Transformers
How a transformer works
Your phone charger turns 220 volts from the socket into a few volts. A power station sends out electricity at hundreds of kilovolts. Both jobs are done by a transformer.
A transformer has two coils wound on the same soft-iron core. The primary coil, with N p turns, is connected to the ac input. The secondary coil, with N s turns, gives the output.
An alternating current in the primary makes an alternating magnetic flux in the core. The core guides almost all of this flux through the secondary too.
This is mutual induction. The changing flux induces an emf in the secondary: ε s equals minus N s dΦ by dt, where Φ is the flux through one turn.
The same changing flux also induces a back emf in the primary: ε p equals minus N p dΦ by dt.
For an ideal transformer, the primary has negligible resistance, so its back emf equals the applied voltage, v p. And with no load drawing much current, the secondary voltage v s equals ε s.
Divide one by the other. dΦ by dt cancels, and we get v s over v p equals N s over N p. The voltage is in the ratio of the turns.
If N s is greater than N p, the output voltage is higher: a step-up transformer. If N s is less than N p, it is lower: a step-down transformer.
Connect the primary to a battery instead, and the flux becomes steady. No change in flux, no induced emf. A transformer works only with ac.
Currents, losses and power lines
If a transformer is 100% efficient, the power going in equals the power coming out: I p V p equals I s V s.
Combine this with the voltage ratio, and I p over I s equals N s over N p. When the voltage is stepped down, the current is stepped up by the same factor, and the other way round.
Example: a transmission line feeds 2300 volts to a step-down transformer whose primary has 4000 turns. To get 230 volts out, N s equals 4000 × 230 over 2300, which is 400 turns.
Real transformers lose a little energy. Some flux leaks out and misses the secondary; winding the two coils over each other reduces this.
The windings have resistance and get hot; thick copper wire keeps this small. The core gets hot too: currents induced in the iron are cut down by making it of thin insulated sheets, or laminations.
Repeated magnetisation of the core also wastes energy as heat. A core material with low hysteresis loss keeps this small. Large transformers are often over 95% efficient.
Now see why transformers matter. A power station sends 1.1 megawatts through lines with a total resistance of 10 ohms.
At 11 kilovolts, the current is 1.1 × 10⁶ over 11 000, which is 100 amperes. The heating loss in the lines is I² R, 100² × 10, which is 100 kilowatts. Almost a tenth of the power is wasted.
Step it up to 110 kilovolts first. The current falls to 10 amperes, and the loss falls to 10² × 10, only 1 kilowatt: a hundred times less.
So the voltage is stepped up at the power station, sent over long distances, and stepped down at sub-stations and on poles near your home, until it reaches you at 220 volts.
Try it: build a transformer
- This is a charger transformer: 2000 turns to 100 turns gives 220 V → 11 V. The flux arrows in the core grow, shrink and reverse.
- Compare the dots in the two circuits: the secondary current (1 A) is 20 times the primary current (0.05 A). Power in = power out = 11 W.
- Press “Swap coils” to make a step-up transformer. What happens to the currents now?
- Tick “Use a battery”: with a steady flux, nothing is induced in the secondary.
Turns for a charger
A step-down transformer converts 220 V mains to 11 V. Its primary coil has 2000 turns. How many turns does the secondary have?
- 100
- 1000
- 2200
- 40 000
Answer
A. V_s/V_p = N_s/N_p, so N_s = N_p × V_s/V_p = 2000 × 11/220 = 100 turns. A step-down transformer always has fewer turns on the secondary. (40 000 comes from turning the ratio upside down, which would make a step-up transformer.)
Primary current
An ideal transformer steps 220 V down to 22 V. The load on the secondary draws 5 A. What current flows in the primary?
- 50 A
- 5 A
- 0.5 A
- 0.05 A
Answer
C. For an ideal transformer, power in = power out: I_p V_p = I_s V_s, so I_p = 5 × 22/220 = 0.5 A. Both sides carry 110 W. The voltage went down by 10, so the current on the primary side is 10 times smaller, not larger.
Summary
- Voltage ratio:
V_s/V_p = N_s/N_p - Ideal transformer:
I_p V_p = I_s V_s, I_p/I_s = N_s/N_p - Line loss:
P_loss = I²R, I = P/V
- A transformer uses mutual induction: an alternating flux in a shared iron core links both coils. It works only with ac.
- Step-up: N_s > N_p, higher voltage and lower current. Step-down: N_s < N_p.
- Energy losses: flux leakage, resistance of the windings, currents induced in the core (reduced by laminating it) and hysteresis.
- Power is transmitted at high voltage so that the current, and the I²R loss in the lines, is small.