The post presents a detailed explanation regarding how to calculate and design customized ferrite core transformers. The content is easy to understand, and can be very handy for engineers engaged in the field of power electronics, and manufacturing SMPS inverters.

## Why Ferrite Core is used in High Frequency Converters

You might have often wondered the reason behind using ferrite cores in all modern switch mode power supplies or SMPS converters. Right, it is to achieve higher efficiency and compactness compared to iron core power supplies, but it would be interesting to know how ferrite cores allow us to achieve this high degree of efficiency and compactness?

It is because in iron core transformers, the iron material has much inferior magnetic permeability than ferrite material. In contrast, ferrite cores possess very high magnetic permeability.

Meaning, when subjected to a magnetic field, ferrite material is able to achieve a very high degree of magnetization, better than all other forms of magnetic material.

A higher magnetic permeability means, lower amount of eddy current and lower switching losses. A magnetic material normally has a tendency to generate eddy current in response to a rising magnetic frequency.

As the frequency is increased, eddy current also increases causing heating of the material and increase in coil impedance, which leads to further switching losses.

Ferrite cores, due to to their high magnetic permeability are able to work more efficiently with higher frequencies, due to lower eddy currents and lower switching losses.

Now you may think, why not use lower frequency as that would conversely help to reduce eddy currents? It appears valid, however, lower frequency would also mean increasing the number of turns for the same transformer.

Since higher frequencies allow proportionately lower number of turns, results in transformer being smaller, lighter and cheaper. This is why SMPS uses a high frequency.

### Inverter Topology

In switch mode inverters, normally two types of topology exits: push-pull, and Full bridge. The push pull employs a center tap for the primary winding, while the full bridge consists a single winding for both primary and secondary.

Actually, both the topology are push-pull in nature. In both the forms the winding is applied with a continuously switching reverse-forward alternating current by the MOSFETs, oscillating at the specified high frequency, imitating a push-pull action.

The only fundamental difference between the two is, the primary side of the center tap transformer has 2 times more number of turns than the Full bridge transformer.

## How to Calculate Ferrite Core Inverter Transformer

Calculating a ferrite core transformer is actually quite simple, if you have all the specified parameters in hand.

For simplicity, we'll try to solve the formula through an example set up, let's say for a 250 watt transformer.

The power source will be a 12 V battery. The frequency for switching the transformer will be 50 kHz, a typical figure in most SMPS inverters. We'll assume the output to be 310 V, which is normally the peak value of a 220V RMS.

Here,the 310 V will be after rectification through a fast recovery bridge rectifier, and LC filters. We select the core as ETD39.

As we all know, when a 12 V battery is used, it's voltage is never constant. At full charge the value is around 13 V, which keeps dropping as the inverter load consumes power, until finally the battery discharges to its lowest limit, which is typically 10.5 V. So for our calculations we will consider 10.5 V as the supply value for *V*_{in(min)} .

## Primary Turns

The standard formula for calculating the primary number of turns is given below:

*N*_{(prim)} = *V*_{in(nom)} x 10^{8} / 4 x *f* x *B*_{max} x *A*_{c}

Here *N*_{(prim)} refers to the primary turn numbers. Since we have selected a center tap push pull topology in our example, the result obtained will be one-half of the total number of turns required.

*Vin*_{(nom) }= Average Input Voltage. Since our average battery voltage is 12V, let's, take*Vin*_{(nom)}= 12.*f*= 50 kHz, or 50,000 Hz. It is the preferred switching frequency, as selected by us.*B*_{max}= Maximum flux density in Gauss. In this example, we'll assume*B*_{max}to be in the range of 1300G to 2000G. This is the standard value most ferrite based transformer cores. In this example, let’s settle at 1500G. So we have*B*_{max}= 1500. Higher values of*B*_{max}is not recommended as this may result in the transformer reaching saturation point. Conversely, lower values of*B*_{max}may result in the core being underutilized.- A
_{c}= Effective Cross-Sectional Area in cm^{2}. This information can be collected from the datasheets of the ferrite cores. You may also find A_{c}being presented as A_{e}. For the selected core number ETD39, the effective cross-sectional area furnished in the datasheet sheet is 125mm^{2}. That is equal to 1.25cm^{2}. Therefore we have, A_{c}= 1.25 for ETD39.

The above figures give us the values for all the parameters required for calcuating the primary turns of our SMPS inverter transformer. Therefore, substituting the respective values in the above formula, we get:

*N*_{(prim)} = *V*_{in(nom)} x 10^{8} / 4 x *f* x *B*_{max} x *A*_{c}

*N*_{(prim)} = 12 x 10^{8} / 4 x 50000 x 1500 x 1.2

*N*_{(prim)} = 3.2

Since 3.2 is a fractional value and can be difficult to implement practically, we'll round it off to 3 turns. However, before finalizing this value, we have to investigate whether or not the value of *B*_{max} is still compatible and within the acceptable range for this new rounded off value 3.

