Calculation of Positive- and Negative-Sequence Current Filters
Abstract Generated abstract
This paper addresses the practical calculation of positive and negative sequence current filters for relay protection circuits, with attention to element matching and the stability of tuning during short circuit conditions. It derives the required relationships among active and reactive filter impedances, discusses the choice of resistances relative to relay winding resistance, and gives a design procedure for chokes with air gaps that limits core saturation under relevant fault currents. The authors provide formulas for the number of turns, air gap, and effective gap cross section, then illustrate the method with a numerical design example for a generator overcurrent protection filter.
Full Text
Calculation of Positive- and Negative-Sequence Current Filters
I. D. Kutyavin and N. V. Lisetskii
In recent years, in the technique of relay protection, there has appeared a tendency to move from protection circuits that respond to the total values of currents, voltages, and powers to circuits that use the separate symmetrical components of the indicated quantities. Such circuits include filters of positive-, negative-, and zero-sequence currents and voltages.
The theory of symmetrical-component filters has been presented in the works of many authors. However, the calculation and manufacture of individual elements, as well as the assembly and adjustment of filters, have been treated insufficiently. In the present work the calculation of positive- and negative-sequence current filters is given.
As is known, the current circuits of relay protections have very small resistances, and for good adjustment of filters it is necessary additionally to match the resistances of their individual elements with the resistances of the protection current circuits. In addition, the inductive resistances of filters, when made in the form of chokes with steel cores, may change their values during short circuits in networks because of saturation of the cores and thereby detune the filters. If the indicated resistances are made without steel cores, they will be excessively bulky and uneconomical. All these circumstances make it necessary to consider in more detail the operation of the indicated filters in the current circuits of relay protection.
Fig. 1
The schematic diagrams of the filters considered are shown in Fig. 1. The circuit of Fig. 1a represents the schematic diagram of a positive-sequence current filter. To obtain the circuit of a negative-sequence current filter, it is sufficient in the indicated figure to interchange the places—
...by the resistances \(Z\) and \(R\). The circuit of Fig. 1b is a combined filter of positive- and negative-sequence currents. In this circuit relay \(A_1\) is traversed by positive-sequence currents, and relay \(A_2\) by negative-sequence currents.
As is known, the current filter shown in the circuit of Fig. 1a must have the following relation between the auxiliary resistances:
\[ Z=(r+jx)=Re^{j60}. \tag{1} \]
The same for the circuit of Fig. 1b:
\[ (Z_p+Z)=(Z_p+R)e^{j60}, \tag{2} \]
where \(Z_p\) is the resistance of the relay current winding.
As is seen from expressions (1) and (2), one of the resistances is purely active, and the other is mixed, consisting of an active resistance \(r\) and a reactive one \(x\). For the circuit of Fig. 1a, \(r=0.5R\) and \(x=0.5\sqrt{3}\,R\). For the circuit of Fig. 1b, the resistances \(Z_p\) may be considered purely active. Then for this circuit
\[ \left. \begin{aligned} r&=(R-Z_p)0.5,\\ x&=(R+Z_p)0.5\sqrt{3} \end{aligned} \right\} \tag{3} \]
To implement the filter according to one of the circuits under consideration, it is necessary to manufacture the resistances \(R\) and \(Z\). As experiments have shown, in order to obtain good tuning of the filter, the resistance \(R>(10\div15)Z_p\). The resistances \(R\) and \(Z\) are calculated to carry the secondary short-circuit current and must not depend on it.
The resistance \(R\) is made of constantan or rheostat wire of diameter \(1.5\div2\) mm in the form of a spiral. This material is adopted because of the small dependence of \(R\) on heating of the element.
The inductive resistance of the filter, from the standpoint of its dimensions, should be made in the form of a choke with an air gap. In order that the tuning of the filter not be disturbed during short circuits, the induction in the steel under these conditions must be allowed to be no more than \(8\text{–}10\) thousand gauss. In connection with this, the question arises of determining the value of the design current. If the condition is imposed that the choke not saturate at all short-circuit currents, then the rms value of the total short-circuit current at the initial instant of time must be taken as the design current. However, this condition is necessary only for instantaneous protections.
