When does the transformer equation hold?

Question

The "transformer equation" is as follows, $$\frac{N_s}{N_p}=\frac{V_s}{V_p}$$ What is assumed when this equation is used?

(Primary coil on the left, secondary on the right, in both cases)

Is transformer 1 an ideal transformer, or transformer 2 (I have seen both in books and I'm not sure whether in the first picture, the other field lines are just not shown because they are irrelevant). It would make sense that transformer 1 is ideal and is 100% efficient because no work is done by the magnetic field in the air, but wouldn't a magnetic field through "ideal air" do no work regardless? How does this relate to flux linkage? Surely the flux linkage in both coils must be the same for the equation to hold, so in transformer 2, conceptually, less magnetic field lines go through the secondary coil than the primary coil, therefore the equation would not hold.

Answer 1

The most basic and important assumption required for the equation to hold is that all of the flux generated in one coil (primary or secondary) passes through the other coil. For this assumption to be true, generally a magnetic core is needed to trap and guide the flux from one coil to the other. Without the core, much of the magnetic flux generated by one coil does not pass through the other coil.

After that, there are many issues that lead to deviation from the ideal behavior, such as various forms of losses.

Answer 2

The transformer equation is for an ideal transformer. A real transformer behaves like an ideal transformer with added components as shown in the equivalent circuit shown below. The added components account for the flux linkage between the primary and secondary being less than 100% as depicted in your transformer 2. It also accounts for the winding resistance and core losses.

It is the winding resistance and core losses that make the transformer less than 100% efficient, not the incomplete flux linkage.

Incomplete flux linkage results in leakage reactance, Xp and Xs in the equivalent circuit. There is no energy lost in those components, but there is a voltage drop across them that is proportional to the current. That means that the expected secondary voltage from the transformer equation could be reduced significantly as the load current increases. Some leakage reactance is unavoidable, but good designs minimize that to the extent required for the type of use intended.

For the equivalent circuit presented below, the transformer equation holds for the ideal transformer part of the circuit: $$\frac{N_s}{N_p}=\frac{E_s}{E_p}$$

For additional information about the above figure, see my answer to: Equivalent ciruit parameters of a 1kVA transformer.

Answer 3

What is assumed when this equation is used?

1. All the magnetic flux produced by the primary couples to all other primary turns i.e. there is no leakage flux that can give rise to leakage inductance that gives rise to formula errors when there is a load or magnetization inductance (which there always will be).
2. All the flux generated by the primary couples to all the turns on the secondary - this ensures that the voltage in the secondary is truly dependent on the turns ratio and the voltage ratio

However, meeting the above (and satisfying the equation) can be done theoretically with what we sometimes refer to as a non-ideal transformer. But only if that non-ideal transformer has zero leakage components (see below for the listings in the equivalent circuit). It can have magnetization current and, it can have core losses and the basic equation still works - this is because core losses and magnetization current are parallel to the ideal "inner transformer".

Transformer equivalent circuit showing leakages and magnetization inductance and losses: -

Picture from here

• \$L_P\$ is the primary leakage inductance

• \$R_P\$ is the primary copper loss

• \$R_C\$ is the core losses due to eddy currents and hysteresis

• \$L_M\$ is the magnetization inductance

• \$L_S\$ is the secondary leakage inductance

• \$R_S\$ is the secondary copper loss

If there is no loading on the secondary, \$R_S\$ and \$L_S\$ won't affect the output voltage. If the magnetization inductance and core losses (\$L_M\$ and \$R_C\$) are both very high impedances, then some leakage components on the primary (\$L_P\$ and \$R_P\$) can be tolerated without much detriment to the equation in question (providing there is still no load on the secondary).

Is transformer 1 an ideal transformer, or transformer 2

T1 might be an ideal transformer, T2 is not an ideal transformer

so in transformer 2, conceptually, less magnetic field lines go through the secondary coil than the primary coil, therefore the equation would not hold.

Correct.