# Electromagnetism - Principle of Transformer - 電磁學 - 變壓器原理

1) 變壓器結構

• A transformer consists of two sets of coils, insulated from each other. They are wound on a soft-iron core as shown in Figure 1.
• Primary coil is where the input alternating voltage is applied.
• Secondary coil is where an increased (step up) or decreased (step down) voltage is output.

2) 變壓器原理 Principle of Transformer

• When an alternating voltage is applied to the primary coil, an alternating magnetic flux is resulted in the primary coil. (Consider the primary coil as a solenoid.)
• The magnetic flux is linked to the secondary coil through the soft iron core.

• In the Secondary Coil:
• By the Faraday's law, an induced e.m.f., $\epsilon_s$ is resulted in the secondary coil due to the alternating magnetic flux in primary coil.
• Hence, $\epsilon_s=N_s\frac{\triangle\phi}{\triangle t}$ ------ (1), where $\phi$ is the magnetic flux through a single coil.
• The number of coil in the secondary coil, $N_s$, (Figure 2.) is the major variable over which the values of the induced e.m.f. can be controlled.
• The induced e.m.f. in the secondary coil is given by $\epsilon_s=V_s + I_s R_s$ (Figure 2.).
• In ideal transformer, where negligible current flows through the secondary coil. We can approximate the equation above such that $\epsilon_s=V_s$------ (2).

• In the Primary Coil:
• The changing magnetic flux in the secondary coil in return induced a "back e.m.f." in the primary coil $\epsilon_p=N_p\frac{\triangle\phi}{\triangle t}$------ (3)
• In Figure 3., the induced back e.m.f. in the primary coil is opposite to the direction of the applied voltage, $\epsilon_p=V_p-I_pR_p$
• In ideal transformer, it is assumed that the resistance of the coil is zero and negligible current flow through the primary coil.
• We can approximate the equation above as $\epsilon_p=V_p$------ (4)
• In an ideal transformer, it is assumed that the soft iron core include all flux in both primary and secondary coil
• Hence, it is assumed that no flux leakage.
• The rate of change of magnetic flux through a single coil $\frac{\triangle\phi}{\triangle t}$ are the same in both equation (1) and (3)
• Equating equation (1) - (4)
• $\frac{V_p}{V_s}=\frac{N_p}{N_s}$
• The ratio of the input voltage to the output voltage is equal to the ratio of number of turns in the Primary Coil to that in Secondary Coil.
• Hence, the input voltage can be amplified.
• More about Principle of Transformer
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3) 變壓器功用

• A step-up transformer is a transformer which raised the input voltage to higher output voltage.
• The numbers of turns in the Primary coil is fewer than that in Secondary coil. (i.e. $N_p
• In power stations, voltage is first stepped up by engineered transformer to minimize the energy loss in the course of electricity transmission through overhead cable.
• A step-down transformer is a transformer which lowered the input voltage to lower output voltage.
• The numbers of turns in the Primary coil is greater than that in Secondary coil. (i.e. $N_p>N_s$)
• In electronic gadgets, applied voltage is stepped down by engineered transformer to better adapt the low voltage requirement in most semiconductors.

4) 變壓器的能量損耗

• There are deviations between practical and ideal transformer.
• In practical transformer, there are energy lost in the following case
1. Flux leakage : Some flux due to primary passes through the secondary coil leaks due to the imperfections in the soft iron cores. This is mitigated by winding the primary coil and secondary coil over each other.
2. Resistance of the coils : The coils has non-zero resistance.
3. Eddy currents  : The alternating magnetic flux induces eddy currents on the surface of the soft iron cores. Unncessary loss in heating is prevented by laminating the soft iron core.

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