Derivation of coupled impedance equation of a transformer and its application

When the load of a resonant transformer is variable, it is complicated to find the value of the quality factor Q of the primary circuit of the transformer because the coupled impedance operated by the secondary circuit of the transformer has a great effect on its primary circuit. Therefore, the only way to find out Q is to get the ratio of the inductor voltage V₁ of the primary of the transformer to the voltage of the source V_g, which must be measured by equipment such as oscilloscopes. In this paper we have derived a new equation for the coupled impedance including the coupling coefficient k, the turns ratio n, the resistive load R₀, and the reactance X_L1 of the primary of the transformer as the main parameters from the existing equation of coupled impedance. And we derived an equation for the quality factor Q, which involves the coupled impedance in series with the resonant circuit of the transformer. When R₀ > 2X_L2, for quality factor Q, the errors in its theoretical values are less than 3.98% of its measured values. And in the case of the load voltage V₀, all errors in the theoretical values are less than 2.65% of its measured values. Moreover, we also analyzed a typical transformer by using the derived equation for coupled impedance. Then, in the case of the load voltage V₀, when the load is variable, the difference of its measured values from the theoretical ones is less than 5.18%. As a result, the newly derived equation for coupled impedance is very useful to analyze the characteristics of a typical transformer as well as a series resonant transformer which has many benefits by virtue of the quality factor Q.