N. F. Voloshin.

Improvement of $\cos \varphi$ at Metalworking Plants.

I. Features of metalworking plants affecting the power factor.

Regardless of the nature of the enterprise, the plant-wide average power factor depends on the following principal factors:

1) the average installed power of motors and other kinds of receivers of electric energy
$\left(\dfrac{\sum P_{\text{inst. n}}}{n}\right)$,

2) the magnitude of their load factor ($\eta_{\text{in}}$),

3) the magnitude of the simultaneity factor of operation of the receivers ($\eta_g$),

4) the operating mode of the receivers (continuous, short-time, intermittent, impact),

5) the operating schedule of the enterprise during the day, by shifts,

6) the kind of receiver (asynchronous motor, ordinary crane motor, electric furnaces, welding transformers, etc.),

7) the necessity of speed regulation within known limits,

8) the correctness of the choice of the installed power of the receiver of electric energy.

Let us determine the influence of each of the factors on the average power factor as applied to metalworking plants.

1) Influence of the average installed power of motors.

Let us assume that the installed power of the motors is used completely, i.e., that the load factor and the simultaneity factor of operation are equal to unity.

In this case the receivers operate at their rated power, and, consequently, the total $\cos \varphi$ of the plant will be determined by the rated power factors of the receivers; its value may be expressed in the form (provided that all receivers are of three-phase current):

\[ \cos \varphi_z = \frac{1}{ \sqrt{ 1+ \left[ \frac{\sum J_n V_n \sin \varphi_n} {\sum J_n V_n \cos \varphi_n} \right]^2 } } \qquad\ldots\ldots\ldots (1) \]

If the insignificant difference in the applied voltage is neglected, i.e., if we put:

\[ V_1 = V_2 = V_3 = \ldots = V_n \]

then:

\[ \cos \varphi_z = \frac{1}{ \sqrt{ 1+ \left[ \frac{\sum J_n \sin \varphi_n} {\sum J_n \cos \varphi_n} \right]^2 } } \qquad\ldots\ldots\ldots (2) \]

Obviously, the larger the ratio

\[ \frac{\sum J_n \sin \varphi_n}{\sum J_n \cos \varphi_n} \tag{3} \]

the smaller the plant-wide power factor will be.

Let us determine what this ratio (3) depends on and under what conditions it increases. Expanding the summation sign, we obtain:

\[ \frac{\sum J_n \sin \varphi_n}{\sum J_n \cos \varphi_n} = \frac{J_1 \sin \varphi_1 + J_2 \sin \varphi_2 + \cdots + J_n \sin \varphi_n} {J_1 \cos \varphi_1 + J_2 \cos \varphi_2 + \cdots + J_n \cos \varphi_n} \]

where

\(J_1 \sin \varphi_1 \ldots J_n \sin \varphi_n\) are the reactive components of the current,

\(J_1 \cos \varphi_1 \ldots J_n \cos \varphi_n\) are the active components of the currents of the receivers.

It is obvious that, when small motors predominate, having a relatively lower rated power factor, we shall obtain an increase in the indicated ratio and, consequently, a decrease in the plant-wide power factor.

Introducing the concept of the average installed motor power, as a quantity which to a certain degree characterizes the average power factor, i.e.

\[ P_{0\ \mathrm{av}}=\frac{\sum P_a}{n} \]

\[ P_{p\ \mathrm{av}}=\frac{\sum P_{pn}}{n} \]

we obtain:

\[ \cos \varphi=\frac{1}{\sqrt{1+\operatorname{tg}^{2}\varphi_{\mathrm{av}}}} \]

Fig. 1 presents the curves of the dependence of \(\cos \varphi\) on the rated motor power for type \(T\), where I is for \(n=1500\), II for \(n=1000\), and III for \(n=750\) rpm.

Table 1 gives a summary of the average installed power, according to design data, for certain metalworking plants.

Table 1

No.	Character of plant	\(P\) inst.	No.	Character of plant	\(P\) inst.
1	Large machine-building	9.7	7	Plant for crane equipment	5.9
2	" "	8.5	8	" " "	3.7
3	" "	7.9	9	Locomotive-repair	4.87
4	Medium machine-building	7.4	10	"	4.85
5	" "	6.2	11	Small machine-building	1.32
6	" "	5.2

It follows from the table that small motors predominate at metalworking plants, since the average installed power does not exceed 10 kW. This predominance of small motors, in turn, is the reason why the average rated power factor does not exceed 0.8, unlike other industrial enterprises with a predominance of large motors (for example, the metallurgical industry).

Fig. 1.

2) Influence of the load factor \((\eta_{\mathrm{B}})\) on the plant power factor.

