ANALYSIS OF DESIGN VARIANTS OF SINGLE-PHASE TRPSH AND THE POSSIBILITIES OF THEIR APPLICATION IN AUTOMATIC INSTALLATIONS

V. P. OBRUSNIK, V. G. KISELEV, Z. N. SOBOLEVA

(Presented by the Scientific and Technical Seminar of the Static Converters Department of the Scientific Research Institute of AEM at TPI)

As is known, single-phase transformers regulated by shunt magnetization (TRPSH) have more than twenty design varieties. The possibilities for their application in automatic installations of various purposes will be clearer if comparative evaluation data for these devices are available in terms of energy and specific-weight indices. In the literature [1, 2, 3], comparisons of TRPSH have been carried out on the basis of circuit and design solutions, as well as according to their energy characteristics. In the authors’ opinion, questions of an in-depth analysis of these devices according to weight and overall-dimension indices have been insufficiently covered.

On the basis of earlier investigations [1], a number of designs from the entire diversity of single-phase TRPSH can be excluded from consideration as having clearly inferior technical-and-economic indices. Among those made on non-twisted magnetic cores with single-sided magnetization, the TRPSH designs shown in Figs. 1 and 2 deserve particular attention. Their comparative evaluation in the present article is carried out on the basis of the method for determining the geometrical dimensions of TRPSH proposed in [4].

If, in contrast to the classical theory of investigating the optimal geometry of ferromagnetic devices (FD), the square root of the cross-sectional area of the latter, rather than the width of the laminations of the main magnetic core, is taken as the base quantity, then the principal dimensions of the magnetic core and coils can be expressed by means of the relations:

\[ a_o=\sqrt{\frac{S_o}{k_s}}, \qquad b_o=\sqrt{k_s S_o}, \qquad h_o=k_h\sqrt{S_o}, \qquad C=k_c\sqrt{S_o}; \tag{1} \]

where

\[ \begin{aligned} a_o,\, b_o,\, h_o &\text{ are the principal dimensions of the main magnetic core,}\\ k_s,\, k_h,\, k_c &\text{ are variable coefficients,}\\ C &\text{ is the base thickness of the coils.} \end{aligned} \]

It is convenient to take as the base thickness of the coils \(C\) the thickness of the outer winding embracing the legs of the main magnetic core and the shunt. If there are gaps between the windings, the latter can also be expressed by means of constant coefficients and the base thickness \(C\).

For the designs shown in Figs. 1 and 2, the value of some gaps is taken equal to \(\delta=0.5C\), and in other cases \(\delta=1C\). The distance between the main magnetic core and the shunt magnetic core is chosen on the basis of design considerations and is also expressed in fractions of the base quantity \(C\).

Fig. 1a

Design 3

Design 4

Fig. 1b

Fig. 1c

Fig. 2a

Design 7

Labels in the diagram: \(W_{\mathrm{н}}\), \(W_{\mathrm{б}}\), \(W_{\mathrm{а}}\); \(b_0\); \(0.5c\); \(0.5b_{\mathrm{ш}}\); \(a_{\mathrm{ш}}\); \(h_0-h_{\mathrm{ш}}\).

a)

Design 8

Labels in the diagram: \(W_{\mathrm{а}}\), \(W_{\mathrm{н}}\), \(W_{\mathrm{б}}\); \(b_0-b_{\mathrm{ш}}\); \(0.5c\); \(a_0\); \(a_{\mathrm{ш}}\); \(h_0\); \(h_{\mathrm{ш}}\).

