On the Reduction of Electrodynamic Forces (EDF) in Transformer Windings
Abstract Generated abstract
This paper examines methods for reducing electrodynamic forces in transformer windings under external short-circuit conditions, with emphasis on high-power and unit transformers. Using expressions for mechanical stress as a function of winding cross section and short-circuit impedance, it analyzes winding splitting, separate operation of split windings, and special connection schemes for two-limb single-phase transformers. The authors show that natural and intentional splitting can substantially reduce calculated winding stresses, in some cases by factors of two, four, or eight relative to conventional arrangements, while also considering the effect of surge magnetizing current on the limiting impedance. The analysis indicates that appropriate winding connections can bring electrodynamic forces in powerful single-phase block transformers within permissible limits and provide additional strength margin, particularly for layer windings.
Full Text
On the Reduction of Electrodynamic Forces (EDF) in Transformer Windings
G. V. Del, V. P. Krasnov, I. D. Kutyavin
Short-circuit currents flowing through transformer windings produce electrodynamic forces between them. These forces increase with an increase in transformer power, reaching dangerous magnitudes. In operating practice there are numerous cases of winding damage during short circuits beyond transformers; in this connection, reduction of EDF in their windings is of interest. This applies especially to modern ultra-high-power transformers.
To clarify methods for reducing EDF, one may use the expression for the mechanical stress in the winding material due to the radial component of the EDF
\[ \sigma = k \frac{\Delta^2 q_{\mathrm{m}} l_{\mathrm{m}}}{h x_{\mathrm{p}}^2}\ \text{kg}/\text{cm}^2, \tag{1} \]
where
\[ k = 6.37\, k_{\mathrm{r}}\, 10^{-8}, \]
\(\Delta\) — current density in the winding, \(\text{A}/\text{cm}^2\),
\(q_{\mathrm{m}}\) — total cross section of the material of the phase winding, \(\text{cm}^2\),
\(l_{\mathrm{m}}\) — length of the mean turn of the winding, \(\text{cm}\),
\(h\) — total axial height of the winding, \(\text{cm}\),
\(x_{\mathrm{p}}\) — minimum calculated impedance of the circuit during a short circuit beyond the transformer (in relative units),
\(k_{\mathrm{r}}\) — Rogowski coefficient.
As is evident from (1), a reduction in the mechanical stress in the winding material can be achieved by changing a number of quantities. However, without impairing the economy and design quality of the transformer, only the quantities \(q_{\mathrm{m}}\) and \(x_{\mathrm{p}}\), which enter into (1) in the form of the ratio
\[ v = \frac{q_{\mathrm{m}}}{x_{\mathrm{p}}^2}. \tag{2} \]
may be used for this purpose.
One of the methods of reducing EDF is splitting transformer windings. In this case, parts of the split winding may be connected to one another inside the transformer tank or
may have independent leads to the tank cover for separate operation.
In the first case, one of the windings of a phase of a two-winding transformer is split into two parts embracing the second winding of this phase, as shown in Fig. 1. Winding II in this case experiences an EDF of repulsion from both parts of winding 1. The inner part of winding 1 is subjected to compression, and the outer part to tension.
The greatest stress in the material during a short circuit beyond the transformer is experienced by the outer half of winding 1. The magnitude of this stress is determined from (1) by substituting \(0.5 q_M\) in the numerator, and \(-x_p = u_k\) in the denominator (\(u_k\) is the short-circuit voltage of the transformer in relative units), and it will be two times less than for a transformer with an unsplit winding. Relation (2) in this case will be equal to
\[ v = 0.5 \frac{q_M}{u_k^2}. \tag{3} \]
Fig. 1.
Recently, the splitting of windings of high-power transformers has found increasingly wide application for the purpose of using them for separate operation. This limits the short-circuit currents in the secondary windings and reduces the EDF in them.
Winding splitting is also used in unit transformers for separate connection of the branches of the generator stator winding with parts of the split primary winding (Fig. 2a), or for an enlarged unit (Fig. 2b).
