Question

The junction of four single-phase two-wire lossless lines, denoted A, B, C, and D, is shown in Figure 13.13. Consider a voltage wave${\mathrm{\vartheta }}_{\mathrm{A}}^{+}$ arriving at the junction from line A. Using (13.3.8) and (13.3.9), determine the voltage reflection coefficient${\mathrm{\Gamma }}_{\mathrm{AA}}$ and the voltage refraction coefficients${\mathrm{\Gamma }}_{\mathrm{BA}}$ , ${\mathrm{\Gamma }}_{\mathrm{CA}}$, and ${\mathrm{\Gamma }}_{\mathrm{DA}}$.

Short answer

The value of voltage reflection coefficient,${\mathrm{\Gamma }}_{\mathrm{AA}}$ is $\frac{{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}-1}{{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}+1}$.

The value of voltage refraction coefficient,${\mathrm{\Gamma }}_{\mathrm{BA}}$ is $\frac{2{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}}{{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}+1}$.

The value of voltage refraction coefficient, ${\mathrm{\Gamma }}_{\mathrm{CA}}$is $\frac{2{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}}{{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}+1}$.

The value of voltage refraction coefficient, ${\mathrm{\Gamma }}_{\mathrm{DA}}$is $\frac{2{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}}{{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}+1}$.

Step-by-step solution

Write the given data from the question.

Consider the following data:

The voltage reflection coefficient for the waves arriving at the junction from A, ${\mathrm{\Gamma }}_{\mathrm{AA}}$.

The voltage reflection coefficient for the waves arriving at the junction from B, ${\mathrm{\Gamma }}_{\mathrm{BA}}$.

The voltage reflection coefficient for the waves arriving at the junction from C, ${\mathrm{\Gamma }}_{\mathrm{CA}}$.

The voltage reflection coefficient for the waves arriving at the junction from D, ${\mathrm{\Gamma }}_{\mathrm{DA}}$.

The voltage wave arriving at the junction of line A, B, C, D from line A, ${\mathrm{V}}^{+}$.

Determine the formula of voltage reflection coefficient and voltage refraction coefficient.

Write the formula ofvoltage reflection coefficient for the waves arriving at the junction from A.

${\mathbf{\Gamma }}_{\mathbf{AA}}\mathbf{=}\frac{{\mathbf{V}}_{\mathbf{A}}^{\mathbf{-}}}{{\mathbf{V}}_{\mathbf{A}}^{\mathbf{+}}}$ …… (1)

Here,${\mathbf{V}}_{\mathbf{A}}^{\mathbf{-}}$ is voltage wave and is${\mathbf{V}}_{\mathbf{A}}^{\mathbf{+}}$ voltage wave arriving.

Write the formula ofvoltage reflection coefficient for the waves arriving at the junction from B.

${\mathbf{\Gamma }}_{\mathbf{BA}}\mathbf{=}\frac{{\mathbf{V}}_{\mathbf{B}}^{\mathbf{+}}}{{\mathbf{V}}_{\mathbf{A}}^{\mathbf{+}}}$ …… (2)

Here,${\mathbf{V}}_{\mathbf{B}}^{\mathbf{+}}$ is voltage wave and is${\mathbf{V}}_{\mathbf{A}}^{\mathbf{+}}$ voltage wave arriving.

(a) Determine the value of voltage reflection coefficient and voltage refraction coefficient.

Assume the following junction of four single phase two-wire lossless line:

Figure 1

Draw the circuit diagram of the junction of four single phase two-wire lossless lines:

Figure 2

Apply Kirchhoff’s voltage law to the junction.

${\mathrm{V}}_{\mathrm{A}}^{+}+{\mathrm{V}}_{\mathrm{A}}^{-}={\mathrm{V}}_{\mathrm{B}}^{+}$ …… (3)

Since, ${\mathrm{V}}_{\mathrm{C}}^{+}={\mathrm{V}}_{\mathrm{D}}^{+}={\mathrm{V}}_{\mathrm{B}}^{+}$

Apply Kirchhoff’s current law equation at the junction.

${\mathrm{I}}_{\mathrm{A}}^{+}+{\mathrm{I}}_{\mathrm{A}}^{-}={\mathrm{I}}_{{\mathrm{B}}_{}^{}}^{+}+{\mathrm{I}}_{\mathrm{C}}^{+}+{\mathrm{I}}_{\mathrm{D}}^{+}$ …… (4)

We know that,

Determine the current wave arriving, ${\mathrm{I}}_{\mathrm{A}}^{+}$.

${\mathrm{I}}_{\mathrm{A}}^{+}=\frac{{\mathrm{V}}_{\mathrm{A}}^{+}}{{\mathrm{Z}}_{\mathrm{A}}}$

Determine the current wave arriving, ${\mathrm{I}}_{\mathrm{A}}^{-}$.

