﻿ Study and Theory about Ideal Transformer

To begin our study of transformers, let's define what an ideal transformer is.

Ideal Transformer - is the transformer where there is no accumulated energy in the magnetic field (losses in iron is zero), there is no inductance, the wire has no resistance (losses in copper is zero) and the coupling coefficient between coils is unitary.

In the ideal transformer the input power is equal to output power.

The winding where we apply the voltage source we call PRIMARY and the winding where we connect the load we call SECONDARY.

We usually represent the number of turns of the primary by N1 and the number of secondary turns by N2, as can be seen in Figure 91-01, where we represent a transformer in its didactic format.

Note that in the construction of the transformer there is no direct contact between the primary and the secondary. They are completely independent circuits. There must also be no electrical contact between the wires that make up the windings and the iron core. They must be isolated from each other. For the transformer to work, we must apply a sinusoidal or cosine voltage to the primary, V1, (attention: transformers do not work with direct current, or DC) and this voltage will generate a variable flux in iron core. This variable flux, which runs through the entire iron core, will induce a voltage, V2, in the secondary winding (Faraday's law See here!). Knowing the value of V1, the voltage V2 in the secondary depends only on the so-called ratio of transformation given by the ratio between the number of turns that make up the primary and secondary, according to eq. 91-01, below. So if N1 > N2 we say that the transformer is a voltage reducer. Otherwise, that is, if we have N1 < N2, then we say that the transformer is an elevator of voltage.

2. Transformer Mathematical Relations

In the Figure 91-02 we present the schematic of a transformer.

Where the variables are:

• V1 - Electrical voltage applied to transformer primary
• V2 - Electrical voltage removed from the transformer secondary
• I1 - Electrical current in transformer primary
• I2 - Electrical current in transformer secondary
• N1 - Number of turns of transformer primary
• N2 - Number of turns of transformer secundary
• ZL - Load connected to secondary
• P1 - Power delivered to transformer primary
• P2 - Power delivered to the load

An important parameter of the transformer is the so-called transformation relation, which is defined as the relation between N1 and N2 and represented by the letter a, given by:

eq.  91-01

On the other hand, there is a direct relation between transformer voltages and transformation ratio, given by:

eq.  91-02

And as said before, the input and output powers in the ideal transformer are equal, that is   P1 = P2 = V1 I1 = V2 I2.   So this lets you write that:

eq.  91-03

These mathematical relationships are of fundamental importance for understanding the operation of a transformer.

3. Impedance Reflection

In the study of transformers we can work with the so-called impedance reflection. That is, we can reflect the impedance from primary to secondary and vice versa. It depends on the convenience of one or the other. Let's look at how these reflections are made.

3.1. Secondary to Primary Reflection

The secondary impedance can be calculated as the ratio of secondary voltage and current. Referring to the circuit above, we can write that:

Zs = V2 / I2 = ZL

But by eq. 91-03 we know that V1 = a V2  and  I1 = I2 / a. Thus, calculating the impedance that the circuit offers to the primary, we find:

Zp = V1 / I1 = a V2 / (I2 / a) = a2 ZL

That is, when we reflect the secondary to primary impedance, we must multiply the secondary impedance by the square of the transform relationship. In short:

eq.  91-04

3.2. Reflection from Primary to Secondary

Just as we reflect the impedance from secondary to primary, we can reflect that from primary to secondary. From eq. 91-03 we concluded that I2 = a I1 and also V2 = V1 / a. So by calculating the impedance that the circuit offers to the secondary, we find:

Zs = V2 / I2 = (V1 / a) / a I1 = V1 / a2 I1 = Zp / a2

We conclude that when we reflect the primary to secondary impedance, we must divide the primary impedance by the square of the transform relation, a. In short:

eq.  91-05