When it comes to transformer design, there is an equation most of us are familiar with. This transformer equation is known as the “transformer design equation”.
This transformer equation is applicable for inputs that are of most any voltage. The usual voltage forms are sinusoids as seen in lower frequency silicon steel type transformers such as UL Class 2 recognized power transformers.
It is also useful for higher frequency switching transformers such as push-pull transformers or flyback transformers.
It is these 3 types of transformers that I will derive this equation for. The transformer design equation only differs between the 3 due to what is known as a “voltage factor”. This equation can be found online in many places but what I’ve yet to see anywhere in the derivation of this voltage factor for these different types of applied voltage input waveforms. We will derive these 3 factors after we do the transformer equation itself first.
We will then show how the transformer design equation can easily be manipulated so as to be applied to inductors.
The Transformer Design Equation:
Let’s start with Faraday’s law of induction…
This law states that an induced voltage across some coil is directly proportional to the turns of that coil and the negative time rate of change in flux through the surface enclosed by the loop.
Written as an equation (ignoring the sign) it is:
The flux density is simply this flux per unit area, so we can re-write the above equation as:
This is the equation we will work with for the different transformers already introduced. In the above transformer design equation, the derivative of the flux density with respect to time is an equation that is a function of time that represents the slope of that flux density versus time… this will become important soon.
Let’s start with a sinusoidal input as in used with our UL Class 2 recognized power transformers.
Since the voltage is sinusoidal, so is the current and so is the flux in the core. Let’s assume the flux density is of the form:
Now, we don’t need this as a function of time so let’s drop the cosine part; this only acts to bring the value between plus and minus the MAX over time. This just leaves:
Substituting back into our transformer equation:
Now the voltage above is a MAX or peak voltage because the flux density is a MAX or peak value… since this is a sinusoidal voltage let’s make the voltage be in rms…
This equation relates the applied volt-sec’s to the turns, core area, and flux density in that core. Often what is of importance in determining the turns for a given input voltage, time period (reciprocal of the frequency), core area, and a desired operating flux density. Let’s arrange it for turns…
This is the equation you can find online in many places. What we’ve done here is derived it along with the correct voltage factor.
The derived equation we just did was for sinusoidal inputs, now let’s look at the case of a PUSH-PULL switching transformer which has a square wave input (we will assume a 50% duty cycle, it could be less but this is usually the MAX duty cycle). It is continually switched between a positive DC voltage value and a negative DC voltage value, and it makes this transition every half period.
Let’s go back to this equation below…
Now since the rate of change in flux density with respect to time cannot be defined as a function to take the derivative of, we will instead change the differentials above to changes in values…
This is perfectly fine to do because the change in flux with time must be triangular. This is because the derivative of a triangle wave is a square wave and the voltage we know is a square wave…
Now it should also be clear that the flux density in the core changes from some positive MAX value to some negative MIN value in just half a period…put another way, the instant before a voltage transition from positive to negative the flux is at a positive MAX, it then goes from that to a negative MIN in half a period of time. The value of the negative MIN is the same as the positive MAX but only the opposite, or negative, in sign. So…
And here’s our equation for a push-pull switching transformer that sees a square wave such that the flux in the core goes from some positive MAX value to some negative MIN value…
Onto our last example which would be for a flyback transformer…
This type of transformer sees only a positive voltage for some amount of time then no voltage as the flux is reduced back toward zero through some means…
Let’s go back to this transformer equation below…
We can again just use this form below because the rate of change in flux with time must be a straight line with non-zero slope due to the applied voltage being a constant DC value.
The time this voltage is applied would be the MAX duty cycle multiplied by the period.
This final equation, as can be seen, is a function of the duty cycle. The less time the voltage is applied the fewer the needed turns, which makes sense…
Ok, so we’ve derived 3 different equations for the MIN number of turns needed for transformers that take in sinusoidal inputs like our UL Class 2 recognized power transformers, those that take in square wave voltage inputs like our push-pull switching transformers and a transformer design equation that takes in an offset square wave input whose applied polarity is always the same, such as for our switching flyback transformers.
In summary:
Finally, I said we’d apply the transformer design equation to be used with inductors. This we can do by knowing the following equation for the voltage across an inductor:
And we already derived/showed this transformer equation below.
Setting them equal we get:
Integrating after getting rid of the time differential from each side yields:
To avoid saturating or surpassing the max flux density one needs to choose turns that are a direct function of the desired inductance and current through this inductance, and then inversely with the core area and the max flux density.