How to Design a Buck Converter Circuit: Formulas and Calculations

In this post I have explained the method of designing buck converter circuits by calculating all the required parameters, in order to ensure an optimal performance from these devices.

We take the example of IC 555 buck converter typologies, and try to understand the optimizing techniques through equations and manual adjustments, for achieving the most optimal output response from these converter designs.

You may want to summarize the details from the following articles, before embarking on the present article which deals with the inductor designing methods.

How Boost Converters Work

How Buck Converters Work

How to Design: Basic Buck Converter Equations

The crucial specifications needed to design a buck converter are as given below::

• Input Voltage (V_in)
• Output Voltage (V_out)
• Output Current (I_out)
• Switching Frequency (f_s)

Duty Cycle Calculation

Now, the crucial element required for calculating the inductor, is the duty cycle. We can use the following formula for calculating the duty cycle:

D = V_out / V_in

The next crucial thing for calculating the buck inductor is the Ripple Current, which can be calculated in the following manner:

Target Inductor Ripple Current (Δ_IL): As a rule of thumb, you can choose 10% of output current.

Calculating the Buck Inductor

Now, finally we can calculate the buck inductor, using the following formula:

L = (V_OUT × (1 - D)) / (Δ_IL × f_s)

Next, we need to calculate the output filter capacitor, which requires the Ripple Voltage ΔV_OUT

As a rule of thumb, we can take the Voltage Ripple (Δ_Vout) as 1% of output voltage.

Now, we can calculate the value of the output capacitor in the following manner:

Output Filter Capacitance Calculation for Buck Converter:

C_OUT = Δ_IL / (8 × f_s × Δ_Vout)

When we design a buck converter, the output ripple voltage Δ_Vout is mainly affected by two things, the inductor ripple current ΔIL and the output capacitor C_out.

You might be wondering where that factor of 8 came from.

It appears when we use an estimate to connect the peak-to-peak ripple voltage, peak-to-peak ripple current, and capacitance value. We assume that the ripple waveform is shaped like a triangle, which is a popular method in these computations.

So, in simple terms, number 8 is simply the result of reducing our arithmetic while preserving everything relevant to how the output ripple acts in continuous conduction mode (CCM).

Choose an output capacitor rated at least 250 μF with a voltage rating above 100V and low ESR.

Diode Selection

Reverse Voltage: Greater than input voltage with a safety margin.
Average Current Rating: Should be 1.5 times higher than the output current.

Switch (MOSFET) Selection

For the MOSFET to work efficiently, make sure to consider the following parameters:

Voltage Rating VDS of the MOSFET should be higher than the input voltage, typically 1.2-1.5 times.

Current Rating I_D of he MOSFET must be selected so that the MOSFET can handle peak current.

The R_DS(on) value must be selected to be minimum to reduce conduction losses.

By appropriately adjusting any one of the above parameters it becomes possible to tailor the output voltage from the converter. This adjustment could be implemented manually or automatically through a self adjusting PWM circuit.

Although the above formulas clearly explain how to optimize the output voltage from a buck converter, we still do not know how the inductor can be built for getting an optimal response in these circuits.

Solving an Example Buck Converter Design

Let's say we want to design a buck converter with 60V input and 12V output with 5 Amp current, let's calculate the above discussed parameters.

Given Specifications

• Input Voltage (V_in): 60V
• Output Voltage (V_out): 12V
• Output Current (I_out): 5A
• Switching Frequency (f_s): Let us assume 100 kHz (a common choice for this power range)

Duty Cycle Calculation

• Formula: Duty Cycle (D) = V_out / V_in
• Calculation: D = 12 / 60 = 0.2
• Result: Duty Cycle (D) = 20%

Inductor Selection

• Target Inductor Ripple Current (ΔIL): Assume 10% of output current, so:
• Δ_IL = 0.1 × I_out = 0.1 × 5 = 0.5 A
• Inductance Calculation:
• Formula:
• L = (V_out × (1 - D)) / (Δ_IL × f_s)
• Calculation:
• L = (12 × (1 - 0.2)) / (0.5 × 100,000) = (12 × 0.8) / 50,000 = 9.6 / 50,000 = 0.000192 H
• Result:
• Inductance (L) = 192 μH
• Choose an inductor with a value close to 192 μH, rated for at least 5A current.

Output Capacitor Selection

• Target Voltage Ripple (Δ_Vout): Let us assume 1% of output voltage, so:
• Δ_Vout = 0.01 × V_out = 0.01 × 12 = 0.12 V
• Output Capacitance Calculation:
• Formula:
• C_out = Δ_IL / (8 × f_s × Δ_Vout)
• Calculation:
• C_out = 0.5 / (8 × 100,000 × 0.12) = 0.5 / 9600 = 52.08 μF
• Result:
• Output Capacitance (Cout) = 52 μF
• Choose an output capacitor rated at least 52 μF with a voltage rating above 12V, and low ESR for handling ripple current.