The conclusions drawn are also valid for units with single-phase transformers. High-power single-phase transformers are manufactured on two-limb cores with additional side yokes. Each of the windings is made in the form of two parallel—
Fig. 2.
parallel branches located on different cores and having weak mutual induction during short circuits. Consequently, single-phase two-core transformers have a natural splitting of the windings. The value of $\text{э}_p$ is determined separately for the windings of each core. This gives a reduction of the EDF, taking (3) into account, by a factor of two. However, it is possible to split the windings of each core in accordance with Fig. 1. Then the electrodynamic forces will be reduced by a factor of four.
Fig. 3.
But for single-phase transformers, one more interesting variant of connecting the windings is possible, as shown in Fig. 3a. The secondary winding of the transformer, consisting of two sections placed on different cores, is connected in series. For better utilization of the window area this winding may be made of four sections connected according to the scheme (Fig. 3b). The current from the system side $I_c$ during a short circuit at point $k_1$ can be determined from the equation (Fig. 3c)
\[ E_c = E_r + I_c (2x''_d + u_k). \tag{4} \]
where
$E_c$ and $E_r$ are the e.m.f. of the system and of the generator,
$x''_d$ and $u_k$ are the resistances, in ohms, in expressions (4)—(6).
Since the winding of one core is short-circuited, the entire e.m.f. of the system is applied to the winding of the second core; therefore in (4) it is necessary to put $E_c = 2E_r$. Then the current from the system side
\[ I_c=\frac{E_{\mathrm{г}}}{2x''_d+u_k}. \tag{5} \]
In the event of a short circuit at point \(k_2\), the current flowing through the transformer from the generator side of the unit is
\[ I_{\delta}=\frac{E_{\mathrm{г}}}{x''_d+u_k}. \tag{6} \]
Consequently, \(x_{p1}=2x''_d+u_k\) and \(x_{p2}=x''_d+u_k\).
The calculated value is \(x_{p2}\). With \(x''_d=u_k\), \(x_{p2}=2u_k\). Then relation (2) will have the following expression:
\[ \nu=\frac{0.5q_{\mathrm{м}}}{x_{p2}^{2}} \le \frac{q_{\mathrm{м}}}{8u_k^{2}} . \tag{7} \]
In this case the stress in the material \(\sigma_p\) will be reduced, in comparison with a three-phase transformer, by a factor of eight and, in comparison with a two-limb single-phase transformer, by a factor of four.
Since the leakage impedances of the windings (Fig. 3) of the limbs are connected in series, the calculated short-circuit voltage of a limb, referred to its power, is \(u_{kc}=0.25u_k\).
In the latter case, in the event of a short circuit at point \(k_1\), the impact magnetizing current of the limb with the undamaged winding will be added to the current of the system, since the magnetization of this limb will increase owing to the application to its winding of a higher voltage, almost double the voltage \((E_c \simeq 2E_{\mathrm{г}})\). As a result, the total current from the system side may turn out to be the calculated one for determining the EDF. To clarify this question, let us find this current, using the equivalent circuit (Fig. 4). First we shall find the expression for the resistance of the magnetizing branch \(x_0\) of the equivalent circuit. Beyond the limits of the nominal induction \(B_n\), the magnetization curve may be straightened:
\[ B=B_n+mI_0, \tag{8} \]
where
\[ \begin{aligned} B_n&=17\ \text{kgf} &&\text{is the nominal calculated induction,}\\ m& &&\text{is the angular coefficient,}\\ I_0& &&\text{is the magnetizing current of the transformer.} \end{aligned} \]
To determine the value of \(m\), let us take on the magnetization curve of cold-rolled steel Э330 two points with coordinates: \((B_n=17\ \text{kgf},\ H_{17}=10\ \text{A/cm})\) and \((B=20\ \text{kgf},\ H_{20}=300\ \text{A/cm})\). Then the value of \(m\) is determined from the expression:
Fig. 4.
