specific input voltage range of the application, VIN (CCM) may appear at a minimum VIN, a maximum VIN, or somewhere in between. Solve equation (5) and find L, and recalculate according to KRF under VIN (CCM).
(14), where VIN(CCM) is the actual operating VIN closest to 2â„3VOUT.
For critical inductance and VIN and IOUT changes, KRF = 2, which can be derived
(15).
Given a VIN and L value, a critical load (ICRIT) occurs when KRF = 2:
(16)
DCM work
As shown in Figure 5, the DCM mode operation remains the same when the inductance at a certain operating VIN and output current (IOUT) is less than LCRIT. For DCM converters, the shortest idle time can be selected to ensure DCM operation over the entire input voltage range. The tidle minimum is typically 3%-5% of the switching period, but may be longer at the expense of increased peak current. The minimum inductance value (LMAX) is then calculated using the tidle minimum. LMAX must be below the minimum LCRIT in the VIN range. For a given VIN, the CrCM is induced when the inductance value is equal to LCRIT (tidle = 0).
Figure 5 – Changes in LCRIT and normalized VIN
To calculate the LMAX of the selected minimum idle time (tidle(min)), first use the DCM volt-second balance equation to find the function of tON(max) (the maximum allowable MOSFET on-time) and VIN, where tdis is the inductor discharge. time.
(17), of which (18)
Can be drawn
(19).
The average (DC) inductor current is equal to the converter's DC input current. By rearranging (17), a function of tdis versus tON is obtained. For the sake of simplicity, we will again assume PIN = POUT.
(20) , where (twenty one).
Substituting equations (3), (5), (10), (19), and (21) into (20), find L under VIN (DCM)
(twenty two).
LMAX follows a curve similar to LCRIT and peaks at VIN = 2â„3VOUT. To ensure a minimum tidle, calculate the lowest LMAX value at the actual operating input voltage (VIN(DCM)) opposite 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 2â„3 VOUT (including 2â„3 VOUT), VIN(DCM) is the input voltage that is furthest from 2â„3 VOUT. If the input voltage range covers 2â„3 VOUT, calculate the inductance at the minimum and maximum VIN and select the lower (worst case) inductance value. Alternatively, the VIN is evaluated graphically to determine the worst case.
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 initiated if the input voltage increases above the upper mode boundary or falls below the lower mode boundary, ie, IOUT is greater than ICRIT. The DCM operation occurs between the mode boundaries of two VINs, that is, when IOUT is less than ICRIT. To graphically present these conduction mode boundaries at VIN, plot the critical load (using the selected inductor) versus the input voltage and associated output current in 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 of the critical load equal to the associated output current to find the intersection:
(twenty three).
This can be rewritten as a cubic equation, and KCM can be calculated by constants.
Here, the three solutions of the cubic equation x3 + ax2 + bx + c = 0 can be obtained by the trigonometric solution of the cubic equation [1] [2]. In this case, the "b" coefficient of the x1 term is zero. We define the solution as a vector VMB.
we know
Due to the physical limitations of the boost converter, any solution with VMB ≤ 0 or VMB > VOUT can be ignored. Both positive solutions are valid values ​​of VIN at the mode boundary.
Pattern boundary – design example
Let's assume a DCM boost converter with the following specifications:
VOUT = 12 V
IOUT = 1 A
L = 6 μH
FSW = 100 kHz
First, calculate KCM and θ from (25) and (28):
Substituting VOUT and the calculated θ value into (29) yields the VIN value at the mode boundary:
Ignoring the pseudo-solution (-3.36 V), we get two input voltage mode boundaries at 4.95 V and 10.40 V. These calculated values ​​correspond to the intersections shown in FIG.
Figure 7 – Calculated mode boundaries
Accelerate design with WebDesignerTM Boost Powertrain
Manually repeating these design calculations can be tedious and time consuming for different boost inductor values. The complex cubic equation also makes the calculation of the input voltage mode boundary quite cumbersome and error-prone. By using online design tools such as ON Semiconductor's WebDesignerTM, design work can be accelerated more easily and dramatically. The Boost Powertrain design module (Figure 8) automatically performs all of these calculations (including the effects of actual energy efficiency) and recommends the best inductance values ​​for your application requirements. You can choose true inductor component values ​​from a wide range of built-in databases, or enter your own custom inductor specifications to instantly calculate ripple current and mode boundaries, and their output capacitance, MOSFET, diode losses, And the impact of overall energy efficiency.
Figure 8 - WebDesignerTM Boost Powertrain
in conclusion
Inductance values ​​can affect many aspects of boost converters. If not chosen properly, it can result in excessive cost, oversize, or poor performance. By understanding the relationship between inductance, ripple current, duty cycle, and conduction mode, designers can ensure the desired performance over the input voltage range.
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