Diode Selection

• Reverse Voltage: Greater than input voltage (60V), with a safety margin.
• Average Current Rating: Should be higher than the output current, so choose a diode rated for at least 5A (ideally 6A for reliability).

Switch (MOSFET) Selection

• Voltage Rating: Higher than the input voltage, typically 1.2 to 1.5 times Vin (60V), so choose a MOSFET rated for at least 80-90V.
• Current Rating: Should be able to handle peak current, so at least 8A (10A is recommended for safety).
• Low RDS(on): To reduce conduction losses and increase efficiency.

Final Results for our example 60V to 12V Buck Converter Design

• Duty Cycle: 20%
• Inductor: 192 μH
• Output Capacitor: 52 μF (voltage rating > 15V)
• Diode: Reverse voltage > 60V, current rating > 5A
• MOSFET: Voltage rating > 80V, current rating > 5A, low R_DS(on)

Using Practical Trial and Error Method

You may find many elaborate and researched formulas for settling this issue, however no new hobbyist or any electronic enthusiast would be interested to actually struggle with these complex formulas for the required values, which could actually have more possibility of providing erroneous results due to their complexities.

The better and more effective idea is to "calculate" the inductor value with an experimental set up and through some practical trial and error process as explained in the following paragraphs.

The inductor L may be initially made arbitrarily.

The rule of the thumb is to use the number of turns slightly higher than the supply voltage, therefore if the supply voltage is 12V, the number of turns could be around 15 turns.

1. It must be wound over a suitable ferrite core, that could be a ferrite ring or a ferrite rod, or over an EE core assembly.
2. The thickness of the wire is determined by the amp requirement which initially won't be a relevant parameter, therefore any relatively thin copper enameled wire would work, may be around 25 SWG.
3. Later on as per the current specs of the intended design, more number of wires could be added in parallel to the inductor while winding it in order to make it compatible with the specified ampere rating.
4. The diameter of the inductor will depend on the frequency, higher frequency would allow smaller diameters and vice versa. To be more precise, the inductance offered by the inductor becomes higher as frequency is increased, therefore this parameter will need to be confirmed through a separate test using the same IC 555 set up.

Basic Circuit Diagram Buck Converter

In-Depth Buck Converter Designing Details

In this section I have explained the various parameters required for designing a correct buck converter inductor, such that the required output is able to achieve maximum efficiency.

In our previous post I explained the basics of buck converters 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.

In this post we'll go a little deeper and try to evaluate the relationship between the input voltage, switching time of the transistor, output voltage and the current of the buck inductor, and regarding how to optimize these while designing a buck inductor.

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.

Minimum Inductor Current, (i_o) = It's the minimum amount of current that may be allowed for the inductor to reach while the inductor is discharging by releasing its stored energy in the form of back EMF.

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.

The above condition is also called the continuous mode for a buck converter design.

If the transistor does not switch ON back before the inductor current has dropped to zero, then the situation may be referred to as the discontinuous mode, which is an undesirable way to operate a buck converter and may lead to an inefficient working of the system.

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

A filter capacitor at the output of the buck converter will normally stabilize this ripple current and help to make it relatively constant.

Duty Cycle, (D = T_on /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_on = 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.

RMS value of Triangle wave 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 I have explained 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

or

i = 1/L 0ʃtVdt + 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

Here Vt is the voltage across the winding multiplied by the time for which it's sustained (in micro-secs)

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*)

Calculating Peak Current

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.

If the expression io is shifted to the LHS we get:

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

Here Vtrans refers to the voltage drop across the transistor's collector/emitter

Recall that the ripple current is also given by Δi = ipk - io, therefore substituting this in the above formula we get:

Δi = (Vin – Vtrans – Vout)Ton / L ------------------------------------- Eq#1
Now let's see the expression for acquiring the current within the inductor during the switch-OFF period of the transistor, it may be determined with the help of the following equation:

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.

In the above discussion we successfully derived the equation for determining the current (amp) factor in a buck inductor.

Determining Voltage

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

Since the Δi is common in both Eq#1 and Eq#2, we can equate the terms with each other to get:

(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
or

D =   Vout - VD / (Vin – Vtrans – VD)

Here VD refers to the voltage drop across the diode.

Calculating Step Down Voltage

If we ignore the voltage drops across the transistor and the diode (since these can be extremely trivial compared to the input voltage), we can trim down the above expression as given below:

Vout  = DVin

The above final equation can be used for calculating the step down voltage that may be intended from a particular inductor while designing a buck converter circuit.