A detailed explanation on how to select inductor value of the DC boost converter.
Forcing topology is very important in field of power electronics, but inductor sizing is not always as easy as it is often assumed. In a DC boost converter, selected inductance value affects input current ripple, size of output capacitor, and transient response. Selecting correct inductor value helps to optimize size and cost of converter, as well as ensuring operation in desired conductive mode. This article describes a method for calculating inductance value to maintain desired ripple current and selected conduction mode over input voltage range, and presents a mathematical method for calculating upper and lower limits of input voltage modes.
The conduction mode of boost converter is determined by magnitude of ripple current of inductor (ΔI) relative to input DC current (I). This ratio can be defined as ripple factor of inductor (K). The higher inductance, lower ripple current and K.
(1), where (2) 在连续导通模式 (CCM) 中，正常开关周期内，瞬时电感电流不会达到零 (图1)。 Therefore, when ΔI is less than 2 times or K<2, The SCM remains unchanged. The MOSFET or diode must conduct in CCM. This mode is typically used in medium and high power drives to minimize peak and RMS currents in components. Discontinuous conduction mode (DCM) occurs when K > 2 and inductor current can decrease to zero in each switching cycle (Fig. 2). Until start of next switching cycle, inductor current remains zero, and neither diode nor MOSFET conducts current. This non-conduction time is called tidle. DCM provides lower inductance values and avoids reverse recovery loss of the output diode.
Figure 1. Working CCM
Fig. 2. DCM operation
When K = 2, transducer is said to be in critical conduction mode (CrCM) or boundary conduction mode (BCM). In this mode, inductor current reaches zero at end of a cycle, just as MOSFET turns on at beginning of next cycle. For applications requiring a range of input voltages (V), fixed frequency converters are typically designed to operate in required single conduction mode (CCM or DCM) with a maximum load within specified V range. As load decreases, CCM will eventually transition to DCM mode. . For a given V, load that changes conduction mode is critical load (I). For a given V, value of inductance that causes CrCM/BCM is called critical inductance (L) and usually occurs at maximum load.
Ripple current and VIN
It is well known that when input voltage is half output voltage (V), that is, when duty cycle (D) is 50% (Figure 3), DC amplifier operates at a fixed output power. voltage in continuous conduction mode The maximum value of voltage converter choke ripple current will appear. This can be expressed mathematically by setting derivative of ripple current with respect to D (tangent slope) to zero and finding D. For simplicity, converter is assumed to be 100% efficient.
according to (3) (4) and (5), 并通过 CCM 或 CrCM 的电感伏秒平衡 (6), 则 (7). 将导数设置为零, (8)We can get (9).
Figure 3 - Inductor ripple current in CCM
In order to select inductance (L) value for CCM boost converter, maximum K value must be selected to ensure that CCM operates over entire input voltage range and avoid current peaks from MOSFET, diode, and output capacitors. . Then calculate minimum inductance value. Kmax is usually chosen between 0.3 and 0.6, but can be as high as 2.0 for CCM. As mentioned earlier, at D = 0.5, maximum ripple current ΔI occurs. So, at what duty cycle is maximum value of K reached? We can get it by inference method.
Suppose that η = 100%, then (10), 然后将(2)、(6)、(7) 和 (10) 代入(1) ，得出: (11) (12). 对 D 求解，可得 (13).The false solution D = 1 can be neglected, since it is practically impossible in steady state (for a boost converter, duty cycle must be less than 1.0). Therefore, ripple factor K is maximum when D = ⅓ or V = ⅔V, as shown in Figure 4. The same method can be used to obtain maximum value of L, L and I at same point.
Figure 4. The largest value of CCM K pulsation coefficient at D = ⅓
For CCM mode, minimum inductance (L) must be calculated to within 2/3V of actual operating input voltage (V). Depending on application's specific input voltage range, V can be at a minimum V, a maximum V, or somewhere in between. Solution of equation (5) for L and recalculation for K at V gives (14) where V is actual working V closest to ⅔V. 对于临界电感与 V 和 I 的变化，KRF = 2，可得出 (15) (16)
As shown in figure 5, when inductance value is less than L at a certain operating V and output current (I), DCM operation remains unchanged. For DCM converters, you can choose shortest idle time to ensure operation of DCM in entire range of input voltages. The minimum tide is typically 3%-5% of switching period, but can be higher by increasing device's peak current. The minimum value of tdle is then used to calculate maximum value of inductance (L). L must be below lowest value of L in V range. For a given value of V, an inductance value of L(tidle=0) starts CrCM.
Figure 5. Changing L and normalized V
To calculate L for a chosen minimum tidle time, first use DCM volt-second balance equation to find t (the maximum allowable MOSFET run time) as a function of V, where t is inductor discharge time. (17) where
(18) 可得出 (19) For simplicity, we will again assume that P = P. (20) , where (21). 将方程 (3)、(5)、(10)、(19) 和 (21) 代入 (20)，求得 V 下的 L (22). L 遵循类似于 L 的曲线，且同在 V = ⅔V时达到峰值。 To ensure minimum time, calculate smallest value of L at actual operating input voltage (V) opposite this operating point. Depending on actual input voltage range of application, V will be equal to minimum or maximum operating V. If total input voltage range is above or below ⅔ V, V is furthest input voltage from ⅔ V. If input voltage range is ⅔ V, calculate inductance at minimum and maximum V and select smaller (worst) value of inductance. Alternatively, evaluate V graphically to determine worst case.
Input voltage mode limit
When output current of boost converter is less than I and V
, if input voltage rises above upper mode limit or falls below lower mode limit, i.e. I is greater than I, CCM operation will be initiated. The operation of DCM, on other hand, occurs between limits of two V modes, i.e., when I is less than I. To graph these conduction mode limits at V, apply a critical load (with selected inductor) as a function of input voltage and corresponding output current on same graph. Then find two V values on x-axis that intersect two curves (Figure 6).
Figure 6. Input voltage mode limits
(23). (24) of which
(25). 这里，三次方程通式 x3 + ax2 + bx + c = 0  。 In this case, coefficient "b" for term x1 is equal to zero. We define solution as a vector V. (26),
(28), (29). V ≤ 0或V > V 的解均可忽略。 Both positive solutions are admissible values of V at mode boundaries.
Chart Border - Design Example
We are going to use a DCM boost converter with following specifications:
V = 12 VI = 1 AL = 6 μHF = 100 kHz 首先，通过(25) 和(28) 计算得出K和θ： . 将V 和计算所得的 θ值代入 (29)，得出模式边界处的V 值： . 忽略伪解 (-3.36V)，我们在 4.95V 和 10.40V 得到两个输入电压模式边界。 These calculated values are consistent with intersections shown in Figure 7. < br>
Figure 7. Computed mode boundaries
The choke value affects many aspects of boost converter, and wrong choice can result in excessive cost, oversizing, or degraded performance. By understanding relationship between inductance value, ripple current, duty cycle, and conduction mode, designers can achieve desired performance over entire input voltage range.
 H. W. Turnbull, Theory of Equations, Chapter IX, Edinburgh & London: Oliver and Boyd, 1952.
 I. J. Zucker, “Cubic Equation—Revisiting Irreducible Case,” The Mathematical Gazette, vol. 92, no. 524, pp. 264–268, July 2008
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