The change in the power factor of an individual motor may be derived from the following: the load factor of a motor is defined as the ratio of the power developed by the motor on the shaft at a given moment to its rated power, i.e.

\[ \eta_{\mathrm{B}}=\frac{P_{mx}}{P_{mn}}=\frac{J_x \cos \varphi_x \eta_x}{J_n \cos \varphi_n \eta_n} \]

whence

\[ \cos \varphi_x=\frac{J_n \eta_n}{J_x \eta_x}\cos \varphi_n \eta_{\mathrm{B}}. \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (5) \]

Applying Goldschmidt’s formula for calculating the efficiency at partial load,

\[ \cos \varphi_x= \frac{J_n \cos \varphi_n}{J_x(1+\alpha)} \left[(1+\alpha)\eta_n\eta_{\mathrm{B}}+(1-\eta_n)(\eta_{\mathrm{B}}^2+\alpha)\right] \ . \ . \ . \ (5\text{-a}) \]

The following notation is adopted:

$J_x$, $J_n$—the motor current at partial and full load,

$\eta_n$, $\cos \varphi_n$—the efficiency and power factor at full load,

$\alpha$—the ratio of the constant losses of the electric motor to the variable losses.

Equation (5-a), while not having direct computational significance, characterizes quite clearly the dependence of a motor’s $\cos \varphi$ on the degree of its loading, since the reduction of the nominal $\cos \varphi_n$ will be the greater, the smaller the degree of loading of the motor. In other words, it indicates the harmful consequences, from the standpoint of $\cos \varphi$, caused by lightly loaded motors.

Figure 2 presents curves of the dependence of the load factor of the motors of metalworking machine tools, obtained as a result of a survey of a group of metalworking plants in Leningrad; the following notation is adopted:

Fig. 2

$P_w$—the mechanical (installed) power of the machine tool. The designations of the curves—

output: I—lathes, II—turret lathes, III—milling machines, IV—drilling machines, V—vertical boring mills, VI—planing machines.

The low values of the load factors of the machine motors determine the low values of \(\cos \varphi\) of the shop.

Thus, for example, for a center lathe with \(P_{\text{n}} = 6.8\) kW, \(n = 950\) rpm, we shall have:

\[ \cos \varphi_{\text{n}} = 0.81 \quad \eta_{\text{v}} = 1 \]

\[ \cos \varphi_{\text{x}} = 0.48 \quad \eta_{\text{v}} = 0.26, \]

i.e., a decrease of \(\cos \varphi\) by 0.31 as compared with the normal value.

It must be noted that the design values of \(\eta_{\text{v}}\) are usually higher. For example, Gipromash takes as the basic value \(\eta_{\text{v}} = 0.6\) for an individual drive, but it is difficult to say how real this value actually is, since, as far as I know, there has been no special study of this question on a broad scale.

One of the reasons for the low load factors of metalworking machines is that the motor installed on them is selected on the basis of the maximum load of the machines, at which the machine practically does not operate. This is especially sharply noticeable at enterprises without a flow-line production system, where there is no assignment of a part to a definite machine. Thus, for example, an investigation of one of the plants showed the following values of \(\eta_{\text{v}}\) for one and the same type of machine (Table 2).

Table 2

No.	\(P\) nom. machine, kW	\(P\) installed motor, kW	Type of machine	\(\eta_{\text{v}}\)	No.	\(P\) nom. machine, kW	\(P\) installed motor, kW	Type of machine	\(\eta_{\text{v}}\)
1	8.6	7.4	Center lathe “Preytus” . . . .	0.18	7	2.36	3.7	Werner turret lathe . .	0.325
2	4.8	”	”	0.13	8	”	”	”	0.21
3	5.5	”	”	0.30	9	3.02	3.7	Universal milling . . .	0.34
4	5.96	”	”	0.38	10	”	”	”	0.13
5	—	”	”	0.17	11	”	”	”	0.22
6	2.36	3.7	Werner turret lathe . .	0.38

Already from this small table it follows that the picture obtained is rather variegated, and moreover

a) the motor power is quite insufficiently substantiated, and
b) the load factors are both low and varied.

Thus, the relatively low load factors of the motors of metalworking machines determine the correspondingly low power factor of metalworking shops.

3) Influence of the coefficient of simultaneity of operation \((\eta_{\text{d}})\).

The coefficient of simultaneity of operation of the receivers of electric power, while not affecting the \(\cos \varphi\) of the receivers of the shop, affects the operating conditions of the plant substations. It is clear that, at an operating plant, a reduction of \(\eta_{\text{d}}\) leads to a reduction in the load of the transformers and, accordingly, to a reduction of their power factor.

