Determination of the Characteristic Impedance and Wavelength of Compensated Lines
Abstract Generated abstract
This paper addresses the determination of characteristic impedance and wavelength in lossless compensated power transmission lines, where distributed line parameters are combined with lumped compensating devices. It critiques existing approaches that either ignore the placement of compensating devices or replace asymmetric line structures with an equivalent universal circuit, arguing that these methods are reliable only for uniformly or symmetrically compensated cases. The author proposes defining the characteristic impedance as the repeated impedance of the corresponding four-terminal network and derives formulas using the network coefficients, with wavelength obtained from the resulting voltage relation. Comparative calculations and model experiments indicate that the proposed method agrees with operating conditions for asymmetric compensated lines and shows that the characteristic impedance is generally complex, with its reactive component depending on the type and location of compensation.
Full Text
DETERMINATION OF THE CHARACTERISTIC IMPEDANCE AND WAVELENGTH OF COMPENSATED LINES
O. B. TOLPYGO
(Presented by the scientific seminar of the Faculty of Power Engineering)
The design and construction of the world’s largest electric power transmissions, Kuibyshev—Moscow and Stalingrad—Moscow, required Soviet scientists and engineers to develop a whole series of problems determining the transmission of large amounts of power by alternating current over long distances.
The fundamental problem of long-distance electric power transmission is the stability of parallel operation of electric stations separated by a long line. One of the principal measures making it possible to increase stability is compensation of the inductance and capacitance of the line.
A compensated line has entirely new properties distinguishing it from an uncompensated line. Important characteristics of a compensated line are its characteristic impedance and wavelength, the determination of which is the subject of the present work.
A compensated electric power transmission line is a device combining distributed resistances and conductances (the line) and lumped ones (compensating devices). The properties of such a line will be determined by the totality of all the elements composing it.
In the literature there is no single established definition of the characteristic impedance and wavelength of lossless compensated lines.
As is known [1], the characteristic impedance of a line is found from the expression
\[ Z_{\mathrm{c}}=\sqrt{\frac{L_0}{C_0}}, \]
where \(L_0\) is the inductance and \(C_0\) the capacitance per unit length of the line.
A number of authors [2; 3], on the basis of this formula, propose determining the characteristic impedance of a compensated line by the formulas
\[
Z_{\mathrm{ve}}=\sqrt{\frac{L-L_k}{L}},
\]
if the inductance of the line is compensated,
\[
Z_{\mathrm{ve}}=\sqrt{\frac{L}{C-C_k}},
\]
if the capacitance of the line is compensated,
\[
Z_{\mathrm{ve}}=\sqrt{\frac{L-L_k}{C-C_k}},
\]
if both types of compensation are used—longitudinal and transverse.
Accordingly, the wavelength of a compensated line is determined by the formula
\[ \lambda_{\mathrm{e}}=\sqrt{LC}\sqrt{\left(1-\frac{L_k}{L}\right)\left(1-\frac{C_k}{C}\right)}, \]
where \(L_k\) and \(C_k\) are, respectively, the compensated inductance and capacitance.
This method (let us provisionally call it “method No. 1”) does not take into account the location of the CUs on the line and, in general, can be applied when the compensating devices are uniformly placed along the line at short intervals.
Wolff [1] quite correctly proposed determining the characteristic impedance and the wavelength of a compensated line on the basis of the coefficients of the electrical calculation of the line, since every compensated line can be represented in the form of a multisection four-terminal network, whose voltages and currents are related by the equation
\[ U_1=A_eU_2+B_eI_2, \]
\[ I_1=C_eU_2+D_eI_2. \]
The characteristic impedance of a compensated line, according to Wolff’s method (let us provisionally call it “method No. 2”), is determined by the expression
\[ Z_{be}=\frac{1}{k}\sqrt{\frac{B_e}{C_e}}, \]
where
\[ k=\sqrt{\frac{A_e}{D_e}} \]
is a coefficient that takes into account the asymmetry of the circuit of the compensated line, which is replaced by the so-called universal circuit (Fig. 1).