For maximum current protections having a large time delay and operating under the steady-state short-circuit process, detuning of the filter in the initial period of time cannot lead to failure of the protection to operate, since the multiple of the short-circuit current during this period of time is large; therefore, as the design current of the chokes of the filters for these protections one should take the steady-state current in a two-pole short circuit.
The current in the chokes of Fig. 1 will have its greatest value for a two-pole short circuit between phases A and C and, for generator protection, will be equal to
\[ i_{p\phi}=\frac{I_n}{n_T}\left(1+2I^{(2)}_{k\infty}\right)\simeq5.7\,\frac{I_n}{n_T}, \tag{4} \]
where
\[ \begin{aligned} I_n&\text{ — rated current of the generator,}\\ n_T&\text{ — transformation ratio of the current transformers, and}\\ I^{(2)}_{k\infty}&\text{ — relative value of the steady-state current of a two-pole}\\ &\text{\quad short circuit.} \end{aligned} \]
When determining \(I_{\kappa}^{(2)}{}_{\infty}\) from the attenuation curves, the calculated reactance of the generator was taken as
\[ x_p=x''_d+x_2=x''_d\cdot 2.22=2.22\cdot 0.125 \simeq 0.28. \]
In addition, in calculating \(i_{p0}\) it was assumed that during the short-circuit in the primary circuit of the protection, the load current preceding the short circuit is preserved. Having made the indicated assumptions, one can carry out an approximate calculation of the choke.
The e.m.f. of the choke during a short circuit in the primary circuit is
\[ E=i_{p0}x=222\, S\, B_k\, W\cdot 10^{-8}. \tag{5} \]
The number of turns of the choke is determined from (5):
\[ W=\frac{i_{p0}x10^8}{222\,S\,B_k} =\frac{2.57\, I_k x}{S n_T B_k}\,10^6, \tag{6} \]
where \(x\) is the inductive resistance of the choke from (3),
\(S\) is the cross section of the choke core,
\(B_k\) is the magnetic induction in the steel of the core.
When determining the number of turns of the choke from (6), the core cross section \(S\) and the magnetic induction \(B_k\) should be taken equal to: \(S=1.5 \div 2\ \mathrm{cm}^2\), \(B_k=8 \div 10\) thousand gauss.
The magnitude of the air gap can be determined from the following:
\[ AW_k=i_{p0}W\sqrt{2}=0.8\,B_k\,\delta\,\frac{S}{S_p}. \tag{7} \]
Then the length of the air gap from (7) is:
\[ \delta=\frac{i_{p0}W\sqrt{2}\,S_p}{0.8\,B_k\,S} =1.77\,\frac{i_{p0}W S_p}{B_k S}, \tag{8} \]
where \(S_p\) is the calculated cross section of the air gap, allowing for the bulging of the lines of force.
The quantity \(S_p\) can be determined from the following expression1:
\[ S_p=S(1+y)^2, \tag{9} \]
where \(y\) is a coefficient that takes into account the bulging of the lines of force in the air gap.
However, the coefficient \(y\) itself depends on the magnitude of the air gap and the dimensions of the core cross section. Thus, for example, according to Fig. 147 of A. Ya. Builov’s cited book1, for an air gap \(\delta=0.5 \div 1.5\ \mathrm{cm}\), the curves \(y=f(\delta)\) may be replaced by straight lines
\[ y=a+b\delta. \tag{10} \]
Then the calculated cross section of the air gap is:
\[ S_p=S(1+a+b\delta)^2. \tag{11} \]
Approximate values of the constants \(a\) and \(b\) are given in Table 1 as a function of the side \(d\) of the core cross section in centimeters (it is assumed that the core has a square cross section).
