Dependence of Transformer Power on Short-Circuit Voltage
Abstract Generated abstract
The article examines how the nominal power of a transformer depends on its short-circuit impedance, a parameter that affects voltage regulation, reactive losses, system stability, and short-circuit current levels. Using standard design relations for transformer limb power and leakage reactance, the authors derive an expression showing that, for fixed principal dimensions, core power is directly proportional to the reactive component of short-circuit impedance. Numerical calculations for a large transformer geometry illustrate this dependence for different interwinding distances and winding types, showing reduced active-material utilization at low impedance and increasing material and compensation costs at higher impedance. The analysis also estimates electrodynamic stresses and suggests that increasing impedance is not limited by radial force stresses, while identifying techno-economic optimization as the basis for selecting an appropriate impedance value.
Full Text
Dependence of Transformer Power on Short-Circuit Voltage
I. D. Kutyavin, S. A. Stepanov
The article presents the results of a study of the dependence of the nominal power of a transformer on its short-circuit voltage (impedance).
The short-circuit impedance of a transformer is a very important parameter. It is known that reducing this impedance 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, short-circuit currents in the system increase, and therefore the operating conditions of electrical equipment deteriorate. A techno-economic evaluation of these factors would possibly make it possible to determine the optimum value of the short-circuit impedance of transformers, but this problem is very complex and cannot be considered within the scope of the present article.
To clarify the dependence under investigation, we shall use the expression for the power per transformer limb, borrowed from [1]:
\[ S_c = k \cdot q_c \cdot B_c \cdot q_m \cdot \Delta \ \text{kVA}, \tag{1} \]
where
\[ k = 4.44 f \cdot 10^{-7}, \tag{2} \]
\(q_c\) and \(q_m\) are the cross-sectional areas of the limb steel and of the winding copper \((q_m = w q_{\mathrm{pr}})\), \(w\) is the number of turns, \(q_{\mathrm{pr}}\) is the cross-sectional area of the winding conductor, and \(B_c\) 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 impedance in relative units [1] is
\[ x_p = \frac{2 k_p q_m \Delta l_m \delta'}{3.33 q_c \cdot B_c \cdot h} \cdot 10^{-4}, \tag{4} \]
where
\(k_p\) is Rogowski’s coefficient;
\(h\) is the axial height of the windings (Fig. 1);
\(l_m\) is the length of the mean turn of the windings (leakage channel);
Fig. 1
\[ l_{\mathrm{M}}=\pi\left(d+2\delta_{01}+2b_1+\delta_{12}\right); \tag{5} \]
\(\delta'\) is the reduced width of the leakage channel;
\[ \delta'=b_1+b_2+3\delta_{12}; \tag{6} \]
\(\delta_{01}\) and \(\delta_{12}\) are the radial distances from the core to the winding and between windings;
\(b_1\) and \(b_2\) are the radial widths of the winding sections.
Substituting the expression for the product \(q_{\mathrm{M}}\Delta\) from (4) into (1), we find
\[ S_{\mathrm{c}}=C\,\frac{\left(q_{\mathrm{c}}B_{\mathrm{c}}\right)^2x_{\mathrm{p}}\cdot h}{l_{\mathrm{M}}\delta'}\ \text{kVA}, \tag{7} \]
\[ C=K\,\frac{3.33\cdot10^{-4}}{2k_{\mathrm{p}}} =7.4\,\frac{f}{k_{\mathrm{p}}}\,10^{-3}, \tag{8} \]
where \(f\) is the number of current periods, cps.
It follows from expression (7) that the core power, for given transformer dimensions, is directly proportional to \(x_{\mathrm{p}}\).
For a quantitative representation of the dependence \(S_{\mathrm{c}}(x_{\mathrm{p}})\), the dimensions of a transformer close to the limiting power were adopted, namely (dimensions in centimeters): \(d=140\); \(h=250\); \(b_1=8\); \(b_2=10\); \(\delta_{01}=3\); \(\delta_{12}=5;\ 10;\ 15;\ 20\); \(k_{\mathrm{p}}=0.95\); \(k_{\mathrm{c}}=0.85\); \(B_{\mathrm{c}}=1.75\ \text{T}\). Windings: the primary is a multi-start helical winding, the secondary is a continuous disk winding.
Items 1—4 of Table 1 give the core-power values in MVA, calculated from (7) in the form of the dependence \(S_{\mathrm{c}}(x_{\mathrm{p}},\delta_{12})\). Values
Table
| No. | \(x_{\mathrm{p}}\) | 0.03 | 0.05 | 0.10 | 0.15 | 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 |
\(\delta_{12}=5 \div 20\) correspond approximately to nominal voltages of the secondary winding (LV) in the range 110–750 kV.
Item 5 gives the dependence \(S_c(x_p)\) for a multilayer secondary winding with the number of layers \(m=8\), the distance between the copper of adjacent 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 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 utilization of active materials is poorer, especially that of the core steel. But as \(x_p\) increases, the costs of active materials and the costs for reactive-power compensation 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 reactive-power compensation.
Let us also determine possible limitations on a wide variation of \(x_p\) from the standpoint of electrodynamic forces. For an approximate estimate of the regularity of the dependence of the stress in the copper of the windings on the radial forces, we use the formula given in [2]:
\[ \sigma_p = 6.4 \frac{(i_k w)^2 l_m k_p}{2\pi q_m h}\cdot 10^{-8}, \tag{9} \]
where
\[ (i_k w)=\frac{1.8\sqrt{2}\,\Delta q_m}{x_p}. \tag{10} \]
Substituting the expressions for \((i_k w)\) 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 as \(x_p\) increases the power \(S_c\), and consequently \(q_m\), increases. This indicates the absence of limitations due to electrodynamic forces when \(x_p\) is increased.
REFERENCES
- I. D. Kutyavin, L. I. Del. On the limiting power of transformers. Izv. TPI, vol. 172. Tomsk, TTU Publishing House, 1967.
- P. M. Tikhomirov. Transformer Design. “Energiya,” 1968.