15.6 Transformers

Although ac electric power is produced at relatively low voltages, it is sent through transmission lines at very high voltages (as high as 500 kV). The same power can be transmitted at different voltages because power is the product $I_{\text{rms}}{V}_{\text{rms}}.$ (For simplicity, we ignore the phase factor $\text{cos}\varphi .)$ A particular power requirement can therefore be met with a low voltage and a high current or with a high voltage and a low current. The advantage of the high-voltage/low-current choice is that it results in lower $I_{\text{rms}}^{2}R$ ohmic losses in the transmission lines, which can be significant in lines that are many kilometers long (Figure 15.20).

Figure 15.20 The rms voltage from a power plant eventually needs to be stepped down from 12 kV to 240 V so that it can be safely introduced into a home. A high-voltage transmission line allows a low current to be transmitted via a substation over long distances.

Typically, the alternating emfs produced at power plants are “stepped up” to very high voltages before being transmitted through power lines; then, they must be “stepped down” to relatively safe values (110 or 220 V rms) before they are introduced into homes. The device that transforms voltages from one value to another using induction is the transformer (Figure 15.21).

Figure 15.21 Transformers are used to step down the high voltages in transmission lines to the 110 to 220 V used in homes. (credit: modification of work by “Fortyseven”/Flickr)

As Figure 15.22 illustrates, a transformer basically consists of two separated coils, or windings, wrapped around a soft iron core. The primary winding has $N_{\text{P}}$ loops, or turns, and is connected to an alternating voltage $v_{\text{P}}(t).$ The secondary winding has $N_{\text{S}}$ turns and is connected to a load resistor $R_{\text{S}}.$ We assume the ideal case for which all magnetic field lines are confined to the core so that the same magnetic flux permeates each turn of both the primary and the secondary windings. We also neglect energy losses to magnetic hysteresis, to ohmic heating in the windings, and to ohmic heating of the induced eddy currents in the core. A good transformer can have losses as low as 1% of the transmitted power, so this is not a bad assumption.

Figure 15.22 A step-up transformer (more turns in the secondary winding than in the primary winding). The two windings are wrapped around a soft iron core.

To analyze the transformer circuit, we first consider the primary winding. The input voltage $v_{\text{P}}(t)$ is equal to the potential difference induced across the primary winding. From Faraday’s law, the induced potential difference is $\text{−}{N}_{\text{P}}\left(d\text{Φ}\text{/}dt\right),$ where $\text{Φ}$ is the flux through one turn of the primary winding. Thus,

Similarly, the output voltage $v_{\text{S}}(t)$ delivered to the load resistor must equal the potential difference induced across the secondary winding. Since the transformer is ideal, the flux through every turn of the secondary winding is also $\Phi ,$ and

Combining the last two equations, we have

Hence, with appropriate values for $N_{\text{S}}\text{and}{N}_{\text{P}},$ the input voltage $v_{\text{P}}(t)$ may be “stepped up” $\left(N_{\text{S}}>N_{\text{P}}\right)$ or “stepped down” ($N_{\text{S}}<N_{\text{P}}$) to $v_{\text{S}}(t),$ the output voltage. This is often abbreviated as the transformer equation,

which shows that the ratio of the secondary to primary voltages in a transformer equals the ratio of the number of turns in their windings. For a step-up transformer, which increases voltage and decreases current, this ratio is greater than one; for a step-down transformer, which decreases voltage and increases current, this ratio is less than one.

From the law of energy conservation, the power introduced at any instant by $v_{\text{P}}(t)$ to the primary winding must be equal to the power dissipated in the resistor of the secondary circuit; thus,

When combined with Equation 15.20, this gives

If the voltage is stepped up, the current is stepped down, and vice versa.

Finally, we can use $i_{\text{S}}(t)=v_{\text{S}}(t)\text{/}{R}_{\text{S}}$, along with Equation 15.20 and Equation 15.22, to obtain

which tells us that the input voltage $v_{\text{P}}(t)$ “sees” not a resistance $R_{\text{S}}$ but rather a resistance

Our analysis has been based on instantaneous values of voltage and current. However, the resulting equations are not limited to instantaneous values; they hold also for maximum and rms values.