Recall and use the equation V_p / V_s = N_p / N_s where p and s refer to primary and secondary

4.5.6 The Transformer

Learning Objective

Recall and use the equation \$\frac{Vp}{Vs} = \frac{Np}{Ns}\$ where the subscripts p and s refer to the primary and secondary windings of a transformer.

Key Concepts

• Definition of a transformer.
• Primary and secondary coils.
• Turns ratio and its effect on voltage and current.
• Ideal vs. real transformer.

How a Transformer Works

A transformer consists of two (or more) coils of insulated wire wound on a common magnetic core. An alternating current (AC) in the primary coil creates a time‑varying magnetic flux in the core. This changing flux links the secondary coil and induces an emf according to Faraday’s law.

Suggested diagram: cross‑section of a transformer showing primary coil, secondary coil, magnetic core and direction of flux.

Derivation of the Turns Ratio Equation

For an ideal transformer:

1. Faraday’s law gives the induced emf in each coil: \$\mathcal{E}= -N\frac{d\Phi}{dt}\$ where \$N\$ is the number of turns and \$\Phi\$ the magnetic flux.
2. Because the same flux \$\Phi\$ links both coils, the ratio of the induced emfs equals the ratio of turns:

   \$\frac{Vp}{Vs}= \frac{Np}{Ns}\$

3. Assuming no losses, the input power equals the output power: \$Vp Ip = Vs Is\$ which leads to the current relationship \$\frac{Is}{Ip}= \frac{Np}{Ns}\$.

Step‑up and Step‑down Transformers

Worked Example

Problem: A transformer has 500 turns on the primary and 200 turns on the secondary. The primary voltage is 240 V. Find the secondary voltage and the secondary current if the primary current is 0.5 A (assume an ideal transformer).

1. Calculate the turns ratio:

   \$\frac{Np}{Ns}= \frac{500}{200}=2.5\$

2. Use the voltage equation:

   \$\frac{Vp}{Vs}= \frac{Np}{Ns}\;\Rightarrow\;Vs = \frac{Vp Ns}{Np}= \frac{240\times200}{500}=96\text{ V}\$

3. Apply the power equality \$Vp Ip = Vs Is\$:

   \$Is = \frac{Vp Ip}{Vs}= \frac{240\times0.5}{96}=1.25\text{ A}\$

Thus \$Vs = 96\,\$V and \$Is = 1.25\,\$A.

Common Mistakes

• Confusing the ratio direction – always write \$\frac{Vp}{Vs}= \frac{Np}{Ns}\$, not the reverse.
• Using the turns ratio for current without inverting it; the current ratio is the inverse of the voltage ratio.
• Neglecting that the equation applies only to ideal (loss‑free) transformers; real devices have efficiency < 100 %.

Quick Revision Checklist

1. Identify primary and secondary windings.
2. Note the number of turns \$Np\$ and \$Ns\$.
3. Write the voltage ratio \$\frac{Vp}{Vs}= \frac{Np}{Ns}\$.
4. If required, use \$Vp Ip = Vs Is\$ to relate currents.
5. Determine whether the transformer is step‑up or step‑down.

Summary

The fundamental relationship for an ideal transformer is \$\frac{Vp}{Vs}= \frac{Np}{Ns}\$. This allows you to predict how the voltage and current will change between the primary and secondary based on the turns ratio. Remember that power is conserved (ignoring losses), so an increase in voltage comes with a decrease in current and vice‑versa.