Nominal T Method of Transmission Lines with Medium Length

The equations and phasor diagram of T method of medium length transmission lines will be obtained. For this purpose the figure shows the one phase of three phase lines.

T Method Equations

The circuit can be separated in three parts based on inputs and outputs to obtain equations for T method.

Basic equation in matrice will be obtained by A, B, C, D parameters.

\[\left( \begin{array}{l}Vs\\{I_S}\end{array} \right) = \left( {\begin{array}{*{20}{c}}A&B\\C&D\end{array}} \right).\left( \begin{array}{l}{V_R}\\{I_R}\end{array} \right)\]

In section 1 output of the circuit will be V1 and Is1 when inputs are Vs and Is. However, consider all the electrical parameters in complex form a + jb, which means there will be a phase angle and magnitude of the currents and voltages. 

\[\begin{array}{l} - {V_S} + {I_{S1}}\frac{Z}{2} + {V_1} = 0\\{V_S} = {V_1} + \frac{Z}{2}{I_{S1}}\\{I_S} = 0.{V_1} + {I_{S1}}\\\left( \begin{array}{l}{V_S}\\{I_S}\end{array} \right) = \left( {\begin{array}{*{20}{c}}1&{\frac{Z}{2}}\\0&1\end{array}} \right)\left( \begin{array}{l}{V_1}\\{I_{S1}}\end{array} \right)\end{array}\]

In section 2, if we use the inputs as V₁ and I_s1 which previously are the outputs of section 1, then the outputs will be V₂ and I_s2. As it is seen V₂=V₀ but no need to use for the matrice. V₀ will be used for phasor diagram later.

\[\begin{array}{l}{V_1} = {V_2} + 0.{I_{S2}}\\{I_{S1}} = {I_Y} + {I_{S2}}\\{I_{S1}} = Y.{V_2} + {I_{S2}}\\\left( \begin{array}{l}{V_1}\\{I_{S1}}\end{array} \right) = \left( {\begin{array}{*{20}{c}}1&0\\Y&1\end{array}} \right)\left( \begin{array}{l}{V_2}\\{I_{S2}}\end{array} \right)\end{array}\]

Remember the basic formula.

\[\begin{array}{l}V = IZ\\I = V/Z\\Y = 1/Z\\I = VY\end{array}\]

In section 3, V₂ and I_s2 , the outputs of section 2, will be the inputs and finally I_R and V_R will be the outputs in similar way.

\[\begin{array}{l}{V_2} = {V_R} + \frac{Z}{2}{I_R}\\{I_{S2}} = 0.{V_R} + {I_R}\\\left( \begin{array}{l}{V_2}\\{I_{S2}}\end{array} \right) = \left( {\begin{array}{*{20}{c}}1&{\frac{Z}{2}}\\0&1\end{array}} \right)\left( \begin{array}{l}{V_R}\\{I_R}\end{array} \right)\end{array}\]

Consider all matrices again, and replace the values into previous ones.

\[\left( \begin{array}{l}{V_1}\\{I_{S1}}\end{array} \right) = \left( {\begin{array}{*{20}{c}}1&0\\Y&1\end{array}} \right)\left( {\begin{array}{*{20}{c}}1&{\frac{Z}{2}}\\0&1\end{array}} \right)\left( \begin{array}{l}{V_R}\\{I_R}\end{array} \right)\]

\[\left( \begin{array}{l}{V_S}\\{I_S}\end{array} \right) = \left( {\begin{array}{*{20}{c}}1&{\frac{Z}{2}}\\0&1\end{array}} \right)\left( {\begin{array}{*{20}{c}}1&0\\Y&1\end{array}} \right)\left( {\begin{array}{*{20}{c}}1&{\frac{Z}{2}}\\0&1\end{array}} \right)\left( \begin{array}{l}{V_R}\\{I_R}\end{array} \right)\]

\[\left( \begin{array}{l}{V_S}\\{I_S}\end{array} \right) = \left( {\begin{array}{*{20}{c}}{1 + \frac{{ZY}}{2}}&{0 + \frac{Z}{2}}\\{0 + Y}&{0 + 1}\end{array}} \right)\left( {\begin{array}{*{20}{c}}1&{\frac{Z}{2}}\\0&1\end{array}} \right)\left( \begin{array}{l}{V_R}\\{I_R}\end{array} \right)\]

\[\left( \begin{array}{l}{V_S}\\{I_S}\end{array} \right) = \left( {\begin{array}{*{20}{c}}{1 + \frac{{ZY}}{2}}&{\left( {1 + \frac{{ZY}}{2}} \right).\frac{Z}{2} + \frac{Z}{2}}\\Y&{1 + \frac{{ZY}}{2}}\end{array}} \right)\left( \begin{array}{l}{V_R}\\{I_R}\end{array} \right)\]

\[\begin{array}{l}\left( {1 + \frac{{ZY}}{2}} \right).\frac{Z}{2} + \frac{Z}{2} = \frac{Z}{2}\left( {1 + \frac{{ZY}}{2} + 1} \right)\\ = Z\left( {1 + \frac{{ZY}}{4}} \right)\end{array}\]

T method equation :

\[\left( \begin{array}{l}{V_S}\\{I_S}\end{array} \right) = \left( {\begin{array}{*{20}{c}}{1 + \frac{{ZY}}{2}}&{Z\left( {1 + \frac{{ZY}}{4}} \right)}\\Y&{1 + \frac{{ZY}}{2}}\end{array}} \right)\left( \begin{array}{l}{V_R}\\{I_R}\end{array} \right)\]

Phasor Diagram of T Method

Equations to obtain phasors. Impedance will be in complex form with angle and magnitude as well. 

\[\left| {\frac{Z}{2}} \right|\angle \theta \]

\[\left| {{I_R}} \right|\angle - {\phi _R}\]

Inductive current lags reference voltage by negative sign angle. V_R receiving voltage is the reference voltage for phasor diagram.

Based on right side of the circuit, voltage drop can be calculated by V₀.

\[\overline {{V_0}} = \overline {{V_R}} + \overline {\Delta {V_2}} = \left| {{V_0}} \right|\angle {\delta _0}\]

Sending current with receiving current can be written in complex form. Notice that admitance with V₀ which means the angle of the current flowing through admitance I_Y angle is 90 degree.

\[\begin{array}{l}\overline {{I_S}} = \overline {{I_Y}} + \overline {{I_R}} = \overline Y .\overline {{V_0}} + \overline {{I_R}} \\\overline {{I_S}} = \left| Y \right|\angle 90^\circ .\left| {{V_0}} \right|\angle {\delta _0} + \left| {{I_R}} \right|\angle - {\phi _R}\\\overline {{I_S}} = \left| {Y{V_0}} \right|\angle \left( {90^\circ + {\delta _0}} \right) + \left| {{I_R}} \right|\angle - {\phi _R}\end{array}\]

Voltage drop in left side of the circuit:

 \[\overline {\Delta {V_1}} = \overline {{I_S}} \frac{{\overline Z }}{2} = \left| {{I_S}} \right|\angle - {\phi _S}\left| {\frac{Z}{2}} \right|\angle \theta = \left| {{I_S}\frac{Z}{2}} \right|\angle \theta - {\phi _S}\]

Sending voltage with left side voltage drop and V₀ . Consider V_S angle with respect to V_R receiving voltage which is reference voltage for phasor diagram.

\[{\overline V _S} = \overline {{V_0}} + \overline {\Delta {V_1}} = \left| {{V_S}} \right|\angle {\delta _{}}\]

Phasor diagram can be drawn now.