${\mathrm{I}}_{\mathrm{A}}^{-}=\frac{{\mathrm{V}}_{\mathrm{A}}^{-}}{{\mathrm{Z}}_{\mathrm{A}}}$

Determine the current wave arriving, ${\mathrm{I}}_{\mathrm{B}}^{+}$.

${\mathrm{I}}_{\mathrm{B}}^{+}=\frac{{\mathrm{V}}_{\mathrm{B}}^{+}}{{\mathrm{Z}}_{\mathrm{B}}}$

Determine the current wave arriving, ${\mathrm{I}}_{\mathrm{C}}^{+}$.

${\mathrm{I}}_{\mathrm{C}}^{+}=\frac{{\mathrm{V}}_{\mathrm{C}}^{+}}{{\mathrm{Z}}_{\mathrm{C}}}$

Determine the current wave arriving, ${\mathrm{I}}_{\mathrm{D}}^{+}$.

role="math" localid="1656669752434" ${\mathrm{I}}_{\mathrm{D}}^{+}=\frac{{\mathrm{V}}_{\mathrm{D}}^{+}}{{\mathrm{Z}}_{\mathrm{D}}}$

Substitute above five equation into equation (4).

role="math" localid="1656670081805" $$\begin{gathered} \frac{{\mathrm{V}}_{\mathrm{A}}^{+}}{{\mathrm{Z}}_{\mathrm{A}}}-\frac{{\mathrm{V}}_{\mathrm{A}}^{-}}{{\mathrm{Z}}_{\mathrm{A}}}=\frac{{\mathrm{V}}_{\mathrm{B}}^{+}}{{\mathrm{Z}}_{\mathrm{B}}}+\frac{{\mathrm{V}}_{\mathrm{C}}^{+}}{{\mathrm{Z}}_{\mathrm{C}}}+\frac{{\mathrm{V}}_{\mathrm{D}}^{+}}{{\mathrm{Z}}_{\mathrm{D}}} \\ \frac{{\mathrm{V}}_{\mathrm{A}}^{+}}{{\mathrm{Z}}_{\mathrm{A}}}-\frac{{\mathrm{V}}_{\mathrm{A}}^{-}}{{\mathrm{Z}}_{\mathrm{A}}}=\frac{{\mathrm{V}}_{\mathrm{B}}^{+}}{{\mathrm{Z}}_{\mathrm{B}}}+\frac{{\mathrm{V}}_{\mathrm{B}}^{+}}{{\mathrm{Z}}_{\mathrm{C}}}+\frac{{\mathrm{V}}_{\mathrm{B}}^{+}}{{\mathrm{Z}}_{\mathrm{D}}} \\ \frac{{\mathrm{V}}_{\mathrm{A}}^{+}}{{\mathrm{Z}}_{\mathrm{A}}}-\frac{{\mathrm{V}}_{\mathrm{A}}^{-}}{{\mathrm{Z}}_{\mathrm{A}}}={\mathrm{V}}_{\mathrm{B}}^{+}\left(\frac{1}{{\mathrm{Z}}_{\mathrm{B}}}+\frac{1}{{\mathrm{Z}}_{\mathrm{C}}}+\frac{1}{{\mathrm{Z}}_{\mathrm{D}}}\right) \end{gathered}$$ …… (5)

From figure 1,

$$\begin{gathered} {\mathrm{Z}}_{\mathrm{eq}}={\mathrm{Z}}_{\mathrm{A}}\mathrm{P}{\mathrm{Z}}_{\mathrm{B}}\mathrm{P}{\mathrm{Z}}_{\mathrm{C}} \\ {\mathrm{Z}}_{\mathrm{eq}}=\frac{1}{{\mathrm{Z}}_{\mathrm{A}}}+\frac{1}{{\mathrm{Z}}_{\mathrm{B}}}+\frac{1}{{\mathrm{Z}}_{\mathrm{C}}} \end{gathered}$$ …… (6)

Substitute equation (6) into equation (5).

$\frac{{\mathrm{V}}_{\mathrm{A}}^{+}}{{\mathrm{Z}}_{\mathrm{A}}}-\frac{{\mathrm{V}}_{\mathrm{A}}^{-}}{{\mathrm{Z}}_{\mathrm{A}}}=\frac{{\mathrm{V}}_{\mathrm{B}}^{+}}{{\mathrm{Z}}_{\mathrm{eq}}}$ …… (7)

Substitute${\mathrm{V}}_{\mathrm{A}}^{+}+{\mathrm{V}}_{\mathrm{A}}^{-}$for${\mathrm{V}}_{\mathrm{B}}^{+}$into equation (7).

