Math of Transformers

Purpose: To understand briefly the math of the forces at work in a transformer, especially an impedance transformer for RF antenna systems.

Scope: AC circuits only, including 50 Hz power, 5 GHz microwave, and everything in between.

Math:

Ohm's law:

    V = I R

    I = V / R

Generalized to complex domain (for alternating currents with reactance) becomes:

    V̇ = Z İ ₍₁₎

    İ =  V̇ / Z ₍₁₎

Faraday's law: A changing magnetic field (ΔΦ/Δt) induces a voltage (Emf = V), and a changing voltage induces a magnetic field based on the number of turns on the ferrite core (N):

    V = -N ΔΦ / Δt (2)

Electrical power:

    P = I V

Combined with Ohm's law:

    P = V² / Z

The magnetic field induced on the ferrite core by the input in turn induces a voltage on the output (see caveats):

    ΔΦ1 /Δt = ΔΦ2 / Δt

Insert Faraday's law:

    -V1 / N1 = -V2 / N2

    V2 =  (N2 / N1) * V1   <-- This is the critical equation for a voltage transformer

Because energy is conserved and transformers are (by design) efficient (with efficiency η):

    η P1 = P2

Insert Ohms law:

    η V1² / Z1 = V2² / Z2

And if you substitute the voltage transformer equation:

    η V1² / Z1 = ((N2 / N1) * V1)² / Z2

    η V1² / Z1 = (N2 / N1)² * V1² / Z2

    η / Z1 = (N2 / N1)² / Z2

    Z2 = (N2 / N1)² * Z1 / η   <-- This is the critical equation for an impedance transformer

Both voltage and impedance transformations occur, but usually only one equation is important for the specific application.

Caveats: Saturation and efficiency of the toroid core are dependent on its size and chemistry and the frequency and amplitude (power) of the driving signal. Saturation is when f1 no longer equals f2.

Example 1: A perfectly efficient (η=1) transformer with 3 primary windings (N1) and 21 secondary windings (N2) has an output voltage that is 7 times the input voltage and an output impedance 49 times the input impedance. If the desired output impedance is 50Ω, the target input impedance is 2450Ω.

Example 2: If the transformer is 95% efficient (η=0.95) with the same setup as Example 1 and an input impedance of 2450Ω, the resulting output impedance will be 52.6Ω.