Fig. 2b

Fig. 2c

Fig. 2г

The encompassing winding may be either the primary winding of the TRPSH or the secondary winding, and, taking into account that \(I_1 w_1 \simeq I_2 w_2\), it would be possible to take the thicknesses of the inner and outer coils of the working windings to be identical, equal to \(C\). But since the cooling of the inner coils is poorer, and therefore, as a rule, the current density in them is approximately 10% lower, it may be assumed that the coefficient relating the thickness of the inner coils and the base thickness \(C\) is equal to 1.1. The coefficient for determining the thickness of the bias-winding coils is equal to 1.5, since it follows from the condition:

\[ I_d w_d = 1.5 I_{\text{outer}} W_{\text{outer}} \]

Let us emphasize that in all subsequent formulas and reasoning it is precisely the quantity \(C\) that will be used, being the thickness of the outer working winding and the base quantity for the relative expression of the thicknesses of all the other coils, air gaps, and windows. This quantity, in comparison with the window width, is more convenient for composing the initial equations and for interpreting the geometry of the magnetic amplifier, described mathematically, which is especially noticeable when the device has a complex design (TRPSH, frequency multipliers, etc.).

The cross section and dimensions of the shunt are related to the cross-sectional value of the main magnetic core \(S_o\), constant for a given depth of regulation, by the coefficient

\[ \beta = \frac{S_{\text{sh}}}{S_o}, \]

where \(S_{\text{sh}}\) is the cross section of the shunt leg. Then there may be two cases, when \(a_{\text{sh}} = a_o,\ b_{\text{sh}} = \beta \cdot b_o\), or \(a_{\text{sh}} = \beta a_o,\ b_{\text{sh}} = b_o\). If, for example, it is assumed that in all the designs shown in Figs. 1 and 2 the value \(\beta\) is determined from the condition of limiting the short-circuit current, and at the same time the value of the maximum induction in the main magnetic core and the shunt must be identical, while the line voltage varies within \((0.85 \div 1.1) U_{1\text{n}}\), then we obtain

\[ \beta = \frac{S_{\text{sh}}}{S_o} = \frac{U_{1\max}}{U_{1\min}} = 1.3 . \]

For other requirements and conditions, the value \(\beta\) can be found in an analogous manner.

Now all the necessary quantities are available in order to describe the volume of any FU design in accordance with [4] and to compare them under equal conditions. The expression for the total volume of steel, independently of the design features of a given apparatus, can in general form be written as follows [4]:

\[ V_c = B \cdot S_o [m'a_o + n'b_o + q'C + ph], \]

where

\[ B,\; m',\; n',\; q',\; p \]

are constant coefficients whose values are not identical for different designs.

Depending on the design features of the FU, when determining the height of the shunt window there may be three cases.

1. \(h_{\text{sh}} = h_o\).

2. \(h_{\text{sh}} = h_o \mp (1.05 \div 1.1)\,2a_{\text{sh}}\), where \(a_{\text{sh}}\) is selected from the conditions:

a) \(a_{\text{sh}} = \beta a_o,\; b_{\text{sh}} = b_o\), or \(\tag{4}\)

b) \(a_{\text{sh}} = a_o,\; b_{\text{sh}} = \beta b_o\), \(\tag{4a}\)

3) \(h_{\text{sh}} = h_o + (1.05 \div 1.1)\,2a_o\). \(\tag{4b}\)

Therefore the expressions for the volume of the coils for the above cases, respectively, will have the form:

\[ 1.\quad V_k = A \cdot C \cdot h_o [m_1 a_o + n_1 b_o + q_1 C]. \tag{5} \]

\[ \begin{aligned} 2.\ \text{a)}\quad V_k = A \cdot C \{&h_o (m_1 a_o + n_1 b_o + q_1 C) + \\ &+ [h_o \mp (1.05 \div 1.1)\,2\beta a_o][m_2 a_o + n_2 b_o + q_2 C]\}, \end{aligned} \tag{6} \]

\[ \begin{aligned} \text{b)}\quad V_k = A \cdot C \{&h_o [m_1 a_o + n_1 b_o + q_1 C] + \\ &+ [h_o - (1.05 \div 1.1)\,2a_o][m_2 a_o + n_2 b_o + q_2 C]\}. \end{aligned} \tag{7} \]

\[ \begin{aligned} 3.\quad V_k = A \cdot C \{&h_o [m_1 a_o + n_1 b_o + q_1 C] + \\ &+ [h_o + (1.05 \div 1.1)\,2a_o][m_2 a_o + n_2 b_o + q_2 C]\}, \end{aligned} \tag{8} \]

where

\[ A,\; m_1,\; m_2,\; n_1,\; n_2,\; q_1,\; q_2 \]

are the same constant coefficients as in (3).