role="math" localid="1656671063619" $$\begin{gathered} \frac{{\mathrm{V}}_{\mathrm{A}}^{+}}{{\mathrm{Z}}_{\mathrm{A}}}-\frac{{\mathrm{V}}_{\mathrm{A}}^{-}}{{\mathrm{Z}}_{\mathrm{A}}}=\frac{{\mathrm{V}}_{\mathrm{A}}^{+}-{\mathrm{V}}_{\mathrm{A}}^{-}}{{\mathrm{Z}}_{\mathrm{eq}}} \\ {\mathrm{V}}_{\mathrm{A}}^{+}\left(\frac{1}{{\mathrm{Z}}_{\mathrm{A}}}-\frac{1}{{\mathrm{Z}}_{\mathrm{eq}}}\right)={\mathrm{V}}_{\mathrm{A}}^{-}\left(\frac{1}{{\mathrm{Z}}_{\mathrm{A}}}+\frac{1}{{\mathrm{Z}}_{\mathrm{eq}}}\right) \\ \frac{{\mathrm{V}}_{\mathrm{A}}^{-}}{{\mathrm{V}}_{\mathrm{A}}^{+}}=\frac{\left[{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}-1\right]}{\left[{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}+1\right]} \end{gathered}$$ …… (8)

Write the expression for voltage reflection coefficient for the waves arriving at the junction from A.

Substitute equation (8) into equation (1).

${\mathrm{\Gamma }}_{\mathrm{AA}}=\frac{\left[{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}-1\right]}{\left[{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}+1\right]}$

Therefore,the value of voltage reflection coefficient,${\mathrm{\Gamma }}_{\mathrm{AA}}$is$\frac{{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}-1}{{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}+1}$.

Substitute ${\mathrm{V}}_{\mathrm{A}}^{+}+{\mathrm{V}}_{\mathrm{A}}^{-}$for ${\mathrm{V}}_{\mathrm{A}}^{+}$into equation (7).

$$\begin{gathered} \frac{{\mathrm{V}}_{\mathrm{A}}^{+}}{{\mathrm{Z}}_{\mathrm{A}}}-\frac{{\mathrm{V}}_{\mathrm{B}}^{-}-{\mathrm{V}}_{\mathrm{A}}^{+}}{{\mathrm{Z}}_{\mathrm{A}}}=\frac{{\mathrm{V}}_{\mathrm{B}}^{+}}{{\mathrm{Z}}_{\mathrm{eq}}} \\ \frac{{\mathrm{V}}_{\mathrm{A}}^{+}}{{\mathrm{Z}}_{\mathrm{A}}}+\frac{{\mathrm{V}}_{\mathrm{A}}^{+}}{{\mathrm{Z}}_{\mathrm{A}}}=\frac{{\mathrm{V}}_{\mathrm{B}}^{+}}{{\mathrm{Z}}_{\mathrm{eq}}}+\frac{{\mathrm{V}}_{\mathrm{B}}^{-}}{{\mathrm{Z}}_{\mathrm{A}}} \\ {\mathrm{V}}_{\mathrm{B}}^{+}=\left[\frac{2{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}}{{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}+1}\right]{\mathrm{V}}_{\mathrm{A}}^{+} \\ \frac{{\mathrm{V}}_{\mathrm{B}}^{+}}{{\mathrm{V}}_{\mathrm{A}}^{+}}=\left[\frac{2{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}}{{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}+1}}\right] \end{gathered}$$

Determine thevoltage refraction coefficient.

Substitute $\left[\frac{2{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}}{{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}+1}\right]$for$\frac{{\mathrm{V}}_{\mathrm{B}}^{+}}{{\mathrm{V}}_{\mathrm{A}}^{+}}$ into equation (2).

${\mathrm{\Gamma }}_{\mathrm{BA}}=\left[\frac{2{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}}{{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}+1}\right]$

We know that, ${\mathrm{V}}_{\mathrm{C}}^{+}={\mathrm{V}}_{\mathrm{D}}^{+}={\mathrm{V}}_{\mathrm{B}}^{+}$. So, the voltage refraction coefficients,${\mathrm{\Gamma }}_{\mathrm{BA}}$ ,${\mathrm{\Gamma }}_{\mathrm{CA}}$ , and${\mathrm{\Gamma }}_{\mathrm{DA}}$ also same.

Therefore, thevalue of voltage refraction coefficient,${\mathrm{\Gamma }}_{\mathrm{BA}}$ is$\left[\frac{2{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}}{{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}+1}\right]$ .

Therefore,the value of voltage refraction coefficient,${\mathrm{\Gamma }}_{\mathrm{CA}}$ is $\left[\frac{2{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}}{{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}+1}\right]$.

Therefore, the value of voltage refraction coefficient,${\mathrm{\Gamma }}_{\mathrm{DA}}$ is $\left[\frac{2{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}}{{\displaystyle \frac{{\mathrm{Z}}_{\mathrm{eq}}}{{\mathrm{Z}}_{\mathrm{A}}}}+1}\right]$.