# 15.6 Transformers  (Page 3/9)

 Page 3 / 9

## Summary

• Power plants transmit high voltages at low currents to achieve lower ohmic losses in their many kilometers of transmission lines.
• Transformers use induction to transform voltages from one value to another.
• For a transformer, the voltages across the primary and secondary coils, or windings, are related by the transformer equation.
• The currents in the primary and secondary windings are related by the number of primary and secondary loops, or turns, in the windings of the transformer.
• A step-up transformer increases voltage and decreases current, whereas a step-down transformer decreases voltage and increases current.

## Key equations

 AC voltage $v={V}_{0}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\omega t$ AC current $i={I}_{0}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\omega t$ capacitive reactance $\frac{{V}_{0}}{{I}_{0}}=\frac{1}{\omega C}={X}_{C}$ rms voltage ${V}_{\text{rms}}=\frac{{V}_{0}}{\sqrt{2}}$ rms current ${I}_{\text{rms}}=\frac{{I}_{0}}{\sqrt{2}}$ inductive reactance $\frac{{V}_{0}}{{I}_{0}}=\omega L={X}_{L}$ Phase angle of an ac circuit $\varphi ={\text{tan}}^{-1}\frac{{X}_{L}-{X}_{C}}{R}$ AC version of Ohm’s law ${I}_{0}=\frac{{V}_{0}}{Z}$ Impedance of an ac circuit $Z=\sqrt{{R}^{2}+{\left({X}_{L}-{X}_{C}\right)}^{2}}$ Average power associated with a circuit element ${P}_{\text{ave}}=\frac{1}{2}{I}_{0}{V}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\varphi$ Average power dissipated by a resistor ${P}_{\text{ave}}=\frac{1}{2}{I}_{0}{V}_{0}={I}_{\text{rms}}{V}_{\text{rms}}={I}_{\text{rms}}^{2}R$ Resonant angular frequency of a circuit ${\omega }_{0}=\sqrt{\frac{1}{LC}}$ Quality factor of a circuit $Q=\frac{{\omega }_{0}}{\text{Δ}\omega }$ Quality factor of a circuit in terms of the circuit parameters $Q=\frac{{\omega }_{0}L}{R}$ Transformer equation with voltage $\frac{{V}_{\text{S}}}{{V}_{\text{P}}}=\frac{{N}_{\text{S}}}{{N}_{\text{P}}}$ Transformer equation with current ${I}_{\text{S}}=\frac{{N}_{\text{P}}}{{N}_{\text{S}}}{I}_{\text{P}}$

## Conceptual questions

Why do transmission lines operate at very high voltages while household circuits operate at fairly small voltages?

There is less thermal loss if the transmission lines operate at low currents and high voltages.

How can you distinguish the primary winding from the secondary winding in a step-up transformer?

Battery packs in some electronic devices are charged using an adapter connected to a wall socket. Speculate as to the purpose of the adapter.

The adapter has a step-down transformer to have a lower voltage and possibly higher current at which the device can operate.

Will a transformer work if the input is a dc voltage?

Why are the primary and secondary coils of a transformer wrapped around the same closed loop of iron?

so each loop can experience the same changing magnetic flux

## Problems

A step-up transformer is designed so that the output of its secondary winding is 2000 V (rms) when the primary winding is connected to a 110-V (rms) line voltage. (a) If there are 100 turns in the primary winding, how many turns are there in the secondary winding? (b) If a resistor connected across the secondary winding draws an rms current of 0.75 A, what is the current in the primary winding?

A step-up transformer connected to a 110-V line is used to supply a hydrogen-gas discharge tube with 5.0 kV (rms). The tube dissipates 75 W of power. (a) What is the ratio of the number of turns in the secondary winding to the number of turns in the primary winding? (b) What are the rms currents in the primary and secondary windings? (c) What is the effective resistance seen by the 110-V source?

a. 45:1; b. 0.68 A, 0.015 A; c. $160\phantom{\rule{0.2em}{0ex}}\text{Ω}$

An ac source of emf delivers 5.0 mW of power at an rms current of 2.0 mA when it is connected to the primary coil of a transformer. The rms voltage across the secondary coil is 20 V. (a) What are the voltage across the primary coil and the current through the secondary coil? (b) What is the ratio of secondary to primary turns for the transformer?

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