The equation for the dependence of the transformer power factor may be represented in the form:

\[ \varphi_1 \simeq \varphi_2 + \arcsin\left(\frac{J_0}{J_1}\cos \varphi_2\right). \]

or, i.e.,

\[ J_1 = J_{\mathrm{n}}\cdot \eta_{\mathrm{vt}} \]

then

\[ \varphi_1 = \varphi_2 + \arcsin \left[\frac{J_0 \cos \varphi_2}{J_x \eta_{\mathrm{vt}}}\right]. \]

If the transformer load factor is expressed through the installed power of the motors, taking certain average values, under the condition of an unchanged \(\cos \varphi_2\) and load of the individual receivers, then

\[ \eta_{\mathrm{v.t.}}=\frac{\eta_{gx}}{\eta_{gn}} \ . . . . . . . . . . . . . . . . . . (6) \]

Finally, we shall have

\[ \varphi_1 \cong \varphi_2 + \arcsin \left[\frac{J_0 \eta_{gn}\cos \varphi_2}{J_{1n}\eta_{gx}}\right] = \varphi_2 + \arcsin \left(\frac{C}{\eta_{gx}}\right) \ . . . . (7) \]

In the derivation the following designations were adopted:
\(\varphi_1\) and \(\varphi_2\)—angles of displacement of the primary and secondary current of the transformer relative to the corresponding e.m.f.
\(J_0\)—no-load current of the transformer.
\(J_{1n}\)—normal load current of the transformer.
\(\eta_{gn}\)—coincidence factor determining the normal load of the transformer.
\(\eta_{gx}\)—coincidence factor at any transformer load.

Equation (7) shows that a decrease in the simultaneity factor of operation of the receivers as compared with the designed one must worsen the operating conditions of the transformers, lowering their load factor and power factor. This decrease in the \(\cos \varphi\) of the transformers themselves must reduce the magnitude of the plant-wide \(\cos \varphi\).

4) Influence of the operating regime of the receivers on the power factor.

If during operation of a receiver there is no constancy of load, then the power factor for any arbitrary interval of time, with any load schedule, can be determined from the general equations. Let the active energy taken by the receiver from the network be equal to:

\[ W_a=\frac{\sum Pm_n\, t_n}{\eta_n} \]

and, correspondingly, the reactive energy will be

\[ W_p=\frac{\sum Pm_n\, t_n\, \operatorname{tg}\varphi_n}{\eta_n}, \]

where \(Pm_n\) is the shaft power of the given motor, assumed constant over the time interval \(t_n\),
\(\operatorname{tg}\varphi_n\) corresponds to the same interval of time.
Then the average value of \(\cos\varphi\) may be determined from the equality:

\[ \operatorname{tg}\varphi_n = \frac{ \sum \dfrac{Pm_n t_n \operatorname{tg}\varphi_n}{\eta_n} }{ \sum \dfrac{Pm_n t_n}{\eta_n} } \ . . . . . . . . . . . . . . . . . . (8) \]

Analysis of equation (8) in its general form is difficult; therefore we shall write it for the particular case of a periodically recurring load with uniform peaks of energy consumption, such as occurs, for example, in forging machines.

Fig. 3.

We shall adopt for calculation the simplified diagram of Fig. 3 and introduce the following notation:

\[ \eta_{\mathrm{v}1}=\frac{P_1}{P_m};\quad \eta_{\mathrm{v}2}=\frac{P_2}{P_m};\quad \gamma=\frac{t_2}{t_1} \]

On the basis of equation (8) we shall have:

\[ \tg\varphi= \frac{ \tg\varphi_1+\dfrac{P_2}{P_1}\dfrac{t_2}{t_1}\dfrac{\eta_1}{\eta_2}\tg\varphi_2 }{ 1+\dfrac{P_2}{P_1}\dfrac{t_2}{t_1}\dfrac{\eta_1}{\eta_2} } \]

or, replacing \(\dfrac{t_2}{t_1}\), \(\dfrac{\eta_1}{\eta_2}\), \(\dfrac{P_2}{P_1}\), we obtain:

\[ \tg\varphi= \frac{\tg\varphi_1+\alpha\beta\gamma\,\tg\varphi_2} {1+\alpha\beta\gamma} \qquad \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (9) \]

\[ \alpha=\frac{\eta_{\mathrm{v}2}}{\eta_{\mathrm{v}1}};\quad \beta=\frac{\eta_1}{\eta_2};\quad \gamma=\frac{t_2}{t_1} \qquad \ldots\ldots\ldots\ldots\ldots\ldots (10) \]

Let us consider equation (9) as applied to various operating modes of electric motors.

