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 N_{1} and the number of secondary turns by N_{2}, 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, V_{1}, (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, V_{2}, in the secondary winding (Faraday's lawSee here!). Knowing the value of V_{1}, the voltage V_{2} 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 N_{1} > N_{2} we say that the transformer is a
voltage reducer. Otherwise, that is, if we have N_{1} < N_{2}, 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:
V_{1} - Electrical voltage applied to transformer primary
V_{2} - Electrical voltage removed from the transformer secondary
I_{1} - Electrical current in transformer primary
I_{2} - Electrical current in transformer secondary
N_{1} - Number of turns of transformer primary
N_{2} - Number of turns of transformer secundary
Z_{L} - Load connected to secondary
P_{1} - Power delivered to transformer primary
P_{2} - Power delivered to the load
An important parameter of the transformer is the so-called transformation relation, which is defined as the relation between N_{1} and N_{2} 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
P_{1} = P_{2} = V_{1} I_{1} = V_{2} I_{2}.
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:
Z_{s} = V_{2} / I_{2} = Z_{L}
But by eq. 91-03 we know that V_{1} = a V_{2} and
I_{1} = I_{2} / a. Thus, calculating the impedance that the circuit offers to the primary, we find:
Z_{p} = V_{1} / I_{1} = a V_{2} / (I_{2} / a) = a^{2} Z_{L}
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 I_{2} = a I_{1} and also V_{2} = V_{1} / a. So by calculating the impedance that the
circuit offers to the secondary, we find:
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: