Why is an ideal transformer's primary voltage in phase with a series resistor for a resistive load?

Question

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1. What the question is really asking

A student has drawn the following circuit

      Vs ──┬── Rs ──(primary winding)───┬───   (ideal transformer)
           │                         │
           └─────────────────────────┘

The primary winding is part of an ideal transformer (no winding resistance, perfect magnetic coupling).
The secondary is terminated by a purely resistive load (R_{\text L}).
Because the resistor (R_s) is in series with the primary, the student observes that the voltage across the resistor, (V_R), is in phase with the primary current (I_1).
The textbook says the primary‑winding voltage (V_1) is also in phase with (I_1), so that the source voltage is just the scalar sum

[ V_s = V_R + V_1 . ]

The student then wonders:

A lone inductor always makes its voltage lead the current by (90^{\circ}) ((v = L\,\frac{di}{dt})).
Why doesn’t the primary winding of the transformer behave the same way?

In other words: Why isn’t the voltage across the primary winding 90° out of phase with the current, even though the winding is an inductor?

2. Full step‑by‑step analysis

2.1 Basic equations of an ideal transformer

For an ideal transformer with (N_1) primary turns and (N_2) secondary turns

Quantity	Expression
Flux linkage (\Phi(t)) (same for both windings)	—
Primary induced emf	(e_1(t)= N_1\frac{d\Phi}{dt})
Secondary induced emf	(e_2(t)= N_2\frac{d\Phi}{dt})
Voltage–turns ratio (holds for the induced emfs)	(\displaystyle \frac{e_1}{e_2}= \frac{N_1}{N_2})
Magnetising (no‑load) current	(I_m = \frac{e_1}{j\omega L_m}) (lags (e_1) by (90^{\circ}))
Reflected secondary current	(I_{\text{ref}} = \frac{N_2}{N_1}\,I_2) (in phase with the secondary voltage)

(L_m) is the magnetising inductance of the core.
The total primary current is the algebraic sum

[ I_1 = I_m + I_{\text{ref}} . \tag{1} ]

2.2 What happens when the secondary is a resistive load

If the secondary is terminated by a resistor (R_L),

[ I_2 = \frac{e_2}{R_L}, \qquad e_2 = \frac{N_2}{N_1} e_1 . ]

Hence the reflected current is

[ I_{\text{ref}} = \frac{N_2}{N_1} I_2 = \frac{N_2}{N_1}\,\frac{e_2}{R_L} = \frac{N_2^2}{N_1^2}\,\frac{e_1}{R_L}. \tag{2} ]

Notice that (I_{\text{ref}}) is **in phase with the primary emf (e_1)** because the secondary load is purely resistive.

2.3 Voltage across the series resistor

The series resistor sees the total primary current (I_1).
Its voltage is therefore

[ V_R = I_1 R_s, \tag{3} ]

which is exactly in phase with the current (I_1).

2.4 Voltage across the primary winding

KVL around the source loop gives

[ V_s = V_R + V_1 . \tag{4} ]

Because the winding itself has zero ohmic resistance, the only voltage that can appear across it is the induced emf (e_1).
Thus

[ V_1 = e_1 . \tag{5} ]

Now, what is the phase of (e_1) relative to the total current (I_1)?

From (1) we have two components of current:

(I_m) lags (e_1) by (90^{\circ}) (purely inductive).
(I_{\text{ref}}) is in phase with (e_1) (purely resistive).

Consequently the vector sum (1) is not 90° out of phase with (e_1); its angle is somewhere between (0^{\circ}) and (-90^{\circ}) depending on how large the load is compared with the magnetising inductance.

If the load is heavy (large (I_{\text{ref}})), the current is almost in phase with (e_1).
If the load is light (small (I_{\text{ref}})), the current is closer to lagging by 90°.

But the voltage (V_1) itself is always exactly the same as the source voltage (minus the tiny drop across (R_s)).
Thus the primary voltage is not forced to lag the current by 90°; it simply follows whatever the source forces, while the current adjusts (splitting into magnetising and reflected parts) to satisfy both the core flux requirement and the load.

2.5 Why the simple “(v = L\,di/dt)” rule does not apply here

The relation

[ v = L\frac{di}{dt} ]

describes the voltage across a single series inductance whose current is the only current flowing through it.
In an ideal transformer the primary winding is not a series inductance in that sense:

Feature	Simple series inductor	Primary winding of ideal transformer
Voltage across element	(v_L = L\frac{di}{dt}) (leads (i) by 90°)	(v_1 = e_1 = N_1\frac{d\Phi}{dt}) (set by the applied source)
Current through element	The only current in the circuit	Total current = magnetising + reflected
Phase relationship between this voltage and total current	Fixed 90° lag	Variable; depends on load

In other words, the primary winding’s inductive reactance is “hidden’’ inside the magnetic core. The induced emf (which we call the primary voltage) is forced by the source, not by the current through a discrete (L). The current reacts to that emf, splitting into two orthogonal components. Therefore the simple 90° rule does not dictate the phase between the applied primary voltage and the total primary current.

2.6 Putting it all together

The source voltage appears across the series combination of (R_s) and the ideal winding.
The resistor voltage (V_R = I_1 R_s) is in phase with the total current.
The winding voltage (V_1) is the induced emf, which is the same phasor as the source voltage (minus (V_R)).
Because the primary current contains a load component that is in phase with (V_1), the overall current is not 90° out of phase with the voltage.
The classic (v = L\,di/dt) lag only applies to a stand‑alone inductor with no coupled secondary and no reflected current.

Hence the primary voltage of an ideal transformer does not lag its current by 90°; it is essentially in phase with the source, while the current’s phase depends on the balance between magnetising and reflected components.

3. Final answer

In an ideal transformer the primary voltage is the induced emf (e_1 = N_1 \, d\Phi/dt).
This emf is forced by the source; it is therefore in phase with the source voltage (apart from the drop across the series resistor).
The primary current is the sum of a magnetising current (which lags the emf by 90°) and a reflected load current (which is in phase with the emf).
Because the reflected component exists whenever the secondary is

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