The presence of several varieties of expressions for the volume of the coils (5)–(8) makes the analysis of FU design variants more difficult and, in certain cases, somewhat complicates their direct calculation.

These difficulties can be eliminated if restrictions are introduced on the height of the apparatus, that is, if one chooses

\[ Z = \frac{h}{a} = 5 \div 10. \]

Then expressions (6)–(8) reduce to a single form:

\[ V_k = A \cdot C \cdot a_o [m a_o + n b_o + q C], \tag{9} \]

where, for condition (4):

\[ m = Zm_1 + m_2 [Z - 2\beta (1.05 \div 1.1)], \tag{10} \]

\[ n = Zn_1 + n_2 [Z - 2\beta (1.05 \div 1.1)], \tag{11} \]

\[ q = Zq_1 + q_2 [Z - 2\beta (1.05 \div 1.1)]; \tag{12} \]

for (4a) and (4b):

\[ m = Zm_1 + m_2 [Z \mp 2(1.05 \div 1.1)], \tag{13} \]

\[ n = Zn_1 + n_2 [Z \mp 2(1.05 \div 1.1)], \tag{14} \]

\[ q = Zq_1 + q_2 [Z \mp 2(1.05 \div 1.1)]. \tag{15} \]

Expression (5) can be brought to the form (9) by the simple substitution \(h_o = Z a_o\); in this case \(m = Zm_1\), \(n = Zn\), \(q = Zq_1\). In addition, limiting the design by height makes it possible to write the variable coefficients \(k_h\) and \(k_c\) in the form

\[ k_h = \frac{Z}{\sqrt{k_s}}, \qquad k_c = \frac{L\sqrt{k_s}}{Z}. \tag{16} \]

and to obtain a form of the apparatus that is not only optimal with respect to specific-weight indicators, but also simpler in manufacturing technology, since the possibility of obtaining an elongated, technologically impractical design is excluded.

In accordance with [4], the cross-sectional area of the main magnetic core is determined as:

\[ S_o = D \sqrt{\frac{P_{1\phi}}{n_k \cdot k_c \cdot k_h}} = D \cdot \sqrt{\frac{P_{1\phi}}{n_k L}}, \tag{17} \]

where

\[ D = \frac{1}{\sqrt{4.44 f B_{mo} \cdot j k_{zk} \cdot k_{zc}}}, \tag{18} \]

\(P_{1\phi}\) — the apparent power of one phase,
\(n_k\) — the number of coils of the primary winding,
\(B_{mo}\) — the operating induction of the main magnetic core,
\(j\) — the average current density in the windings.

Substituting (1), (16), and (17) into (3) and (9), we finally obtain expressions for the volume of the coils and cores:

\[ V_k = \sqrt{D^3}\,\sqrt[4]{P_{1\phi}^3}\cdot A \sqrt[4]{\frac{L}{n_k^3}} \left( \frac{m}{\sqrt{k_s}} + n\sqrt{k_s} + q\frac{L\sqrt{k_s}}{Z} \right). \tag{19} \]

\[ V_c = \sqrt{D^3}\,\sqrt[4]{P_{1\phi}^3}\, \frac{B}{\sqrt[4]{(n_k L)^3}} \left( \frac{m'}{\sqrt{k_s}} + n'\sqrt{k_s} + q'\frac{L\sqrt{k_s}}{Z} + \frac{pZ}{\sqrt{k_s}} \right). \tag{20} \]

In abbreviated notation these expressions have the form:

\[ V_k = \sqrt{D^3}\cdot \sqrt[4]{P_{1\phi}^3}\, v_k, \tag{21} \]

\[ V_c = \sqrt{D^3}\cdot \sqrt[4]{P_{1\phi}^3}\, v_c, \tag{22} \]

where

\(v_k\) and \(v_c\) are the dimensionless volumes of the coils and steel, depending only on the ratio of the geometric dimensions of the FU.

