created at December 3, 2022

# Power Transformer calculation

A power transformer takes input AC voltage U1 and current I1, and outputs another AC voltage U2 and current I2. It has at least two windings, which are inductively connected (winded around a common core). The relation between the input and output currents could be difficult in general, but in case of Power Transformer, the source voltage (and primary winding current) are sinusoidal, and the output voltage and current are sinusoidal as well. The output frequency is the the same as input.

## How Transformer works?

The AC voltage U1 is plugged to the primary winding. The current I1 flows there and generates a magnetic flux inside the core, which is also sinusoidal. The flux is inducing the Voltage in the secondary winding coils, since it is winded on the same core. Then a consumer with resistance Rload is connected to the secondary winding. The electromotive force, induced in the secondary winding coils, becomes a voltage source for the consumer. Current I2 flows in the secondary circuit, and it also produces its magnetic flux, which increases the total magnetic flux in the core. As a result, the primary current I1 increases.

## Transformation coefficient

Since the single coil induced Voltage is the same for every coil, the number of coils ratio for both windings defines the Voltage ratio. We consider the ideal Transformer, without any looses, therefore the input power (consumed from the input Voltage source) equals to the output power (taken by consumer). Then the transformation coefficient:

$n = \frac{W_2}{W_1} = \frac{U_2}{U_1} = \frac{I_1}{I_2}$

## Transformer simple mathematical model

Consider the general case transformer, when we don't know the purposes it will be used for. Consider the given input voltage, frequency, and desired output voltage.

Consider the core, made of ferromagnetic material, having a rectangle a x b cross-section. The winding span length is lw. Consider the number of coils for the primary winding and the wire diameters for both windings, dw1 and dw2 accordingly.

Core cross-section perimeter is 2(a + b), primary winding total length: l1 = W1·2(a + b). Wire cross-section area is S1 = π·dw12/4. Then the primary winding active resistance is:

$R_1 = \rho \frac{l_1}{S_1} = \rho \frac{W_1\cdot 8\cdot (a + b)}{\pi \cdot dw_1^2}$

The same is actual for the secondary winding

Primary winding inductance:

$L_1 = \frac{\mu_0\cdot \mu\cdot W_1^2\cdot A}{lw}$

where A = a·b (core cross-section Area)

Primary winding inductive resistance:

$XL_1 = \omega\cdot L_1 = 2\cdot \pi \cdot f \cdot L_1$

Primary winding total resistance (impedance):

$Z_1 = \sqrt{R_1^2 + XL_1^2}$

The open circuit primary winding current (when no load is connected to the secondary winding):

$I_{11} = \frac{U_1}{Z_1}$

Secondary winding voltage:

$U_2 = n\cdot U_1$

Then after the consumer is connected to the secondary winding, the secondary current:

$I_2 = \frac{U_2}{(R_2 + R_{load})}$

That current is contributing the total magnetic flux in the core, thus the primary current increases as well in order to sustain the balance between the source voltage and the induced electromotive force. Then resulting current in the primary winding is:

$I_1 = I_{11} + I_2\cdot n$

The power consumed from electrical network:

$P_1 = U_1\cdot I_1$

The power taken by consumer:

$P_2 = U_2\cdot I_2$

The model described above is implemented in the document "power_transformer". The are also the variables named with postfix "margin", which are intended to show the margin for such parameters as winding wire diameters, core size, number of coils. The margins are obtained by comparing the actual values to those ones obtained from the empirical formulas, which are described below. If the margin happens to be "too negative", one should increase the according parameter value.

## What's next?

That model could be used for preliminary calculation of already given transformer, but what if we have to design the transformer for very specific needs (for the given consumer)? Let's take a look at the very common transformer calculation method

## Power Transformer Calculation

A set of empirical formulas is used for that purpose. Let's find out first what output current and voltage are required by the consumer. Then knowing the power P2 will take into account the real transformer internal looses with some coefficient:

$P_1 = 1.2\cdot P_2$

The core cross-section area (in square centimeters) can be evaluated as:

$A = 1.2\cdot \sqrt{P_1}$

The number of coils per single Volt:

$Nv = \frac{55}{A}$

Knowing this, we can now obtain the number of coils for primary and secondary windings:

$W_1 = Nv\cdot U_1$
$W_2 = Nv\cdot U_2 = Nv\cdot U_1\cdot n$

Primary and secondary winding currents:

$I_1 = \frac{P_1}{U_1};\quad I_2 = \frac{P_2}{U_2}$

The wire diameter for every winding is evaluated according to:

$D = 0.8\sqrt{I}$