On the Design Parameters and Operating Modes of Autotransformers
Abstract Generated abstract
This paper examines design parameters and operating modes of three winding autotransformers, including block step-up units, step-up units with medium-voltage primaries, and step-down units with high-voltage primaries. Using approximate formulas based on equal current density, it derives winding currents, design powers, conductor cross sections, copper weight, and copper losses as functions of transformation ratio and relative winding loading. Tabulated comparisons with equivalent three-winding transformers indicate that autotransformer windings generally require less total design power and copper, with associated reductions in losses and core dimensions, although block autotransformers can show higher copper losses under some load distributions. The analysis supports replacing three-winding transformers with autotransformers where medium- and high-voltage networks have compatible neutral operating modes.
Full Text
ON THE DESIGN PARAMETERS AND OPERATING MODES OF AUTOTRANSFORMERS
I. D. Kutyavin, V. A. Zorin, D. I. Anderman
In the design and operation of autotransformers (AT), it becomes necessary to determine their design parameters and operating modes. In particular, it is desirable to know the distribution of currents in the windings under different loading in the mode of a three-winding transformer, the design currents and powers of the windings, the ratios of conductor cross sections, copper weights, losses in the windings, etc.
Below, in the mode of a three-winding transformer, the following are considered:
a) step-up block ATs (Fig. 1a) with a low-voltage primary winding (LV),
b) step-up ATs with a medium-voltage primary winding (MV) (Fig. 1b),
c) step-down ATs with a high-voltage primary winding (HV) (Fig. 1c).
Fig. 1
The conventional designations of the currents are shown in the diagrams of Fig. 1. The design powers, copper weights, and losses are accompanied by the same indices.
If, for a preliminary approximate estimate of the parameters of an AT, the same current density \(\Delta\) is assumed for all windings, then the cross sections of the winding conductors will be proportional to their calculated currents and are found from the expression
\[ q_n=\frac{I_p}{\Delta}=\frac{S_p}{\sqrt{3}U\Delta}. \tag{1} \]
The copper cross-sectional area of the entire winding with number of turns \(w\) is
\[ q_m=q_n w=\frac{S_p}{\sqrt{3}E\Delta}. \tag{2} \]
The weight of copper in the winding is
\[ Q_m=q_m l\gamma=\frac{l\gamma S_p}{\sqrt{3}E\Delta}. \tag{3} \]
The copper losses in the windings are
\[ P_m=a\Delta^2 Q_m=a\Delta \frac{l\gamma S_p}{\sqrt{3}E}; \qquad a=\frac{\rho}{\gamma}, \tag{4} \]
where
\[ \begin{aligned} S_p &\text{ is the calculated power of the winding,}\\ l &\text{ is the length of the mean turn of the winding,}\\ E &\text{ is the emf per turn,}\\ \gamma &\text{ is the specific density, and}\\ \rho &\text{ is the specific resistance of the winding material.} \end{aligned} \]
The calculated power of the HV windings of all three types of AT is usually equal to the through power \(S_p\); the MV winding for the first type of AT may have a calculated power within \((0.5 \div 1.0)S_p\), and for the latter two types it is equal to the through power \(S_p\). The calculated power of the LV winding for the first type is equal to the through power, and for the latter two types may lie within \((0.25 \div 1.0)S_p\).
The calculated power of the additional (series) part of the autotransformer (AT) winding can be determined from the expression
\[ S_{pg}=\sqrt{3}I_g(U_v-U_c)=S_n(1-N), \tag{5} \]
where
\[ N \]
is the transformation ratio of the AT winding, equal to the ratio of the medium voltage \(U_c\) to the high voltage \(U_v\).