Table 1
| \(d_{\mathrm{cm}}\) | 1.0 | 1.2 | 1.4 | 1.6 |
|---|---|---|---|---|
| \(a\) | 0.27 | 0.25 | 0.21 | 0.18 |
| \(b\) | 0.35 | 0.28 | 0.25 | 0.23 |
Substituting the value of \(S_p\) from (11) into (8), for the air gap \(\delta\) we obtain the following expression:
\[ \delta=\frac{c-2b(a+1)-\sqrt{c[c-4b(a+1)]}}{2b^2}, \tag{12} \]
where
\[ c=\frac{B_k}{1.77\, i_{p0} W}\simeq \frac{n_r B_k}{10 I_n W}. \tag{13} \]
In view of the fact that, in determining \(\delta\), it is necessary to use formula (12), in whose numerator there is a very small difference of comparatively large quantities, it becomes necessary to check the results of the calculation. For this one may use the known formula for the inductance of a toroid with the length of the mean line of force equal to \(\delta\):
\[ x=\omega L=\omega \frac{4\pi W^2 S_p}{\delta}\,10^{-9}\simeq 4\frac{S_p W^2}{\delta}\,10^{-6}\ \text{ohm}. \tag{14} \]
Then the number of turns of the choke is
\[ W\simeq 100\sqrt{\frac{25x\delta}{S_p}}, \tag{15} \]
where \(x\) is the inductive reactance of the choke.
Example. Calculate the resistances of the filter \(R\) and \(Z\), made according to the circuit of Fig. 1a and intended for protecting a generator against overcurrents. For protection it is proposed to use a relay of type ET-5216 with an operating current of \(2.5\ \text{A}\).
The resistance of the relay winding, with its coils connected in series to obtain an operating-current range of \(1.5\text{--}3\ \text{A}\), is
\[ Z_p=\frac{0.1}{3^2}=0.0111\ \text{ohm}. \]
The active resistance of the filter is
\[ R=15Z_p=0.167\ \text{ohm}. \]
The active resistance of the choke is
\[ r=0.5R=0.5\cdot 0.167=0.0835\ \text{ohm}. \]
The inductive reactance of the choke is
\[ x=0.5\sqrt{3}\,R=0.5\sqrt{3}\cdot 0.167=0.145\ \text{ohm}. \]
We shall take the following dimensions of the choke core: cross section \(S=2\ \text{cm}^2\), side of the core cross section \(d=1.40\ \text{cm}\), window dimensions \(2.2\times 2.2\ \text{cm}\), and outside dimensions \(5\times 5\ \text{cm}\) (see Fig. 2).
The number of turns of the choke from (6) is
\[ W=\frac{2.57\cdot 5\cdot 0.145}{2\cdot 10^1}\,10^3=93\ \text{turns}. \]
The constant is
\[ C=\frac{n_r B_k}{10 I_n W}=\frac{10^4}{10\cdot 7\cdot 93}=2.15. \]
From Table 1, for \(d=1.4\), we find \(a=0.21\) and \(b=0.25\). The air gap of the choke, from (12), is
\[ \delta=\frac{2.15-2\cdot 0.25(1+0.21)-\sqrt{2.15[2.15-4\cdot 0.25(1+0.21)]}}{2\cdot 0.25^2}=1\ \text{cm}. \]
To check the obtained values of \(W\) and \(\varepsilon\), from (11) we find the calculated core cross sections:
\[ S_p = 2(1 + 0.21 + 0.25 \cdot 1)^2 = 4.28. \]
Then the number of turns of the choke from (15) is:
\[ W = \sqrt{\frac{25 \cdot 0.145 \cdot 1}{4.28}} = 92. \]
We take \(W = 92\) turns.
The active resistances of the filter \(R\) can be made of constantan wire of diameter \(d = 1.8\ \text{mm}\) and length \(0.915\ \text{m}\).
Fig. 2
The choke winding may be made with PELShO wire of diameter \(d = 1.8\ \text{mm}\). The diameter of the wire with insulation is \(d_{\text{ins}} = 1.92\ \text{mm}\). The number of turns per layer is 10; the number of layers is 9.2. The length of the mean turn is \(13\ \text{cm}\). The total wire length is \(12.1\ \text{m}\). The active resistance of the choke is \(r = 0.085\ \Omega\), which may be considered permissible.