Because, decreasing the number of turns will cause a proportionate increase in the *B*_{max}, therefore it becomes imperative to check if the increased *B*_{max} is still within acceptable range for our 3 primary turns.

Counter checking *B*_{max} by substituting the following existing values we get:*Vin*_{(nom)} = 12, *f* = 50000, *N*_{pri} = 3, *A*_{c} = 1.25

*B*_{max} = *V*_{in(nom)} x 10^{8} / 4 x *f* x *N*_{(prim)} x *A*_{c}

*B*_{max} = 12 x 10^{8} / 4 x 50000 x 3 x 1.25

*B*_{max} = 1600

As can be seen the new *B*_{max} value for *N*_{(pri)} = 3 turns looks fine and is well within the acceptable range. This also implies that, if anytime you feel like manipulating the number of *N*_{(prim)} turns, you must make sure it complies with the corresponding new *B*_{max} value.

Oppositely, it may be possible to first determine the *B*_{max} for a desired number of primary turns and then adjust the number of turns to this value by suitably modifying the other variables in the formula.

## Secondary Turns

Now we know how to calculate the primary side of an ferrite SMPS inverter transformer, it's time to look into the other side, that is the secondary of the transformer.

Since the peak value has to be 310 V for the secondary, we would want the value to sustain for the entire battery voltage range starting from 13 V to 10.5 V.

No doubt we will have to employ a feedback system for maintaining a constant output voltage level, for countering low battery voltage or rising load current variations.

But for this there has to be some upper margin or headroom for facilitating this automatic control. A +20 V margin looks good enough, therefore we select the maximum output peak voltage as 310 + 20 = 330 V.

This also means that the transformer must be designed to output 310 V at the lowest 10.5 battery voltage.

For feedback control we normally employ a self adjusting PWM circuit, which widens the pulse width during low battery or high load, and narrows it proportionately during no load or optimal battery conditions.

This means, at low battery conditions the PWM must auto adjust to maximum duty cycle, for maintaining the stipulated 310 V output. This maximum PWM can be assumed to be 98% of the total duty cycle.

The 2% gap is left for the dead time. Dead time is the zero voltage gap between each half cycle frequency, during which the MOSFETs or the specific power devices remain completely shut off. This ensures guaranteed safety and prevents shoot through across the MOSFETs during the transition periods of the push pull cycles.

Hence, input supply will be minimum when the battery voltage reaches at its minimum level, that is when *V*_{in} = *V*_{in(min)} = 10.5 V. This will prompt the duty cycle to be at its maximum 98%.

The above data can be used for calculating the average voltage (DC RMS) required for the primary side of the transformer to generate 310 V at the secondary, when battery is at the minimum 10.5 V. For this we multiply 98% with 10.5, as shown below:

0.98 x 10.5 V = 10.29 V, this the voltage rating our transformer primary is supposed to have.

Now, we know the maximum secondary voltage which is 330 V, and we also know the primary voltage which is 10.29 V. This allows us to get the ratio of the two sides as: 330 : 10.29 = 32.1.

Since the ratio of the voltage ratings is 32.1, the turn ratio should be also in the same format.

Meaning, x : 3 = 32.1, where x = secondary turns, 3 = primary turns.

Solving this we can quickly get the secondary number of turns

Therefore secondary turns is = 96.3.

The figure 96.3 is the number of secondary turns that we need for the proposed ferrite inverter transformer that we are designing. As stated earlier since fractional vales are difficult to implement practically, we round it off to 96 turns.

This concludes our calculations and I hope all the readers here must have realized how to simply calculate a ferrite transformer for a specific SMPS inverter circuit.

### Calculating Auxiliary Winding

An auxiliary winding is a supplemental winding that a user may require for some external implementation.

Let's say, along with the 330 V at the secondary, you need another winding for getting 33 V for an LED lamp. We first calculate the **secondary : auxiliary** turn ratio with respect to the secondary winding 310 V rating. The formula is:

N_{A} = V_{sec} / (V_{aux} + V_{d})

N_{A} = secondary : auxiliary ratio, V_{sec} = Secondary regulated rectified voltage, V_{aux} = auxiliary voltage, V_{d} = Diode forward drop value for the rectifier diode. Since we need a high speed diode here we will use a schottky rectifier with a V_{d} = 0.5V

Solving it gives us:

N_{A} = 310 / (33 + 0.5) = 9.25, let's round it off to 9.

Now let's derive the number of turns required for the auxiliary winding, we get this by applying the formula:

N_{aux} = N_{sec} / N_{A}

Where N_{aux} = auxiliary turns, N_{sec} = secondary turns, N_{A} = auxiliary ratio.

From our previous results we have N_{sec} = 96, and N_{A} = 9, substituting these in the above formula we get:

N_{aux} = 96 / 9 = 10.66, round it off gives us 11 turns. So for getting 33 V we will need 11 turns on the secondary side.

So in this way you can dimension an auxiliary winding as per your own preference.