Fig. 1. Universal equivalent circuit of an electric transmission line.
It seems to us that replacing the actual circuit of the line by such a universal circuit is a departure from reality, since the asymmetry in fact extends over the entire line, and the line does not consist, as shown in the universal circuit, of one asymmetric section (transformer) and a symmetric section (line), whose coefficients \(A\) and \(D\) are equal. Such a departure from reality, as will be shown below, in the case of an asymmetric arrangement of compensating devices leads to an inaccurate determination of the characteristic impedance and wavelength of the line.
In our opinion, the characteristic impedance and wavelength of a compensated line must be determined in another way, set forth below.
It is known [4] that every four-terminal network—and a compensated line is a four-terminal network—is characterized by the so-called iterative impedance, which in the natural operating mode of the line is equal to its characteristic impedance.
For the line we may write
\[ \dot U_1=A_eU_2+B_e\dot I_2, \]
\[ \dot I_1=C_eU_2+D_e\dot I_2. \]
In the natural-power mode, which is possible when the load resistance is equal to the characteristic impedance of the compensated line,
\[ Z_2=Z_{be} \]
the load current is
\[ \dot I_2=\frac{U_2}{Z_{be}}, \]
will be valid
\[ \dot U_1=A_e U_2+B_e\frac{U_2}{Z_{ve}}=U_2\left(A_e+\frac{B_e}{Z_{ve}}\right), \]
\[ \dot I_1=C_e U_2+D_e\frac{U_2}{Z_{ve}}=U_2\left(C_e+\frac{D_e}{Z_{ve}}\right). \tag{2} \]
But, according to the definition of the characteristic impedance as a repeated impedance, for the four-terminal network
\[ Z_{ve}=\frac{\dot U_1}{I_1} \tag{3} \]
and, proceeding from (2),
\[ Z_{ve}=\frac{A_e+\dfrac{B_e}{Z_{ve}}}{C_e+\dfrac{D_e}{Z_{ve}}} \tag{4} \]
or
\[ Z_{ve}=\frac{A_e Z_{ve}+B_e}{C_e Z_{ve}+D_e}, \tag{5} \]
whence
\[ Z_{ve}^2 C_e+Z_{ve}(D_e-A_e)-B_e=0 \tag{6} \]
and finally
\[ Z_{ve}=\frac{-(D_e-A_e)+\sqrt{(D_e-A_e)^2+4B_eC_e}}{2C_e}. \tag{7} \]
After transformations we obtain
\[ Z_{ve}=\frac{-(D_e-A_e)+\sqrt{(D_e+A_e)^2-4}}{2C_e}, \tag{8} \]
(the solution
\[ Z_{ve}=\frac{-(D_e-A_e)-\sqrt{(D_e+A_e)^2-4}}{2C_e} \]
has no physical meaning).
In the case of a compensated line with a symmetrical arrangement of the CU with respect to the ends of the line, \(A=D\), and therefore for such a line
\[ Z_{ve}=\sqrt{\frac{A_e^2-1}{C_e}}. \tag{9} \]
For an uncompensated line the equality
\[ Z_{ve}=Z_v \]
must hold. Indeed,
\[ Z_{ve}=\sqrt{\frac{B}{C}}=\sqrt{\frac{jZ_v\sin\lambda}{j\dfrac{\sin\lambda}{Z_v}}}=Z_v. \]
It follows from expression (2) that
\[ \dot U_1=U_2\left(A_e+\frac{B_e}{Z_{ve}}\right)=U_2 e^{\lambda_e}, \]
and therefore
\[ \lambda_e=\ln\left(A_e+\frac{B_e}{Z_{ve}}\right), \]
whence the wavelength of the compensated line is determined.
Calculations of the operating modes of compensated lines with characteristic impedance determined by our method, carried out in this work for a large number of circuits, give complete agreement of the magnitudes of the voltage and current vectors at the beginning and end of the line, which has also been confirmed experimentally on an experimental setup—a model of a long compensated line.