The calculated power of the common (parallel) part of the AT winding (Fig. 1) for a block AT is
\[ S_{p01}=\sqrt{3}U_c(I_c+I_v) =U_c\left[\frac{p_c S_n}{U_c}-\frac{(1-p_c)S_n}{U_v}\right] =S_n\left[p_c+N(1-p_c)\right]. \tag{6} \]
The same for a step-up AT with primary MV winding is
\[ S_{p02}=\sqrt{3}U_c(I_c-I_v) =U_c\left[\frac{S_n}{U_c}-\frac{S_n(1-p_n)}{U_v}\right] =S_n\left[1-N(1-p_n)\right]. \tag{7} \]
The same for a step-down AT with primary HV winding is
\[ S_{p03}=\sqrt{3}U_c(I_c-I_v) =U_c\left[\frac{S_n(1-p_n)}{U_c}-\frac{S_n}{U_v}\right] =S_n(1-p_n-N), \tag{8} \]
where
\[ \begin{aligned} p_c &\text{ is the relative calculated power of the MV winding,}\\ p_n &\text{ is the same for the LV winding.} \end{aligned} \]
Table 1 gives the dependence of the ratio of the calculated power of the common part of the AT winding \(S_{p0}\) to the through power \(S_p\) on the variable parameters \(P\) and \(N\), calculated from expressions (6)--(8). For block ATs (items
1–6) the design power \(S_{\mathrm{po1}}\) increases with increasing \(p_{\mathrm{c}}\) and \(N\). The total design power of the windings of a block AT can be computed, taking into account (5) and (6), from the expression
\[ S_{\mathrm{c1}} = S_{\mathrm{n}} + S_{pg} + S_{\mathrm{po1}} = S_{\mathrm{n}}\bigl[2 + p_{\mathrm{c}}(1 - N)\bigr], \tag{9} \]
Table 1
| Type of AT | Item no. | \(p \backslash N\) | \(1/4\) | \(1/3\) | \(1/2\) | \(2/3\) | \(3/4\) | \(1,0\) |
|---|---|---|---|---|---|---|---|---|
| Block | 1 | 0,00 | 0,25 | 0,33 | 0,50 | 0,67 | 0,75 | 1,00 |
| Block | 2 | 0,25 | 0,44 | 0,50 | 0,63 | 0,75 | 0,81 | 1,00 |
| Block | 3 | 0,50 | 0,63 | 0,67 | 0,75 | 0,83 | 0,88 | 1,00 |
| Block | 4 | 0,60 | 0,70 | 0,73 | 0,80 | 0,87 | 0,90 | 1,00 |
| Block | 5 | 0,75 | 0,81 | 0,83 | 0,88 | 0,92 | 0,94 | 1,00 |
| Block | 6 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 |
| Step-up | 7 | 0,00 | 0,75 | 0,67 | 0,50 | 0,33 | 0,25 | 0 |
| Step-up | 8 | 0,25 | 0,81 | 0,75 | 0,63 | 0,50 | 0,44 | 0,25 |
| Step-up | 9 | 0,50 | 0,88 | 0,83 | 0,75 | 0,67 | 0,62 | 0,50 |
| Step-up | 10 | 0,60 | 0,90 | 0,88 | 0,80 | 0,73 | 0,70 | 0,60 |
| Step-up | 11 | 0,75 | 0,94 | 0,92 | 0,88 | 0,83 | 0,81 | 0,75 |
| Step-up | 12 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 |
| Step-down | 13 | 0,00 | 0,75 | 0,67 | 0,50 | 0,33 | 0,25 | 0 |
| Step-down | 14 | 0,25 | 0,50 | 0,42 | 0,25 | 0,08 | 0 | 0,25 |
| Step-down | 15 | 0,50 | 0,25 | 0,17 | 0 | 0,17 | 0,25 | 0,50 |
| Step-down | 16 | 0,60 | 0,15 | 0,07 | 0,10 | 0,27 | 0,35 | 0,60 |
| Step-down | 17 | 0,75 | 0 | 0,08 | 0,25 | 0,42 | 0,50 | 0,75 |
| Step-down | 18 | 1,00 | 0,25 | 0,33 | 0,50 | 0,67 | 0,75 | 1,00 |
For step-up ATs (items 7–12), the power \(S_{\mathrm{po2}}\) decreases with increasing \(N\) and increases with increasing \(p_{\mathrm{n}}\). The total power of the windings of an AT of this type, taking into account (5) and (7), is
\[ S_{\mathrm{c2}} = p_{\mathrm{n}}S_{\mathrm{n}} + S_{pg} + S_{\mathrm{po2}} = S_{\mathrm{n}}\bigl[2(1 - N) + p_{\mathrm{n}}(1 + N)\bigr]. \tag{10} \]
For step-down ATs (items 13–18), the dependence of \(S_{\mathrm{po3}}\) on \(p_{\mathrm{n}}\) and \(N\) proves to be more complex, because the power \(S_{\mathrm{po3}}\) may have a negative sign and its algebraic summation with the powers of the other windings will lead to an error. From Table 1 (items 13–18) it is seen that the largest value of the power \(S_{\mathrm{po3m}}\) (which is also the design value) occurs at \(p_{\mathrm{n}} = 0\) or 1. Then the total design power of the windings of a step-down AT is
\[ S_{\mathrm{c3}} = p_{\mathrm{n}}S_{\mathrm{n}} + S_{pg} + S_{\mathrm{po3m}}. \tag{11} \]
The total design power of the windings of a three-winding transformer, when the design power of one of the secondary windings is \(pS_{\mathrm{n}}\), is
\[ S_{\mathrm{ct}} = S_{\mathrm{n}}(2 + p). \tag{12} \]
The data of Table 2 for the total design powers of AT windings in relative units, computed from (9)—(12) for \(p = 0{,}5\) and 1,0, give an idea of the copper consumption for manufacturing the windings of ATs and three-winding transformers of the same throughput power, since, in accordance with (3), copper consumption is proportional to the design power of the winding. The copper saving in manufacturing AT windings that follows from Table 2 is underestimated, since it corresponds to the same length of the mean turn \(l\), whereas for AT windings the value of \(l\) will be smaller. In addition, the window dimensions are reduced and, consequently, the weight of steel; losses in copper and steel are reduced.
