Induced EMF in a transformer

In a transformer a sinusoidally varying voltage is applied to the primary coil of the transformer which give rise to a time varying flux which induce another sinusoidally varying EMF in the primary coil. The induced EMF has a 180° phase difference with the applied voltage and hence opposes it due to which the voltage across the coil should decrease leading to a decrease in the max value of the time varying flux. So the decrease in flux should then also decrease the max value of induced EMF.

How can the EMF across the primary then continue to be sinusoidal when the max value of sinusoidally varying out of phase induced EMF change with time? Shouldn't it be represented by a sinusoidal curve whose max value change with time?

• it's not 180 phase difference. The induced voltage is $\dfrac{d\Psi}{dt}$ if $\Psi=\Psi_{max}cos(\omega t)$ then $u=U_{max}sin(\omega t)$ in both windings, primary and secondary. Commented Jun 21, 2016 at 8:18
• the flux is in phase with the current which lags the applied voltage by 90 degree(if we assume the primary winding is purely inductive). And as you have proved above the induced emf further lags the flux by 90 degree . So there is a net 90+90=180 degree phase difference between the applied and induced emf. Commented Jun 21, 2016 at 8:42
• No, it doesn't add 90+90. Both voltages: secondary and primary are in phase. Both voltages lags the flux by 90deg. You sholud twist the logics: the voltages are induced due to the alternating flux, of course the flux is due to the applied voltage on the primary, but if you want to understand the transformers basics you should skip that fact. Not an easy machine this transformer, right? Commented Jun 21, 2016 at 9:08