Explain the operating principle of a transformer and derive the transformer equation
The operating principle of a transformer is based on electromagnetic induction. A transformer consists of two coils, known as the primary coil and the secondary coil, which are wound on a core (usually made of iron) that serves to concentrate the magnetic flux.
When an alternating current (AC) passes through the primary coil, it generates a changing magnetic field around the coil. According to Faraday's law of electromagnetic induction, the changing magnetic field induces a voltage in the secondary coil. This voltage is then available for use in an electrical circuit connected to the secondary coil.
The derivation of the transformer equation starts with the principle of conservation of energy. The power input to the primary coil of a transformer must be equal to the power output from the secondary coil. Mathematically, this can be expressed as:
P_in = P_out
Since power is given by the equation P = VI, where V is the voltage and I is the current, we have:
V_in x I_in = V_out x I_out
Where V_in and V_out are the primary and secondary voltages, respectively, and I_in and I_out are the primary and secondary currents, respectively.
Now, we can define the turns ratio (N) as the ratio of the number of turns in the secondary coil to the number of turns in the primary coil:
N = N_out / N_in
Using the turns ratio, we can express the secondary voltage in terms of the primary voltage:
V_out = N x V_in
Similarly, the primary current can be expressed in terms of the secondary current:
I_in = (1 / N) x I_out
Substituting these expressions back into the power equation, we get:
V_in x I_in = (N x V_in) x ((1 / N) x I_out)
Simplifying further, we arrive at the transformer equation:
V_in / V_out = N_in / N_out = I_out / I_in
This equation shows that the voltage ratio is equal to the turns ratio, and the current ratio is the inverse of the turns ratio in a transformer.