Table 2
| No. | Designation | $P \backslash N$ | $\dfrac{1}{4}$ | $\dfrac{1}{3}$ | $\dfrac{1}{2}$ | $\dfrac{2}{3}$ | $\dfrac{3}{4}$ | $1$ |
|---|---|---|---|---|---|---|---|---|
| 1 | $S_{c1}$ | 0,5 | 2,38 | 2,33 | 2,25 | 2,17 | 2,13 | 2,00 |
| 2 | $S_{c2}$ | » | 2,13 | 2,00 | 1,75 | 1,5 | 1,38 | 1,00 |
| 3 | $S_{c3}$ | » | 2,00 | 1,84 | 1,5 | 1,16 | 1,00 | 0,50 |
| 4 | $S_{\mathrm{ct}}$ | » | 2,50 | 2,50 | 2,50 | 2,50 | 2,50 | 2,50 |
| 5 | $S_{c1}$ | 1,0 | 2,75 | 2,67 | 2,50 | 2,33 | 2,25 | 2,00 |
| 6 | $S_{c2}$ | » | 2,75 | 2,67 | 2,50 | 2,33 | 2,25 | 2,00 |
| 7 | $S_{c3}$ | » | 2,50 | 2,33 | 2,00 | 1,67 | 1,50 | 1,00 |
| 8 | $S_{\mathrm{ct}}$ | » | 3,00 | 3,00 | 3,00 | 3,00 | 3,00 | 3,00 |
| 9 | $2S_{c3}$ | 0,5 | 4,00 | 3,68 | 3,00 | 2,33 | 2,00 | 1,00 |
| 10 | — | — | 1,25 | 1,00 | 0,50 | 0 | −0,25 | −1,00 |
Because of the absence of block autotransformers at present, step-down autotransformers of double capacity with $p_{\mathrm{н}}=0{,}5$ are used for blocks. The total capacity of the autotransformer windings, found by doubling the data of item 3 (Table 2), is given in item 9. The overuse of copper in this case will be characterized by the difference between the data of items 9 and 5 and is given in item 10.
When determining the total copper losses in the autotransformer windings from expression (4), it is necessary to take into account the distribution of load between the secondary windings. Suppose that the relative loading of the LV winding of the block autotransformer is $x$ fractions of unity; then the relative loading of the HV winding will be $(1-x)$.
As a guide, the relative total copper losses in the windings of a block autotransformer are determined from the expression
\[ P_{\mathrm{mx}1}=1+(1-x)^2(1-N)+\left(\frac{I_c}{I_{\mathrm{po}1}}\right)^2[x+N(1+x)]^2. \tag{13} \]
For a step-up autotransformer with the primary LV winding,
\[ P_{\mathrm{mx}2}=\frac{x^2}{p_{\mathrm{н}}}+(1-x)^2(1-N)+\left(\frac{I_c}{I_{\mathrm{po}2}}\right)^2\{1-N(1-x)\}^2. \tag{14} \]
For a step-down autotransformer with the primary HV winding,
\[ P_{\mathrm{mx}3}=\frac{x^2}{p_{\mathrm{н}}}+(1-N)+\left(\frac{I_c}{I_{\mathrm{po}3}}\right)^2(1-x-N)^2. \tag{15} \]
For a three-winding transformer,
\[ P_{\mathrm{mx}}=1+\frac{x^2}{p_{\mathrm{н}}}+\frac{(1-x)^2}{p_c}. \tag{16} \]
The design current of the common part of the autotransformer winding (by which the cross section of the winding conductor is determined), for a block autotransformer, taking (6) into account, is
\[ I_{\mathrm{po}1}=\frac{S_{\mathrm{n}}}{\sqrt{3}U_c}[p_c+N(1-p_c)]. \tag{17} \]
The same for a step-up autotransformer, taking (7) into account,
\[ I_{\mathrm{po}2}=\frac{S_{\mathrm{n}}}{\sqrt{3}U_c}[1-N(1-p_{\mathrm{н}})]. \tag{18} \]
The same for a step-down autotransformer, taking (8) into account,
\[ I_{\mathrm{po}3}=\frac{S_{\mathrm{n}}}{\sqrt{3}U_c}(1-N-p_{\mathrm{н}}). \tag{19} \]
Table 3 shows the ratio of losses in the windings of autotransformers (items 1–18) and of a three-winding transformer (item 19) for a combination of through capacities of the windings of 100/100/100%. The total relative
power losses in the AT windings were calculated from expressions (13)–(15), and for the three-winding transformer from (16) at \(p=1\).