**Wrapping up**

In this post we learned how to calculate and design ferrite core based inverter transformers, using the following steps:

- Calculate primary turns
- Calculate secondary turns
- Determine and Confirm
*B*_{max} - Determine the maximum secondary voltage for PWM feedback control
- Find primary secondary turn ratio
- Calculate secondary number of turns
- Calculate auxiliary winding turns

Using the above mentioned formulas and calculations an interested user can easily design a customized ferrite core based inverter for SMPS application.

For questions and doubts please feel free to use the comment box below, I'll try to solve at an earliest

Ferrite cores are used NOT because of their high permeability. Actually, ironcores, and especially permalloy cores, etc could showcase permeability exceeding that of typical transformer ferrite cores.

For the reference, effective permeability of typical transformer core ferrites used in SMPS would usually be about 1000 to 2500 or so, depending on application. Gapped cores, like those in flybacks would have effective permeability much lower than that (though it “coupled inductor” rather than “transformer”). However, transformer-grade steel alloys and especially permalloy-like “low-frequency” cores can showcase permeability well around 20 000, more than 10x higher compared to typical transformer ferrites. Some “special” ferrites also expose permeability about 15 000 or more, but are NOT used for transformers, being high-loss things, so only suitable for ferrites. Who needs high-loss transformer? It would just heat itself up, wasting power on that. Avoiding this scenario is pretty much goal of transformer calculations, especially when it comes to ferrite type selection, etc.

But what makes ferrite transformers small? And why we need ferrite? Its mostly about high frequency operation and eddy currents.

Generally, the higher operating frequency is, the more power could be pushed through same transformer size. So if we want small but powerful supply, we have to go for high frequency. At high frequency one can make far less turns in windings, yet they still would have enough inductance to “resist” incoming voltage. The higher frequency, the less turns we need, the smaller transformer we can afford. To certain extent.

What’s the problem with steel, permalloy, etc? Being metallic they are inherently conductive. At which point, core could act a bit like some kind of (shori-circuited) winding, heating itself by eddie currents. This effect could be put in use, giving a rise to “induction heating” – but we do not want it in transformers. So transformer steels are optimized to have relatively high resistance and also split into separate thin plates to reduce eddie currents. However, the higher frequency is, the more pronounced this “induction heating” gets – and even special steel alloys and very thin plates aren’t enough to prevent core heating. That’s where ferrites come into play. They are ceramic materials, so unlike metals they show rather poor electric conductivity. At which point eddie current loses are mostly mitigated, so we can increase operational frequency a lot compared to steel designs. This implies far smaller windings, so we can push way more power without increasing transformer size. So SMPS would have unusually small ferrite transformer for given power, compared to low-frequency steel transformer designs.

As for topologies, there’re a bit more options. Out of bridge-like, there’re three: half-bridge, full-bridge and push-pull. Half-bridge and push-pull are similar in overall idea, but implementation is different, half-bridge uses just 1 winding and some capacitors to expose winding to both polarities. Push-pull rather uses different halves of winding to achieve same result (in terms of magnetic field behavior). This brings different tradeoffs, and full bridge lacks most shortcomings of both – but being more expensive due to 4 high-voltage transistors reqirement.

Some relatively recent power supply designs could also use e.g. “2-switch forward” approach, which also uses 2 MOSFETs but operates in somewhat different manner. It underuses transformer core, magnetizing it in just 1 polarity, but on other hand it gets favorable MOSFETs voltage and current tradeoffs, pretty much like full bridge, being cheaper solution. So it fairly usual to see 2-switch forward in e.g. ATX power supplies up to about 800-1000W or so. Transformer computed much like “usually”, but with some special considerations in regard of saturation & demagnetizing attached.

There’re also flybacks, that aren’t even transformers by their actions, though turns ratio still haves some meaning. These have their own limitations but are cheapest option up to around ~100W of output power. Above this point different topology could be better consideration though.

Thank you for the nice explanation!

> being high-loss things, so only suitable for ferrites.

Oh, I meant high-loss high-permeability ferrites are used in EMI filters, etc, where turning noise into heat is rather something good.

Also it worth to mention there is approximately the following trend: the higher intended opreational frequency of ferrite compound is, the less permeability it would get, the higher resistivity it would enjoy, it would have lower core losses but require more turns.

As example, think of N49 vs N67, for example. N49 is tailored up to about ~1MHz or even more, while N67 is barely ok up to maybe 200kHz at very most. N67 would have higher permeability, but would get far higher core losses at high frequencies. Overall it comes to trading copper loses vs core loses, in perfect world we would try to hit minimum of their sum, however numerous secondary considerations make it challenging.

How do we determine the swg of the coil used

SWG is related to current and can be found by estimating how much current a copper wire with particular thickness can carry. Probably this article can help:

https://www.homemade-circuits.com/56492-2/

Explained in simple and lucid manner, thank you, sir.

My pleasure Ajay, Glad you liked it!