The expressions for the economic indicator can now be written as

\[ E = \frac{k_{zc}}{P_{n,\mathrm{out}}} \sqrt{D^3}\sqrt[4]{P_{1\phi}^3}\,(a v_k + v_c), \tag{23} \]

where \(P_{n,\mathrm{out}}\) is the rated output power of the FU,
\(k_{zc}\) is the weight or cost of a unit volume of cores,
\(a\) is a coefficient characterizing the ratio of the weight or cost of a unit volume of coils to the weight (cost) of a unit volume of cores.

Expressions (21), (22), and (23) are products of two functions, one of which depends on the geometric forms of the apparatus, while the other is associated only with physical quantities.

When comparatively evaluating different design variants of the FU by one of the economic indicators (weight, volume, or cost), the function associated with physical quantities may be considered

Table 1

\[ \beta = 1,3;\qquad Z = 5;\qquad \alpha = 0,5 \]

as a constant quantity, since the values \(B_{m0}\), \(j\), \(P_{1\phi}\), \(k_{z\kappa}\), \(k_{zc}\) may be taken to be the same for all structural varieties.

However, in the practice of calculating an FU, two cases are often encountered:

1. When the induction value \(B_{m0}\) and the current density in the windings \(j\) are considered specified.

2. When the induction value \(B_{m0}\) is specified or is selected within the knee of the saturation region of the magnetization curve, while the current density in the apparatus windings \(j\) is unknown and is determined in the course of the calculation. In the first case, when all the quantities \(B_{m0}\), \(j\), \(P_{1\phi}\), \(k_{zc}\), \(k_{z\kappa}\) entering into the function that depends on the physical parameters are constant, the analysis of the weight indices of the structural variants of the FU may be carried out using a function that depends only on the ratios of the geometrical dimensions of the apparatus. The latter, after substituting the coefficient \(k_S\) from expression (24), obtained in [4],

\[ k_S = \frac{(xmL + m' + pZ)} {\left[(xnL+n')+\frac{L(xqL+q')}{Z}\right]} \tag{24} \]

with allowance for certain simplifications, takes the form:

\[ v_{\Sigma}=(av_{\kappa}+v_c)= \frac{2B}{\sqrt[7]{n_{\kappa}^{6}}}\cdot \frac{1}{\sqrt[4]{L^{3}}}\cdot \sqrt{\left[(xmL+m')+pZ\right]\cdot \left[(xnL+n')+\frac{L\left[(xqL+q')\right]}{Z}\right]}, \tag{25} \]

where

\[ x=\frac{aA}{ZB}. \]

According to this expression, for the constructions shown in Figs. 1, 2 and for the comparison conditions adopted above for them, which gave the constant coefficients presented in Table 1, graphs of the function \(V_{\Sigma}=f(L)\) were plotted; these are shown in Fig. 3.

In the second case, considering the current density to be unspecified and dependent on the geometrical ratio of the FU dimensions, one may, in accordance with [4], write:

\[ j=\sqrt[7]{\frac{M_j^{4}\cdot n_{\kappa}\cdot k_h}{D_j^{2}\cdot P_{1\phi}\cdot k_c^{3}}}, \tag{26} \]

where

\[ D_j=\frac{1}{\sqrt{4.44 f B_{m0} k_{z\kappa} k_{zc}}}, \tag{27} \]

\[ M_j=\frac{\alpha_{\kappa}\cdot \Delta\tau_{\kappa}\,10^{6}} {\rho_{\kappa}\cdot k_{zc}}; \tag{28} \]

\(\alpha_{\kappa}\) — averaged coefficient of heat transfer of the coils,
\(\Delta\tau_{\kappa}\) — steady-state permissible temperature rise of the coils,
\(\rho_{\kappa}\) — specific resistance of the coil material.

