Transformer (In phase or out of phase?)

Question: Use Faraday’s law of electromagnetic induction to explain whether the input potential difference and the output e.m.f. of an ideal transformer are in phase.

            
       Some physics teachers may incorrectly use the equation V_s = N_s dΦ/dt to explain that the input potential difference and the output e.m.f. are in phase. A more reasonable answer is: “the output e.m.f. (induced e.m.f. in the secondary coil) and the input potential difference (applied e.m.f. in the primary coil) are 180^o out of phase. According to Faraday’s law of electromagnetic induction, the magnitude of induced e.m.f. in the secondary coil is directly proportional to the time rate of change of magnetic flux linkage of the transformer (E_s = -N_s dΦ/dt). Furthermore, the induced current in the secondary coil generates magnetic fields such that the effect is to oppose the changing magnetic fields in the primary coil (Lenz’s law).” 

          The answer is elaborated below (Whelan & Hodgson, 1989): 

Let E_p and E_s be the input potential difference and the output potential difference respectively. 

By using Faraday’s law of induction, the induced e.m.f. in the primary coil is -N_p dΦ/dt and the induced e.m.f in the secondary coil is -N_s dΦ/dt. 

That is, E_s = -N_s dΦ/dt------ Equation (1)

By applying Kirchhoff’s voltage law to the primary circuit,

E_p + (-N_pdΦ/dt) = IR = 0   (assuming electrical resistance of the primary is negligible)

Therefore, E_p = N_pdΦ/dt ------ Equation (2)

Eqn (1) / Eqn (2) ⇒ E_s/E_p = -N_s/N_p

The minus sign indicates that the input potential difference (E_p) and the output potential difference (E_s) are in anti-phase. This is analogous to saying action and reaction are equal and opposite. 

How would Feynman answer? 

       Feynman would answer that the induced e.m.f in the secondary coil and the applied e.m.f. in the primary coil are neither exactly in phase nor out of phase by π radians. Generally speaking, the induced e.m.f. in the secondary coil could be expressed as E_s = -L_s dIs/dt ± M dI_p/dt  (Feynman et al., 1964, section 22-8 Other circuit elements). The sign of the second term can be plus or minus because it is dependent on the types of winding connections. Furthermore, the primary and secondary coils are not pure inductors and they do have electrical resistance (Feynman et al., 1964, section 23-1 Real Circuit Elements).

Importantly, Feynman might explain the operations of a transformer as follows: “[a] ‘transformer’ is often made by putting two coils on the same torus — or core — of a magnetic material. Then a varying current in the ‘primary’ winding causes the magnetic field in the core to change, which induces an emf in the ‘secondary’ winding. Since the flux through each turn of both windings is the same, the emf’s in the two windings are in the same ratio as the number of turns on each. A voltage applied to the primary is transformed to a different voltage at the secondary. Since a certain net current around the core is needed to produce the required change in the magnetic field, the algebraic sum of the currents in the two windings will be fixed and equal to the required ‘magnetizing’ current. If the current drawn from the secondary increases, the primary current must increase in proportion — there is a ‘transformation’ of currents as well as voltage (Feynman et al., 1964, section 36–4 Iron-core inductances).” It is worth mentioning that Feynman does not cite Faraday’s law of electromagnetic induction or Lenz’s law in his explanation.

However, Feynman would discuss problems of defining a transformer or problems of defining a coil or an inductor of the transformer. For example, in the real world, the inductor has some electrical resistance, and a resistor has some inductance. As an analogy, all of the mass of a mechanical oscillator is not actually located at the mass; some of the mass is in the inertia of the spring (Feynman et al., 1963, section 23-3 Electrical resonance). Similarly, all of the spring is not located at the spring; the mass is not absolutely rigid and it has a little elasticity. The idea of the mechanical oscillator being a mass on the end of a spring is only an approximation or idealization. 

Note:

1. In Feynman's words, “[i]n the electrical world, there are a number of objects which can be connected to make electric circuits. These passive circuit elements, as they are often called, are of three main types, although each one has a little bit of the other two mixed in. Before describing them in greater detail, let us note that the whole idea of our mechanical oscillator being a mass on the end of a spring is only an approximation. All the mass is not actually at the ‘mass’; some of the mass is in the inertia of the spring. Similarly, all of the spring is not at the ‘spring’; the mass itself has a little elasticity, and although it may appear so, it is not absolutely rigid, and as it goes up and down, it flexes ever so slightly under the action of the spring pulling it. The same thing is true in electricity. There is an approximation in which we can lump things into ‘circuit elements’ which are assumed to have pure, ideal characteristics (Feynman et al., 1963, section 23-3 Electrical resonance).”

2. “If we think of a real resistor, we know that the current through it will produce a magnetic field. So any real resistor should also have some inductance. Also, when a resistor has a potential difference across it, there must be charges on the ends of the resistor to produce the necessary electric fields. As the voltage changes, the charges will change in proportion, so the resistor will also have some capacitance. We expect that a real resistor might have the equivalent circuit shown in Fig. 23-1. In a well-designed resistor, the so-called ‘parasitic’ elements L and C are small, so that at the frequencies for which it is intended, ωL is much less than R, and 1/ωC is much greater than R (Feynman et al., 1964, section 23-1 Real Circuit Elements).”

References:
1. Feynman, R. P., Leighton, R. B., & Sands, M. L. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.
2. Feynman, R. P., Leighton, R. B., & Sands, M. L. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.
3. Whelan, P. M., & Hodgson, M. J. (1989). Essential Principles of Physics (2nd ed.). London: John Murray.