Below is a table of characteristic impedances and wavelengths for several compensated-line circuits, as determined by all three methods.
Table No. 1
| No. | Type of line circuit | Characteristic impedance, 1st method | Characteristic impedance, 2nd method | Characteristic impedance, 3rd method | Wavelength, 1st method | Wavelength, 2nd method | Wavelength, 3rd method |
|---|---|---|---|---|---|---|---|
| 1 | [[circuit diagram: series inductance–capacitance–inductance section]] | 0.705 | 0.74 | 0.74 | 42°15′ | 46°30′ | 46°30′ |
| 2 | [[circuit diagram: series capacitance–inductance section]] | 0.705 | 1.32 | 0.88∠14°30′ | 42°15′ | 48°50′ | 46°20′ |
| 3 | [[circuit diagram: series inductance–capacitance section]] | 0.705 | 0.65 | 0.88∠14°30′ | 42°15′ | 48°50′ | 46°20′ |
| 4 | [[circuit diagram: cascaded series inductance–capacitance–inductance–capacitance–inductance section]] | 0.705 | 0.745 | 0.745 | 42°15′ | 45°30′ | 45°30′ |
| 5 | [[circuit diagram: line section with shunt inductance at the input and series inductance]] | 1.41 | 0.88 | 1.16∠14°30′ | 42°15′ | 52° | 47° |
| 6 | [[circuit diagram: series inductance section with shunt inductance at an intermediate node]] | 1.41 | 1.34 | 1.34 | 42°15′ | 46°30′ | 46°30′ |
| 7 | [[circuit diagram: shunt inductance at the input with series inductance–capacitance–inductance section]] | 1 | 0.78 | 0.9∠11°10′ | 30° | 39°20′ | 42°50′ |
| 8 | [[circuit diagram: series inductance section with shunt inductance at the output]] | 1.41 | 1.56 | 1.16∠14°30′ | 42°15′ | 52° | 47° |
As can be seen from the table, the values of the characteristic impedances obtained by method No. 2 and by method No. 3, proposed by us, differ substantially for all asymmetrical circuits. The characteristic impedances are the same only for symmetrical circuits.
As the asymmetry of the circuits increases, the difference increases significantly. The difference in the values of the characteristic impedances is especially large for circuits with an end placement of the compensating devices.
The value of the characteristic impedance is complex, consisting of an active component and a reactive component, the sign of which depends on the location of the compensating device on the line and on the type of compensating device. Moreover, when the compensating device in circuits with single compensation is located before or after the midpoint of the line, the sign of the reactive component changes.
On the basis of our calculations, confirmed by experiment on a model of a long line, the following conclusions may be drawn.
-
Determination of the characteristic impedance and wavelength of a compensated line according to the methods described in the literature is inaccurate and does not take into account the special features of the combination of distributed and lumped line constants.
-
The method proposed by Wulff takes into account the specific character of the combination of distributed and lumped parameters, but gives correct values of the characteristic impedance and wavelength only for the case of a symmetrical arrangement of the compensating devices relative to the ends of the line. Allowance for asymmetry by introducing the auxiliary parameter \(k = \sqrt{\dfrac{A}{D}}\) gives results that are not confirmed by experiment and are incorrect in principle.
-
The method for determining \(Z_{be}\) as the repeated impedance of a four-terminal network, proposed by the author, reflects all the features of compensated lines and gives results confirmed by experiment.
-
The value of the characteristic impedance of compensated lines in the general case is a complex quantity, the reactive component of which depends both in magnitude and in sign on the type of compensating device and on its location on the line.
References
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Wulff A. A. The Problem of Transmitting Electrical Energy over Ultra-Long Distances by Compensated Lines. Gosenergoizdat, 1941.
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Zalesskii A. M. Transmission of Electrical Energy. Gosenergoizdat, 1948.
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Rosenberg R. Electrical Transmission of Large Powers over Long Distances. Energoizdat, 1934.
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Kalantarov P. L. and Neiman L. R. Theoretical Foundations of Electrical Engineering. Gosenergoizdat, 1951.