Taking (26)–(28) into account, we obtain a new expression for the economic index:

\[ \mathcal{E}= \frac{k_{zc}}{P_{n,\mathrm{out}}}\cdot \sqrt[7]{\frac{D_j^{12}\cdot P_{1\phi}}{M_j^{3}}}\cdot \frac{1.4\,B(a_1+pZ)} {\sqrt[7]{\left[\frac{2Z(a_1+pz)}{5(a_2Z+a_3L)}\right]^2 (n_{\kappa}^{2}xzL)^3}} \tag{29} \]

where \(a_1=xmL+m'\), \(a_2=xnL+n\), \(a_3=xqL+q'\).

In expression (29), as in (23), one can also single out a function depending only on the ratio of the geometric dimensions of the FU. It may be called the dimensionless volume of the apparatus at an unspecified current density, and it will have the form:

\[ v_{\Sigma j}= \frac{1.4\,B(a_1+pZ)} {\sqrt[7]{\left[\frac{2Z(a_1+pZ)}{5(a_2Z+a_3L)}\right]^2 (n_k^2 ZL)^3}} . \tag{30} \]

The graphs of this function for the single-phase TRPSh shown in Figs. 1, 2 and with constant coefficients—in Table 1—are given in Figs. 4, 5, 6, 7.

Fig. 3.

To make possible an all-round comparison of the specific-weight indices of the TRPSh in Figs. 1, 2, the authors, with the aid of the “Promin” computer, calculated from expressions (25, 30) the curves of the dimensionless volumes \(v_{\Sigma}\) and \(v_{\Sigma j}\) for three criteria: minimum weight with copper windings, which corresponds to the value \(\alpha=0.5\); minimum volume, when \(\alpha=1\); and minimum cost, which corresponds to \(\alpha=2\). For each of the criteria, different relative cross sections of the shunts \(\beta\) from 1.3 to 0.6 were specified. The length of the article does not permit all the curves obtained to be presented, and only some of them are given here, those necessary to illustrate the main conclusions obtained in the analysis performed. Since the qualitative picture for the graphs \(v_{\Sigma j}=f(L)\) with an unspecified current density and \(v_{\Sigma}=f(L)\), when it is specified, was in most cases the same, the curves of dimensionless volumes for the independent geometry are presented by us only in one figure, Fig. 3.

From the results of the investigations carried out, it may be said that, other conditions being equal and if the voltage of the primary windings changes-

Fig. 4

Fig. 5

Fig. 6.

Fig. 7.

...only slightly, approximately within ±5% (the reservation will be explained below), designs Nos. 2, 8, 9 always have the best data. Designs Nos. 1, 3, 5, 13 have indicators 1.3–1.5 times worse in comparison with them. The remaining TRPSH occupy an intermediate position and are grouped approximately as follows: after Nos. 2, 8, 9 come Nos. 7, 10, 11, then Nos. 4, 6, 12.

Analyzing the specific-weight indicators of the TRPSH separately by criteria, it should be noted that they have certain features.

First of all, the extrema of the curves \(v_{\Sigma}\) and \(v_{\Sigma j}\) for some designs...

tions are shifted in their mutual arrangement. For example, the extremum \(v_{\Sigma}\) of design No. 2, with independent geometry, lies above the extremum of the same curves for designs No. 8 and 9 (Fig. 3). With geometry having an unspecified current density, the picture changes to the opposite one: the extremum of the curve \(v_{\Sigma j}\) under number 2 lies considerably lower than for curves No. 8, 9 (Fig. 4). The same picture is observed in Figs. 3 and 4 for curves 5, 13. This is explained by the fact that the designs being compared have different cooling surfaces, and if, with independent TRPSH geometry, this circumstance has no effect on the specific weight indices, then, with geometry having an unspecified current density, for apparatus having a larger cooling surface at the same coil heating, the weight, dimensions, and cost will be smaller because of the increase in the permissible current density.