Table 3
| AT type | Item no. | \(x/N\) | 0,0 | 0,2 | 0,4 | 0,5 | 0,6 | 0,8 | 1,0 |
|---|---|---|---|---|---|---|---|---|---|
| Block | 1 | \(1/2\) | 1,81 | 1,64 | 1,57 | 1,58 | 1,61 | 1,75 | 2,00 |
| Block | 2 | \(1/3\) | 1,78 | 1,65 | 1,60 | 1,62 | 1,65 | 1,77 | 2,00 |
| Block | 3 | \(1/2\) | 1,75 | 1,68 | 1,67 | 1,69 | 1,72 | 1,83 | 2,00 |
| Block | 4 | \(2/3\) | 1,78 | 1,75 | 1,76 | 1,77 | 1,80 | 1,90 | 2,00 |
| Block | 5 | \(3/4\) | 1,81 | 1,80 | 1,81 | 1,82 | 1,85 | 1,93 | 2,00 |
| Block | 6 | 1,0 | 2,00 | 2,00 | 2,00 | 2,00 | 2,00 | 2,00 | 2,00 |
| Step-up | 7 | \(1/4\) | 1,31 | 1,16 | 1,15 | 1,21 | 1,29 | 1,57 | 2,00 |
| Step-up | 8 | \(1/3\) | 1,12 | 1,02 | 1,06 | 1,11 | 1,22 | 1,54 | 2,00 |
| Step-up | 9 | \(1/2\) | 0,75 | 0,72 | 0,83 | 0,94 | 1,08 | 1,47 | 2,00 |
| Step-up | 10 | \(2/3\) | 0,44 | 0,47 | 0,64 | 0,78 | 0,95 | 1,40 | 2,00 |
| Step-up | 11 | \(3/4\) | 0,31 | 0,36 | 0,55 | 0,70 | 0,89 | 1,37 | 2,00 |
| Step-up | 12 | 1,0 | 0,00 | 0,08 | 0,32 | 0,50 | 0,72 | 1,28 | 2,00 |
| Step-down | 13 | \(1/4\) | 1,31 | 1,09 | 1,03 | 1,06 | 1,13 | 1,39 | 1,81 |
| Step-down | 14 | \(1/3\) | 1,12 | 0,93 | 0,90 | 0,95 | 1,04 | 1,33 | 1,78 |
| Step-down | 15 | \(1/2\) | 0,75 | 0,63 | 0,67 | 0,75 | 0,85 | 1,23 | 1,75 |
| Step-down | 16 | \(2/3\) | 0,44 | 0,39 | 0,49 | 0,61 | 0,76 | 1,19 | 1,77 |
| Step-down | 17 | \(3/4\) | 0,31 | 0,31 | 0,43 | 0,56 | 0,73 | 1,19 | 1,81 |
| Step-down | 18 | 1,0 | 0,00 | 0,018 | 0,32 | 0,50 | 0,72 | 1,28 | 2,00 |
| Three-winding transformer | 19 | — | 2,00 | 1,68 | 1,52 | 1,50 | 1,52 | 1,68 | 2,00 |
As can be seen from Table 3, the step-up and step-down ATs, for all ratios of the loads of the secondary windings, have substantially lower copper losses than the three-winding transformer. Only the block AT, when the LV winding is loaded within the range of (40–80)%, has somewhat greater losses owing to the reduced weight of copper in the windings.
Despite the approximate and incomplete nature of the study presented, the very substantial advantage of replacing three-winding transformers with autotransformers is confirmed. Such a replacement can be made in all cases where the medium- and higher-voltage networks have the same neutral operating mode.