Dependence of Transformer Power on Short-Circuit Voltage
Abstract Generated abstract
The paper examines how a transformer’s rated limb power depends on short-circuit voltage, treated through the reactive component of short-circuit resistance. Using standard transformer design relations, the authors derive an expression linking limb power to core dimensions, winding geometry, magnetic induction, current density, and leakage-channel parameters, then calculate representative values for large transformers with different interwinding distances and winding arrangements. The results indicate that, for fixed principal dimensions, limb power is directly proportional to short-circuit resistance, while small resistance values lead to poor utilization of active materials. The authors argue that an optimum short-circuit resistance should be found by techno-economic comparison of transformer cost and reactive-power compensation, and they note that increased resistance does not appear to be limited by radial electrodynamic winding stresses in the considered range.
Full Text
Dependence of Transformer Power on Short-Circuit Voltage
I. D. Kutyavin, S. A. Stepanov, L. I. Voronova
The article presents the results of a study of the dependence of the nominal power of a transformer on its short-circuit voltage (resistance).
The short-circuit resistance of a transformer is a very important parameter. It is known that reducing this resistance improves the conditions for maintaining voltage constancy in the system, reduces reactive-power losses, and increases the dynamic stability of the system. At the same time, however, the short-circuit currents in the system increase, and therefore the operating conditions of electrical equipment deteriorate. A techno-economic evaluation of these factors might make it possible to determine the optimum value of the short-circuit resistance of transformers, but this problem is very complex and cannot be considered within the scope of the present article.
To clarify the dependence under study, we shall use the expression for the power per transformer limb, borrowed from [1]:
\[ S_c = k q_c B_c q_m \Delta \ \mathrm{kVA}, \tag{1} \]
\[ k = 4.44 f \cdot 10^{-7}, \tag{2} \]
where
\(q_c\) and \(q_m\) are the cross-sectional areas of the limb steel and of the winding copper \((q_m = w q_{\text{п}})\), \(w\) is the number of turns, and \(q_{\text{п}}\) is the cross-sectional area of the winding conductor;
\(B_c\) and \(\Delta\) are the induction in the limb, T, and the current density in the winding, A/cm\(^2\);
\[ q_c = k_c \frac{\pi d^2}{4}, \tag{3} \]
where
\(d\) is the diameter of the circle circumscribed around the limb,
\(k_c\) is the coefficient of filling of the limb with steel.
The reactive component of the short-circuit resistance in relative units [1]:
\[ x_p = \frac{2 k_p q_m \Delta l_m \gamma'}{3.33 B_c q_c h} \cdot 10^{-4}, \tag{4} \]
where
\(k_p\) — Rogovskii coefficient;
\(h\) — axial height of the windings (Fig. 1);
\(l_m\) — length of the mean turn of the windings (leakage channel)
\[ l_m=\pi(d+2\delta_{01}+2b_1+\delta_{12}), \tag{5} \]
\(\delta'\) — reduced width of the leakage channel
\[ \delta'=b_1+b_2+3\delta_{12}; \tag{6} \]
\(\delta_{01}\) and \(\delta_{12}\) — radial distances from the limb to the winding and between windings;
\(b_1\) and \(b_2\) — radial widths of the winding sections.
Fig. 1
Substituting the expression for the product \(q_m\Delta\) from (4) into (1), we find
\[ S_c=C\frac{(q_cB_c)^2x_ph}{l_m\delta'}\kappa\theta a, \tag{7} \]
\[ C=k\frac{3.33\cdot10^4}{2k_p}=7.4\frac{f}{k_p}10^{-3}, \tag{8} \]
where:
\(f\) — number of current periods, Hz.
It is evident from expression (7) that the limb power, for given transformer dimensions, is directly proportional to \(x_p\).
For a quantitative representation of the dependence \(S_c(x_p)\), the dimensions of a transformer close to the limiting power were adopted, namely (dimensions in centimeters):
\(d=140;\quad h=250;\quad b_1=8;\quad b_2=10;\)
\(\delta_{01}=3;\quad \delta_{12}=5;\ 10;\ 20;\quad k_p=0.95;\)
\(k_c=0.85;\quad B_c=1.75\) T. Windings: primary — helical, multistart; secondary — continuous coil. Winding material — copper.
In items 1–4 of the table, the values of the limb power in MVA are given, calculated from (7) in the form of the dependence \(S_c(x_p,\delta_{12})\). The values
Table
| No. | \(\delta_{12}\) | \(x_p=0.03\) | \(x_p=0.05\) | \(x_p=0.10\) | \(x_p=0.15\) | \(x_p=0.20\) |
|---|---|---|---|---|---|---|
| 1 | 5 | 88 | 147 | 295 | 442 | 590 |
| 2 | 10 | 59 | 99 | 198 | 296 | 395 |
| 3 | 15 | 44 | 73 | 146 | 218 | 292 |
| 4 | 20 | 34 | 57 | 114 | 172 | 229 |
| 5 | 5 | 66 | 109 | 218 | 327 | 435 |
\(\delta_{12}=5\div20\) correspond approximately to the rated voltages of the secondary winding (LV) within the range 110–750 kV.
Item 5 gives the dependence \(S_c(x_p)\) for a multilayer secondary winding with the number of layers \(m=8\), with the distance between the copper co-
of the middle layers \(\delta=2.4\) cm and the radial thickness of the copper of a layer \(x=0.6\) cm. With these data, the radial thickness of the secondary winding is \(b_2=21.6\) cm. For a multilayer winding, one end of the inner layer is grounded; therefore the value \(\delta_{12}=5\) cm and does not depend on the voltage of this winding.
For all the powers \(S_c\) given in items 1–4 of the table, the principal overall dimensions of the transformer \((d, h, b_1, b_2)\) remain unchanged; therefore, at small values of \(x_p\), the active materials are used poorly, especially the core steel. But as \(x_p\) increases, the expenditures of active materials and of funds for compensation of reactive power will increase. Consequently, there must exist an optimum value of \(x_p\), which can be determined by carrying out a techno-economic assessment of the calculated costs for the transformer and for compensation of reactive power.
Let us also clarify the possible limitations on a wide variation of \(x_p\) from the standpoint of electrodynamic forces. For an approximate estimate of the pattern of dependence of the stress in the winding copper on radial forces, we shall use the formula given in [2]:
\[ \sigma_p=6.4\frac{(i_k w)^2 l_m k_p}{2\pi q_m h}\,10^{-8},\ \text{kg}/\text{cm}, \tag{9} \]
\[ (i_k w)=\frac{1.8\sqrt{2q_m\Delta}}{x_p}. \tag{10} \]
Substituting the expressions \((i_k w)\) and \(x_p\) into (9), we find
\[ \sigma_p=18.2\frac{(B_c q_c)^2 h}{q_m l_m k_p(\delta')^2}. \tag{11} \]
The numerator of expression (11) in our problem is constant, while the denominator increases with increasing \(x_p\) and \(\delta'\), because with increasing \(x_p\) the power \(S_c\) and, consequently, \(q_m\) increase; this indicates the absence of limitations due to electrodynamic forces when \(x_p\) is increased.
LITERATURE
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I. D. Kutyavin, L. I. Del. On the limiting power of transformers. Izv. TPI, vol. 172, 1967.
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P. M. Tikhomirov. Calculation of Transformers. “Energiya,” 1968.