# Tapped and Multiple Load Transformers

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For the center-tapped (primary) transformer of Fig. 1, where the voltage from the center tap to either outside lead is defined as $E_p/2$, the relationship between $Ep$ and $Es$ is
$$\bbox[10px,border:1px solid grey]{{ E_p \over E_s} = { N_p \over N_s}}$$
Fig. 1: Ideal transformer with a center-tapped primary.
For each half-section of the primary,
$$\mathbf{Z}_{1 / 2}=\left(\frac{N_{p} / 2}{N_{s}}\right)^{2} \mathbf{Z}_{L}=\frac{1}{4}\left(\frac{N_{p}}{N_{s}}\right)^{2} \mathbf{Z}_{L}$$
with
$$\mathbf{Z}_{i_{(A-B)}}=\left(\frac{N_{p}}{N_{s}}\right)^{2} \mathbf{Z}_{L}$$
Therefore,
$$\mathbf{Z}_{1 / 2}=\frac{1 / 4}{Z_{i}}$$
Fig. 2: Ideal transformer with multiple loads.
For the multiple-load transformer of Fig. 2, the following equations apply:
$$\bbox[10px,border:1px solid grey]{\frac{\mathbf{E}_{i}}{\mathbf{E}_{2}}=\frac{N_{1}}{N_{2}} \quad \frac{\mathbf{E}_{1}}{\mathbf{E}_{3}}=\frac{N_{1}}{N_{3}} \quad \frac{\mathbf{E}_{2}}{\mathbf{E}_{3}}=\frac{N_{2}}{N_{3}}} \tag{1}$$
The total input impedance can be determined by first noting that, for the ideal transformer, the power delivered to the primary is equal to the power dissipated by the load; that is,
$$P_{1}=P_{L_{2}}+P_{L_{3}}$$
and, for resistive loads $$\mathbf{Z}_{i}=R_{i}, \mathbf{Z}_{2}=R_{2}, \text{ and } \mathbf{Z}_{3}=R_{3}$$
$$\frac{E_{i}^{2}}{R_{i}}=\frac{E_{2}^{2}}{R_{2}}+\frac{E_{3}^{2}}{R_{3}}$$
or, since
$$\quad E_{2}=\frac{N_{2}}{N_{1}} E_{i} \quad$$
and
$$\quad E_{3}=\frac{N_{3}}{N_{1}} E_{1}$$
then
$$\frac{E_{i}^{2}}{R_{i}}=\frac{\left[\left(N_{2} / N_{1}\right) E_{i}\right]^{2}}{R_{2}}+\frac{\left[\left(N_{3} / N_{1}\right) E_{i}\right]^{2}}{R_{3}}$$
and
$$\frac{E_{i}^{2}}{R_{i}}=\frac{E_{i}^{2}}{\left(N_{1} / N_{2}\right)^{2} R_{2}}+\frac{E_{i}^{2}}{\left(N_{1} / N_{3}\right)^{2} R_{3}}$$
Thus,
$$\bbox[10px,border:1px solid grey]{ \frac{1}{R_{i}}=\frac{1}{\left(N_{1} / N_{2}\right)^{2} R_{2}}+\frac{1}{\left(N_{1} / N_{3}\right)^{2} R_{3}} } \tag{2}$$
Fig. 3: Ideal transformer with a tapped secondary and multiple loads.
indicating that the load resistances are reflected in parallel. For the configuration of Fig. 3, with $E_{2}$ and $E_{3}$ defined as shown, Equations (1) and (2) are applicable.

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