The mutual arrangement and the form of the graphs of the functions of the weight indices also depend on the values of \(\alpha\). This can be seen by comparing these curves for designs No. 6, 12 and 13, 5 in Figs. 4, 5, 6. It is characteristic that, as \(\alpha\) increases, the functions \(v_{\Sigma}=f(L)\) and \(v_{\Sigma j}=f(L)\) have a more clearly expressed region of minimum values.

The depth of regulation, i.e., the value \(\beta\), has a more substantial effect on the arrangement of the graphs of dimensionless volumes (compare Fig. 5 and Fig. 7). Here attention is drawn to the fact that, as the depth of regulation decreases, the specific weight indices of a number of designs become equalized (in Fig. 7, curves 4, 5, 13; 1, 3; 2, 7). Therefore, for regulators with a small depth of regulation, the number of TRPSH design variants with similar or even identical economic indices will be greater than for regulators with deep regulation. It is characteristic that, at small \(\beta\), the best design in terms of weight indices becomes TRPSH No. 7, which previously occupied an intermediate position. TRPSH No. 2 still has good data here, and designs No. 1, 3, 5, 13 remain clearly worse; TRPSH No. 4 now also joins the latter.

Let us analyze in somewhat greater detail the indices of each of the TRPSH designs in Figs. 1, 2.

First of all, it should be noted that all of them can be divided into two groups, one of which consists of devices developed on the basis of the shell-type TRPSH of design No. 1 (see No. 1, 3, 4, 5, 6), and the other on the basis of the core-type TRPSH of design No. 2 (see No. 7--13). Let us consider whether it is expedient to use in automatic installations all the TRPSH designs we have presented, if one weighs not only their specific weight indices but also other possible considerations.

It is obvious that the expediency of using TRPSH design No. 2 is beyond doubt, since by all comparison criteria it has the best indices and is made on simple U- or L-shaped magnetic cores.

A similar conclusion was obtained earlier by other authors.

TRPSH design No. 1, although it is inferior to TRPSH No. 2 in all weight indices by approximately a factor of 1.3--1.5, may also be recommended for use, taking into account that it has the minimum number of coils, is highly manufacturable, has good structural forms, and can be made on magnetic cores from E-shaped laminations used for saturable reactors or ordinary transformers.

The situation is different with derivative designs of the TRPSH named above. One can confidently refrain from recommending TRPSH No. 4 for use. Evidently, this apparatus was created from the motive of eliminating the induction of the electromotive force of odd harmonics in the DC bias winding, and it turned out not to be very successful. In Fig. 1 the ex-

a combination, equivalent to this TRPSH, consisting of an ordinary transformer and a saturating reactor connected in series with it. As can be seen, the latter has the same volume of active materials as TRPSH No. 4 and a simpler winding design. But if in this combination the width of the transformer window is reduced, it will already gain in the weight of the magnetic cores and will gain still more if a reactor of a more successful design is used in it. Replacing the transformer–reactor system by a biased transformer with poorer indicators is clearly not rational.

The biased transformer No. 5, described in [5], was also created with the aim of having in it a bias winding without induced e.m.f. of odd harmonics. In addition, provision was made to reduce the dimensions of the alternating-current windings and to have one “linked” magnetic core. However, the effect obtained led to a considerable worsening of the weight indices (according to the results of our analysis they are the worst for this apparatus), and this occurred because of the increase in the volume of the magnetic core. An idea of the latter is easily obtained by joining it from two magnetic cores of TRPSH No. 3, “pulling” the main one of them onto the shunt and leaving windows for the windings (see the magnetic core of design No. 5 in Fig. 1). Possessing double magnetic leakage [6], TRPSH No. 5 has a large depth of regulation under load, owing to which the cross-section of its shunt can be reduced (for equal comparison conditions), and consequently also the weight indices. Nevertheless, this gain is not so large as to make the weight indices of TRPSH No. 5 better than those of TRPSH No. 4. Thus, the use of TRPSH No. 5 in automatic installations is also inexpedient.

