Transformer Frequency Considerations

For certain frequency ranges, the effect of some parameters in the equivalent circuit of the iron-core transformer of Fig. 1 should not be ignored. Since it is convenient to consider a low-, mid-, and high-frequency region, the equivalent circuits for each will now be introduced and briefly examined.
Equivalent circuit for the practical iron-core transformer
Fig. 1: Equivalent circuit for the practical iron-core transformer.
For the low-frequency region, the series reactance ($2\pi f L$) of the primary and secondary leakage reactances can be ignored since they are small in magnitude. The magnetizing inductance must be included, however, since it appears in parallel with the secondary reflected circuit, and small impedances in a parallel network can have a dramatic impact on the terminal characteristics. The resulting equivalent network for the low-frequency region is provided in Fig. 2(a).
Fig. 2: (a) Low-frequency reflected equivalent circuit;
(b) mid-frequency reflected circuit.
As the frequency decreases, the reactance of the magnetizing inductance will reduce in magnitude, causing a reduction in the voltage across the secondary circuit. For $f = 0 Hz$, $Lm$ is ideally a short circuit, and $V_L = 0$.
As the frequency increases, the reactance of $Lm$ will eventually be sufficiently large compared with the reflected secondary impedance to be neglected. The mid-frequency reflected equivalent circuit will then appear as shown in Fig. 2(b). Note the absence of reactive elements, resulting in an in-phase relationship between load and generator voltages.
Fig. 3: High-frequency reflected equivalent circuit.
For higher frequencies, the capacitive elements and primary and secondary leakage reactances must be considered, as shown in Fig. 3. For discussion purposes, the effects of $Cw$ and $Cs$ appear as a lumped capacitor $C$ in the reflected network of Fig. 3; $Cp$ does not appear since the effect of $C$ will predominate. As the frequency of interest increases, the capacitive reactance ($X_C = 1/2\pi f C$) will decrease to the point that it will have a shorting effect across the secondary circuit of the transformer, causing $V_L$ to decrease in magnitude.
Fig. 4: Transformer-frequency response curve.
A typical iron-core transformer-frequency response curve appears in Fig. 4. For the low- and high-frequency regions, the primary element responsible for the drop-off is indicated. The peaking that occurs in the high-frequency region is due to the series resonant circuit established by the inductive and capacitive elements of the equivalent circuit. In the peaking region, the series resonant circuit is in, or near, its resonant or tuned state.

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