Mastering Transformer Basics: Ideal Transformers & The EMF Equation

Transformer Basics: Ideal Transformers & The EMF Equation

Demonstrative Video

Ideal Transformer

An ideal transformer is one which has no losses, i.e. its windings have no ohmic resistance, there is no magnetic leakage, and hence has no \(I^2R\) and core losses
In other words, an ideal transformer consists of two purely inductive coils wound on a loss-free core
It is impossible to realize such a transformer in practice
Consider an ideal transformer whose secondary is open and primary is connected to sinusoidal alternating voltage \(V_1\)
Potential difference cause an alternating current to flow in the primary
Since, the primary coil is purely inductive and the secondary is open (no output) the primary draws a magnetizing current \(I_\mu\) only
\(I_\mu\) is very small in magnitude, magnetize the core, and lags \(V_1\) by \(90^0\)
The alternating current \(I_\mu\) produces an alternating flux \(\Phi\), which at all times is proportional to the current and is in phase with it.
This changing flux is linked with both \(N_1\) and \(N_2\)
Therefore, it produced self induced emf in the primary (\(E_1\)) which at every instant oppose \(V_1\) known as counter emf or back emf of the primary
Similarly, there is secondary mutually induced emf (\(E_2\)), antiphase with \(V_1\)
The magnitude of \(E_2\) is proportional to the rate of change of flux and the number of secondary turns

EMF Equation of a transformer

Let,

\[ \begin{aligned} N_1 & = \text{No. of turns in primary}\\ N_2 & = \text{No. of turns in secondary}\\ \Phi_m & = B_m \times A ~\text{Maximum flux in core in webers}\\ f & = \text{Frequency of a.c. input in Hz} \end{aligned} \]

Flux increases from zero to maximum value \(\Phi_{m}\) in one quarter of the cycle i.e. \(1/4~f\) s

\(\therefore\) Average rate of change of flux	\(=\dfrac{\Phi_{m}}{1/4f}\)
	\(=4f\Phi_{m}\) volt

Rate of change of flux per turn means induced emf in volts

\(\therefore\) Average emf/turn

\(=4f\Phi_{m}\) volt

If flux varies sinusoidally, then rms value of induced emf is obtained by multiplying the average value with form factor

Form factor =

\(\dfrac{\mbox{r.m.s value}}{\mbox{average value}}\)=

1.11

\(\therefore\) r.m.s value of e.m.f/turn =

\(1.11 \times 4f\Phi_{m} = 4.44f\Phi_{m} volt\)

Now, r.m.s value of the induced e.m.f in the whole primary winding

\[E_1 = 4.44fN_1\Phi_m = 4.44fN_1B_mA\]

Similarly, for secondary winding

\[E_2 = 4.44fN_2\Phi_m = 4.44fN_2B_mA\]

From above two equations

\[E_1/N_1 = E_2/N_2 = 4.44f_m\]

emf/turn is same for both the primary and secondary winding

Voltage Transformation Ratio

\[\begin{aligned} V_{1}I_{1} & =V_{2}I_{2}\\ \Rightarrow\dfrac{I_{2}}{I_{1}} & =\dfrac{V_{1}}{V_{2}}=\dfrac{1}{K} \end{aligned}\]

For an ideal transformer, input VA = output VA