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_FC voltage on cable moving towards the inductor

V_RC voltage on cable moving away from the inductor

Z_C characteristic impedance of the cable

V_FE voltage traveling in the forward direction in the even mode

V_FO voltage traveling in the forward direction in the odd mode

V_RE voltage traveling in the reverse direction in the even mode

V_RO 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_FS secondary voltage moving away from the transformer

V_RS secondary voltage moving towards the transformer

From the Cable into the Inductor

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

1)

2)

3)

Transforming (2) into voltages gives,

4)

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

5)

6)

From (1),

7)

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

8)

From (3), V_FO = V_FE . Substituting this into (8) gives,

8)

9)

Substituting V_FE and V_FO from (8) and (9) into (7) gives the reflection coefficient,

10)

From the Inductor into the Cable

Going from the even and odd modes in the inductor to the cable, the following equations hold,

1)

2)

3)

Reflection coefficients can be derived using the same procedure as in the Cable to Inductor section,

4)

5)

6)

7)

Using these, the transmission coefficients to the cable can be derived,

8)

9)

From the Transformer into the Secondary

Going from the even and odd modes in the transformer to the secondary cable, the following equations hold,

1)

2)

3)

Two new ratios emerge in the derivation of the reflection and transmission coefficients,

4)

The resulting reflection coefficients are,

5)

6)

7)

8)

And the resulting transmission coefficients are,

9)

10)

From the Secondary into the Transformer

Going from the secondary into the transformer, the following equations hold,

1)

2)

3)

1)

2)

3)