# Two Turn Inductor, One Turn Transformer Derivation of Reflection and Transmission Coefficients

By Mike Gibson and Ivor Catt, June 1986

Here is a description of the symbols used in the derivations:

$$V_{F C}$$ voltage on cable moving towards the inductor
$$V_{R C}$$ voltage on cable moving away from the inductor
$$Z_C$$ characteristic impedance of the cable
$$V_{F E}$$ voltage traveling in the forward direction in the even mode
$$V_{F O}$$ voltage traveling in the forward direction in the odd mode
$$V_{R E}$$ voltage traveling in the reverse direction in the even mode
$$V_{R O}$$ voltage traveling in the reverse direction in the odd mode
$$Z_E$$ characteristic impedance of even mode
$$Z_O$$ characteristic impedance of odd mode
$$V_{F S}$$ secondary voltage moving away from the transformer
$$V_{R S}$$ secondary voltage moving towards the transformer

Going from the cable to the inductor, the following basic equations hold,

\begin{equation} \begin{aligned} &1) \qquad V_{F C}+V_{R C}=V_{F E}+V_{F O}=V_{A B} \\ &2) \qquad I_{F C}+I_{R C}=I_{F E}+I_{F O}=I_{A B} \\ &3) \qquad V_{F E}-V_{F O}=V_{P Q}=0 \end{aligned} \end{equation}

Transforming (2) into voltages gives,

$$4) \qquad \frac{V_{F C}}{Z_C}-\frac{V_{R C}}{Z_C}=\frac{V_{F E}}{Z_E}+\frac{V_{F O}}{Z_O}$$

Multiplying through by ZC and defining new terms for the resulting ratios yields,

$$5) \qquad V_{F C}-V_{R C}=r_E V_{F E}+r_O V_{F O}$$

6) \qquad \boxed{ \begin{aligned} &r_E=\frac{Z_C}{Z_E}\ &r_O=\frac{Z_C}{Z_O} \end{aligned}}

From (1),

$$7) \qquad V_{R C}=V_{F E}+V_{F O}-V_{F C}$$

Substituting (7) into (5) and gathering terms,

$$7) \qquad V_{F C}-V_{F E}-V_{F O}+V_{F C}=r_EV_{F E}+r_OV_{F O}$$