The biased transformer of design No. 3 differs from TRPSH No. 4 and No. 5 in that it has one winding fewer, but in return it loses the property of compensating the odd harmonics of the flux in the shunt, and its main magnetic core is heavier than that of TRPSH No. 4, since it contains a middle limb. Naturally, the use of TRPSH No. 2 is inexpedient, since it is the worst variant of TRPSH No. 4.

The design of TRPSH No. 6 deserves definite attention. This is one of the first domestic biased transformers. It has one linked magnetic core, a small number of windings, and possesses fairly good weight indices. According to the data of our analysis, this apparatus, for relative shunt cross-sections \(\beta \le 0.7\), differs little in weight indices from the best designs No. 2, 8, 9. If one also takes into account that TRPSH No. 6 possesses double leakage of the working flux and therefore has additional advantages in regulation depth, it will be clear that this transformer deserves serious attention with respect to its application. Practice has shown that it is especially good as a regulator of current, voltage, or power in low-voltage installations.

The biased transformers No. 10, 11, 12, 13 are a variety of TRPSH No. 2 and differ from it in that their main magnetic core and shunts are located in one plane. In this case it is possible to reduce the number of coils of the bias winding to two and to decrease the dimensions of the enclosing alternating-current winding. At the same time, the volume of the shunt cores is doubled, which on the whole leads to a deterioration of the specific-weight indices of these designs in comparison with those for TRPSH No. 2. Even with the optimal geometry of the compared TRPSH, this deterioration is \(24 \div 78\%\), which can be calculated from the levels of the minima of the functions \(v_{\Sigma j}=f(L)\) in Figs. 4, 5, 6.

Of the four named TRPSH, only No. 10 and No. 12 should be recommended for practical application. TRPSH No. 11 differs from

No. 10 in that the coils of the biasing windings are moved out from under the encompassing working windings onto the outer legs of the shunts. In the latter, this reduces the average turn length and, consequently, the volume and weight. Overall, TRPSh No. 11 is 3% lighter than No. 10; however, it is less structurally sound, and most importantly, here the biasing windings have poor flux linkage with the working windings. The latter circumstances force preference to be given to TRPSh No. 10.

The biased transformer of design No. 12 will be obtained from TRPSh No. 11 if one shunt core is introduced under the inner working winding and “glued” to the core of the main magnetic circuit, while the outer working winding encompasses all three cores (one main and two shunt cores) together with the windings \(w_b\) and \(w_d\) located on them. Naturally, now the average turn length of \(w_b\) and \(w_d\) will become greater than it would have been in TRPSh No. 11 or No. 10, and the volume of the coils of these windings will increase. Overall, TRPSh No. 12, even with optimal geometry, has specific weight-and-size indicators 6% worse than TRPSh No. 10. But this does not mean that it should not be used in automatic installations. Having a single articulated magnetic circuit, TRPSh No. 12 is more compact, smaller in overall dimensions, more manufacturable, and possesses the property of magnetic-flux leakage through both working windings; that is, it will provide the same depth of load voltage regulation with a smaller relative cross section of the shunts than designs No. 10 and No. 11. The TRPSh No. 13 design can confidently not be recommended for broad practical use. In principle, it differs in no way from TRPSh No. 12, but because here the shunts are not “glued” to the side of the main magnetic circuit, but are built into it (see the configuration of the magnetic circuits of TRPSh No. 12 and No. 13 in Fig. 2), the sharp increase in the average length of the latter increases the total volume of steel and worsens all specific weight-and-size indicators. This is confirmed by the curves of dimensionless volumes in Figs. 4–7, which for TRPSh No. 13 are everywhere among the uppermost.

Special attention should be given to the TRPSh designs No. 7, 8, and 9. In the literature they are recommended as among the best [7]. This conclusion can also be reached by analyzing the graphs in Figs. 4–7. At first glance everything is logical, since in these TRPSh the shunts consist not of two but of one pair of legs. Hence the expected gain in weight indicators follows.

However, if for all other TRPSh designs in Figs. 1 and 2 the maximum line voltage determines the total apparent power of the parallel and series elements (the main part and the shunt), then for designs No. 7, 8, and 9 it determines only the power of the parallel element, regardless of whether there is a shunt or not. This means that TRPSh No. 7, 8, and 9 are disadvantageous to use as line-voltage stabilizers.

