Calculating Voltage, Current in a Buck Inductor

在这篇文章中，我们将尝试了解设计正确的降压转换器电感器所需的各种参数，以便所需的输出能够达到最大效率。

In our previous post we learned the雄鹿转换器的基础知识and realized the important aspect regarding the transistor's ON time with respect to the periodic time of the PWM which essentially determines the output voltage of the buck converter.

在这篇文章中，我们将更深入一些，并尝试评估输入电压，切换时间，输出电压和降压电感器的电流之间的关系，以及如何在设计降压电感器时如何优化这些关系。

Buck Converter Specifications

Let's first understand the various parameters involved with a buck converter:

Peak inductor current, (i_PK)= It's the maximum amount of current that an inductor can store before getting saturated. Here the term "saturated" means a situation where the transistor switching time is so long that it continues to be ON even after the inductor has crossed its maximum or peak current storing capacity. This is an undesirable situation and must be avoided.

最小电感器电流（（i_o)=这是电感器以返回EMF的形式释放其存储的能量时，可以允许电感器到达的最小电流量。

Meaning, in the process when the transistor is switched OFF, the inductor discharges its stored energy to the load and in the course its stored current drops exponentially towards zero, however before it reaches zero the transistor may be supposed to switch ON again, and this point where the transistor may switch ON again is termed as the minimum inductor current.

以上条件也称为A的连续模式buck converter design.

如果晶体管在电感器电流下降到零之前未打开，则情况可能称为不连续模式，这是操作降压转换器的不受欢迎的方法，并可能导致系统的效率低下。

Ripple Current, (Δi =i_PK-i_o)= As may be seen from the adjoining formula, the rippleδi is the difference between the peak current and minimum current induced in the buck inductor.

降压转换器输出处的滤波器电容器通常会稳定此连锁电流，并有助于使其相对稳定。

占空比，（d =T_上/T)= The duty cycle is calculated by dividing the ON time of the transistor by the periodic time.

Periodic time is the total time taken by one PWM cycle to complete, that is the ON time + OFF time of one PWM fed to the transistor.

ON time of the Transistor (T_上= D/F）= The ON time of the PWM or the "switch ON" time of the transistor may be achieved by dividing the duty cycle by the frequency.

Average output current or the load current, (i_ave=Δi / 2 = i_load) =It's obtained by dividing ripple current by 2. This value is the average of the peak current and the minimum current that may be available across the load of a buck converter output.

三角波的RMS值irms =√{i_o²+(Δi)²/ 12} =This expression provides us the RMS or the root mean square value of all or any triangle wave component that may be associated with a buck converter.

OK, so the above were the various parameters and expressions essentially involved with a buck converter which could be utilized while calculating a buck inductor.

Now let's learn how the voltage and current may be related with a buck inductor and how these may be determined correctly, from the following explained data:

Remember here we are assuming the switching of the transistor to be in the continuous mode, that is the transistor always switches ON before the inductor is able to discharge its stored EMF completely and become empty.

This is actually done by appropriately dimensioning the ON time of the transistor or the PWM duty cycle with regard to the inductor capacity (number of turns).

V and I Relationship

The relationship between voltage and current within a buck inductor may be put down as:

V = L di/dt

或者

i = 1/l0ʃTVDT + i_o

The above formula may be used for calculating the buck output current and it holds good when the PWM is in the form of an exponentially rising and decaying wave, or may be a triangle wave.

However if the PWM is in the form of rectangular waveform or pulses, the above formula can be written as:

i = (Vt/L) + i_o

这里的VT是绕组的电压乘以其持续时间的时间（在微秒中）

This formula becomes important while calculating the inductance value L for a buck inductor.

The above expression reveals that the current output from a buck inductor is in the form of a linear ramp, or wide triangle waves, when the PWM is in the form of triangular waves.

Now let's see how one may determine the peak current within a buck inductor, the formula for this is:

IPK =（VIN - VTRANS - VOUT）TON / L + I_o

The above expression provides us the peak current while the transistor is switched ON and as the current inside the inductor builds up linearly (within its saturation range*)

计算峰值电流

Therefore the above expression can be used for calculating the peak current build-up inside a buck inductor while the transistor is in the switch ON phase.

如果将IO表达转移到LHS，我们将得到：

i_PK- i_o= (Vin – Vtrans – Vout)Ton / L

在这里，vtrans指的是晶体管收集器/发射器上的电压下降

回想一下，连锁电流也由ΔI= ipk -io给出，因此在上述公式中代替了：我们得到：

δi = (Vin – Vtrans – Vout)Ton / L ------------------------------------- Eq#1
现在，让我们看一下在晶体管关闭期内获取电感器内电流的表达式，可以在以下方程的帮助下确定它：

i_o=i_PK- (Vout – VD)Toff / L

Again, by substituting ipk - io by Δi in the above expression we get:

δi = (Vout – VD)Toff / L ------------------------------------- Eq#2

The Eq#1 and Eq#2 can be used for determining the ripple current values while the transistor is supplying current to the inductor, that is during it's ON time..... and while the inductor is draining the stored current through the load during the transistor switch OFF periods.

在上面的讨论中，我们成功得出了确定降压电感器中电流（AMP）因子的方程式。

Determining Voltage

Now let's try to find a expression which may help us to determine the voltage factor in a buck inductor.

由于ΔI在EQ＃1和EQ＃2中都是常见的，因此我们可以将术语等同于彼此以获取：

(Vin – Vtrans – Vout)Ton / L = (Vout – VD)Toff / L

VinTon – Vtrans – Vout = VoutToff – VDToff

VinTon – Vtrans – VoutTon = VoutToff - VDToff


VoutTon + VoutToff = VDToff + VinTon – VtransTon


Vout = (VDToff + VinTon – VtransTon) / T

Replacing the Ton/T expressions by duty cycle D in the above expression, we get

Vout = (Vin – Vtrans)D + VD(1 – D)

Processing the above equation further we get:

Vout + VD = (Vin – Vtrans + VD)D
或者

D = Vout - VD / (Vin – Vtrans – VD)

Here VD refers to the voltage drop across the diode.

计算往下电压

如果我们忽略了跨晶体管和二极管的电压下降（因为与输入电压相比，它们可能非常微不足道），我们可以按照以下给出的方式修剪上述表达式：

vout = dvin

上面的最终方程可用于计算在设计降压转换器电路时特定电感器可能来自特定电感器的降低电压。

The above equation is the same as the one discussed in the solved example of our previous article "how buck converters work.

In the next article we'll learn how to estimate the number of turns in a buck inductor....please stay tuned.