Energy Diagrams of a Lossless Long Line When Connected to a Constant Voltage

V. D. Klement’ev

For a visual representation of the energy aspect of the process of connecting a lossless long line to a constant voltage, one may use the diagrams described below.

Let us consider the construction of diagrams for the following cases frequently encountered in practice:

1. Connection of a line open at the receiving end (no-load operation).
2. Connection of a short-circuited line.
3. Connection of a line loaded by an active resistance.

1. Connection of a line open at the receiving end

We shall carry out the construction of the energy diagram for this case in detail.

It is known that, when a lossless long line is connected to a constant voltage \(U\), the transient process lasts indefinitely and has a cyclic character with cycle duration:

\[ T = 4\tau = \frac{4l}{v}, \tag{1} \]

where \(l\) is the length of the line, \(v\) is the propagation velocity of electromagnetic energy.

Each cycle consists of four stages.

The energy reserve in the entire line for the instants of time \(t = 0, \tau, 2\tau, 3\tau, 4\tau\) is characterized by the data given in Table 1.

Table 1

Total energy reserve in a line open at the end

For the sake of brevity, let us introduce the notation

\[ W_1=\frac{L_0 l i_0^2}{2}=\frac{C_0 l U_0^2}{2}, \tag{2} \]

where
\(L_0\) — inductance per unit length of the line,
\(C_0\) — capacitance per unit length of the line,
\(l\) — length of the line,
\(U_0, i_0\) — values of the voltage and current of the traveling wave.

The validity of the expressions given in Table 1 is readily seen from Fig. 1, where, for all four stages of the process, the motion of the current and voltage waves along the line is represented.

Fig. 1

We construct the energy diagram in the following way. Above the axis of abscissas we plot the magnitude of one kind of energy (for example, the energy of the electric field), and below it—the other kind of energy (for example, the energy of the magnetic field). The total store of energy \(W_{\mathrm{tot}}=W_{\mathrm{e}}+W_{\mathrm{m}}\) may be plotted in either direction.

The data of the table in Fig. 1 are plotted for the time values:

\[ \tau, 2\tau, 3\tau, 4\tau. \]

During each stage of the process the store of energy \(W_{\mathrm{e}}\) and \(W_{\mathrm{m}}\) changes according to a linear law, since

\[ t=\frac{l}{v}\quad \text{and}\quad l=vt, \]

and therefore

\[ W_{\mathrm{m}}(t)=\frac{L_0 l i_0^2}{2} =\frac{L_0 l t i_0^2}{2}=K_1 t . \]

\[ W_{\mathrm{e}}(t)=\frac{C_0 l U_0^2}{2}=\frac{C_0 v t U_0^2}{2}=i_0^2 t. \]

The change in the stored energy of the electric and magnetic fields as a function of time is indicated in Table 2.

Table 2

Change in the stored energy of a line open at the end

Note. Time is reckoned from the beginning of each stage.

On the basis of the expressions given, a diagram has been constructed (Fig. 1); in the figure, in addition, graphs are given of the distribution of current and voltage along the line.

For the second stage of the process, the term \(\displaystyle \frac{C_0 v t(2U_0)^2}{2}\), corresponding to the reflected voltage wave 2, is shown by the line \(ab\), and the term \(\displaystyle \frac{C_0 v t U_0^2}{2}\), corresponding to the incident voltage wave \(U\), is shown by the line \(cd\).

Thus, the proposed diagram is very simply constructed and gives a clear representation of the energy stored in the entire line, both as the energy of the electric field and also as the total stored energy at any instant of time.

From the diagram, in particular, the direction of the energy flow is visible: during the first half of the cycle the energy is taken from the generator uniformly \(\left(\dfrac{dW_{\mathrm{ob}}}{dt}=\mathrm{const}\right)\), and during the second half of the cycle it is returned to the generator \(\left(\dfrac{dW_{\mathrm{ob}}}{dt}=-\mathrm{const}\right)\).

2. Connecting a Short-Circuited Line to a Constant Voltage

By the method indicated above, an energy diagram is also constructed for the case of connecting a short-circuited line to a constant voltage.

The diagram is constructed on the basis of the data given in Table 3, in which the notation

\[ W_1=\frac{C_0 l U_0^2}{2}=\frac{L_0 l i_0^2}{2}. \]

Table 3

Total energy reserve in the short-circuited line

The process as a whole does not have a cyclic character. Only the reserve of energy of the electric field changes cyclically. The reserve of energy of the magnetic field increases continuously.

Fig. 2

The consumption of energy from the generator also increases continuously. All this is evident from the constructed diagram (Fig. 2).

3. Switching on a line with an active load at the end

Using the indicated method, one can construct energy diagrams for switching on a line with an active load at the end.

Let us consider two cases: 1) \(R_2 = Z_\lambda\); 2) \(R_2 \ne Z_\lambda\).

1) The first case, when \(R_2 = Z_\lambda\), is very simple, since here no reflection of current and voltage waves occurs.

The energy diagram will have the form shown in Fig. 3.

Fig. 3

2) The case \(R_2 \ne Z_\lambda\).

Let us take the regime in which \(R_2 = \dfrac{1}{3} Z_\lambda\), i.e., the reflection coefficient is

\[ n=\frac{R_2-Z_\lambda}{R_2+Z_\lambda}=-\frac{1}{2}. \]

The diagram is constructed on the basis of the data of Table 4, which is compiled according to the propagation graphs of current and voltage waves for the adopted

\[ n=-\frac{1}{2}. \]

Table 4

General store of energy in the line with an active load at the end

At \(t = \infty\), \(W_m = 9W_1\), since \(I_{\mathrm{st}} = 3i_0\),

and \(W_e = W_1\), since \(U_{\mathrm{st}} = U_0\).

The diagram will have the following form.

Fig. 4

As can be seen, the process theoretically continues infinitely long, and the diagram can be constructed for any number of cycles.

For other values of the load magnitude, the energy diagram is constructed in exactly the same way.