Let us show this by an example. Suppose the line voltage varies within the limits \((0.65 \div 1.15) U_{1n}\). Then for design No. 10 the apparent power will be equal to:

\[ p_{\mathrm{dim}.10}=p_g+p_{\mathrm{sh}}=0.65p+(1.15-0.65)p_{1n}=1.15p_{1n}. \]

For design No. 8 we obtain

\[ p_{\mathrm{dim}.8}=p_g+p_{\mathrm{sh}}=1.15p_{1n}+(1.15-0.65)p_{1n}=1.65p_{1n}. \]

Thus, the apparent power and, consequently, the specific weight-and-size indicators of TRPSh No. 8 will be greater in comparison with TRPSh No. 10 by a factor of \(1.65-1.15=1.5\). Even taking into account that TRPSh No. 8 at constant line voltage has, compared with TRPSh No. 10, a smaller relative volume by approximately a factor of 1.24. Overall, for our example it

will have worse specific weight indices by a factor of \(1.5 : 1.24 = 1.21\). It should be assumed that the recommendations on the use of TRPSh Nos. 7, 8, 9 as the best designs for voltage stabilizers, given in [7], are highly questionable.

Conclusions

1. Of the entire variety of existing design variants of single-phase TRPSh, only a few can be recommended for practical application in automatic installations:

a) design No. 2, as having the best indices according to all comparison criteria;

b) TRPSh Nos. 6 and 12, which possess the property of double scattering of the working flux, provide an additional increase in the depth of regulation, and have good specific weight indices; they are manufacturable, structurally sound, and compact;

c) TRPSh design No. 10, which has the properties of TRPSh No. 12, but which can be manufactured on magnetic cores made from standard plates of П-, Г-, and С-shape, used for ferromagnetic devices of other types (transformers, saturable reactors, etc.).

1. All other TRPSh designs have noticeably worse indices, and their application may be expedient only in specially stipulated cases. For example, it is sometimes convenient to use the armored TRPSh No. 1, which has a small number of coils, is manufacturable, and can be made on magnetic cores from standard Ш-shaped plates, which substantially facilitates construction of the TRPSh under the conditions of electrical workshops and experimental laboratories.

2. The method used in this work for the comparative evaluation of the specific weight indices of TRPSh, supplemented by analysis of these devices according to other criteria, made it possible to make a theoretically substantiated selection of the most rational designs of single-phase shunt-biased transformers. This method can subsequently be used for comparative evaluation of three-phase TRPSh, as well as ferromagnetic devices of another type—multipliers, shunt-biased saturable reactors, etc.

REFERENCES

1. A. M. Bamdas, S. V. Shapiro, O. D. Getmonenko. Determination of the optimal designs of transformers and autotransformers regulated by bias magnetization, and their calculation. Proceedings of the GPI named after Zhdanov, vol. XVIII, issue 1, 1962.

2. A. M. Bamdas, S. V. Shapiro. Transformers Regulated by Bias Magnetization. “Energiya,” 1965.

3. A. Kh. Tigranyan. On the optimal design of transformers regulated by bias magnetization. INBUZ. — Elektromekhanika, 1964, No. 4.

4. V. P. Obrusnik, V. G. Kiselev. Questions of the optimal geometry of single-phase shunt-biased transformers. Proceedings of the Sixth Scientific and Technical Conference on Problems of Production Automation, vol. II, Publishing House of TSU, 1969.

5. V. P. Obrusnik. A new design variant of a transformer regulated by bias magnetization. Izvestiya TPI, vol. 161, 1966.

6. V. P. Obrusnik, A. I. Zaitsev. Questions of the theory and application of transformers regulated by bias magnetization of shunts. Izvestiya TPI, vol. 117, 1963.

7. A. M. Bamdas, S. V. Shapiro. Stabilizers with Bias-Magnetized Transformers. “Energiya,” 1965.