\[ m=\frac{W(B-B_{\mathrm{n}})}{l_c(H_{20}-H_{17})}=10.3\,\frac{W}{l_c}, \tag{9} \]
where
\[ \begin{aligned} W&\text{ is the number of turns of the low-voltage winding,}\\ l_c&\text{ is the length of the mean line of force.} \end{aligned} \]
The emf of the low-voltage winding of the transformer at \(t=0\) is
\[ E_{\mathrm{t}}=kB=k(B_{\mathrm{n}}+mI_0)=U_{\mathrm{n}}+kmI_0. \tag{10} \]
The resistance of the magnetizing branch is
\[ x_0=\frac{E_{\mathrm{t}}}{I_0}=\frac{U_{\mathrm{n}}}{I_0}+a, \tag{11} \]
where
\[ a=km=10.3\,\frac{W}{l_c}\cdot\frac{U_{\mathrm{n}}}{B_{\mathrm{n}}}. \]
Let us use the following equations for the circuit (Fig. 4):
\[ \left. \begin{aligned} I_c&=I_1+I_0,\quad E_c=0.75x_{\mathrm{t}}I_c+x_0I_0;\\ E_c&=E_{\mathrm{t}}=0.75x_{\mathrm{t}}I_c+I_1(0.25x_{\mathrm{t}}+2x''_d) \end{aligned} \right\} \tag{12} \]
At the same time,
\[ E_c=2E_{\mathrm{t}}\quad\text{and}\quad E_{\mathrm{t}}\simeq U_{\mathrm{n}}. \]
The simultaneous solution of equations (12) gives
\[ I_0=\frac{U_{\mathrm{n}}-0.75x_{\mathrm{t}}I_1}{a+0.75x_{\mathrm{t}}};\qquad I_1=\frac{U_{\mathrm{n}}z}{(x_{\mathrm{t}}+2x''_d)(z+1)-0.75x_{\mathrm{t}}}. \tag{13} \]
The total current from the system side during a short circuit at point \(K_1\) is
\[ I_c=\frac{U_{\mathrm{n}}\left|az+(x_{\mathrm{t}}+2x''_d)(z+1)-0.75x_{\mathrm{t}}\right|} {(a+0.75x_{\mathrm{t}})\left|(x_{\mathrm{t}}+2x''_d)(z+1)-0.75x_{\mathrm{t}}\right|} =\frac{U_{\mathrm{n}}}{x_{p1}}; \tag{14} \]
where
\[ z=\frac{a}{0.75x_{\mathrm{t}}};\qquad x_{\mathrm{t}}=\frac{U_k\%\,U_{\phi}^{2}}{100S_{\phi}};\qquad x''_d=\frac{x''_d\%\,U_{\mathrm{n}}^{2}}{100S_{\mathrm{ng}}}, \]
\[ x_{p1}= \frac{(a+0.75x_{\mathrm{t}})\left|(x_{\mathrm{t}}+2x''_d)(z+1)-0.75x_{\mathrm{t}}\right|} {az+(x_{\mathrm{t}}+2x''_d)(z+1)-0.75x_{\mathrm{t}}}. \tag{15} \]
Here all resistances are in ohms.
\[ \begin{aligned} S_{\phi}&\text{ is the rated power of a single-phase transformer,}\\ S_{\mathrm{ng}}&\text{ is the rated power of the generator,}\\ U_{\mathrm{n}}&\text{ is the rated voltage of the generator.} \end{aligned} \]
Studies show that the resistance \(x_{p1}\), found from (15), decreases as the unit power increases (Fig. 3), reaching, for a unit with single-phase transformers of \(400\ \mathrm{MVA}\), \(x_p=2.4x_{\mathrm{t}}\), and for a unit with single-phase transformers of \(600\ \mathrm{MVA}\), \(x_{p1}=2.16x_{\mathrm{t}}\). Consequently, taking account of the surge magnetizing current, the resistance \(x_{p1}\) for single-phase transformers with a power of up to \(400\ \mathrm{MVA}\) (per phase) will be greater than \(x_{p2}\). Therefore, for determining the electrodynamic forces, \(x_{p2}\) will be the calculated value. And only for high-power single-phase transformers should the design resistance be taken as \(x_{p1}\), determined from (15).
Thus, the EDFs of powerful single-phase block transformers not only can be reduced to permissible values, but a substantial margin of strength can also be created in their windings. This is especially important for high-power transformers with layer windings, which have lower electrodynamic stability.