a) Short-time operation; evidently, in this case

\[ P_2=0,\quad \alpha=0,\quad \tg\varphi=\tg\varphi_1 \]

and, consequently, the deterioration of \(\cos\varphi\) will depend only on the overload of the motor, which in this case is usually on the order of 150–200% of nominal. Usually the maximum \(\cos\varphi\) for normal motors occurs at \(P=(1.25\text{--}1.40)P_m\), and beyond this its decrease proceeds, not so significantly that it need be especially taken into account. The short-time operating mode

at metalworking enterprises is encountered comparatively rarely, and therefore a slight deterioration of the $\cos \varphi$ of receivers under such a regime may be disregarded, especially since the specific weight of the energy consumed by the receivers is usually very small.

b) Repeated short-time operation, to which crane motors may be assigned first of all, plays a more substantial role; moreover, $P_0 < P_2 < P_{\mathrm{m}}$, and $P_1$ is determined by the magnitude of the power at starting, with the calculation allowing for not less than a 1.5-fold overload. To this it must be added that the $\cos \varphi$ of crane motors is less than the $\cos \varphi$ of normal motors, and therefore the average power factor of crane installations alone is extremely low.

The matter is somewhat different if one considers the consumption of energy by crane installations operating jointly with normal motors.

Equation (9) may be regarded as an equation in the presence of two types of load:

$P_1$ — normal consumers of electric energy,
$P_2$ — crane motors,
$t_1$ — operating time of normal consumers,
$t_2$ — operating time of crane motors.

Table 3 gives, for individual plants, a summary of the specific weight of crane motors and their annual consumption, from which one may draw a conclusion regarding the relative influence of crane load on the plant’s $\cos \varphi$.

If the average value is taken as $\cos \varphi_1 = 0{,}707$ ($\operatorname{tg}\varphi_1 = 1$), $\cos \varphi_2 = 0{,}40$ ($\operatorname{tg}\varphi_2 = 2{,}3$), then the values of the plant-wide $\cos \varphi$ shown in Table 3 are obtained.

Table 3

No.	Shop / plant	Normal motors, $P$ installed, average	Crane motors, $P'$ installed, average	$\dfrac{\sum P' \text{ inst.}}{\sum P \text{ inst.}}$, %%	Average power of crane transformers	Average power of transformers for normal motors	Plant $\cos \varphi$	Energy consumption per year by cranes as % of consumption by normal motors
1	Mechanical-assembly shops . . .	17,7	8,4	44	551	2460	0,625	22,5
2	”	15	17,7	68	84	241	0,595	35
3	”	12,5	7,4	42	70	331	0,63	21
4	”	9,0	6,2	33	93	574	0,643	16
5	”	7,7	5,2	34	38,4	224	0,641	17
6	”	7,4	7,9	51	135	539	0,618	25
7	”	5,2	3,7	44	28	128	0,627	22
8	Plants . . . . . . .	6,7	4,9	32	119	811	0,649	14,7
9	”	8,5	4,9	41,5	176	925	0,634	19,1
10	”	—	—	72,0	247	750	0,599	33
11	”	5,9	2,76	25,0	16,5	251	0,68	6,5
12	”	7,9	3,5	75,0	84	313	0,618	25

The maximum deviation at

\[ \frac{\sum P_2' t_2}{\sum P_1 t_1} = 0{,}35 \]

from the adopted power factor of the normal motors is equal to 0.11, while for the calculation only the load of the normal motors was taken and lighting, electric furnaces, and other receivers that raise the plant power factor were not taken into account. Hence one may draw the conclusion that the influence of crane load on the power factor does not exceed 15–20% of the motor load and, of the plant-wide load, not more than 10–15%.

c) Continuous operation will evidently occur at $\gamma = 0$; $\alpha = 1$; $\beta = 1$, and equation (9) becomes

\[ \operatorname{tg}\varphi = \operatorname{tg}\varphi_1 \quad \cos \varphi = \cos \varphi_1. \]

This regime has been examined in sufficient detail above and therefore is not discussed here.

d) Impact or peak load, alternating with idling of the motor, leads to the average $\cos \varphi$ having a low value. Its magnitude can be determined from equation (9) for one cycle of operation of the machine tool.

Let us present data characterizing the impact regime, taking forging machines (Table 4).

Table 4

No.	Name of machines	$P_{\text{inst}}$	$P_{\max}$	$P_1$	$P_2$	$P_{\text{avg}}$	$\eta_{\text{v max}}$	$\eta_{\text{v1}}$	$\eta_{\text{v2}}$	$t_1$	$t$	$\gamma=\dfrac{t_1}{t_2}$	$\cos \varphi_{\text{avg}}$
1	Forging “National”	55	120	40.5	7.0	28	2.00	0.67	0.12	13	15	1.15	0.60
2	”	37	100	40.7	5.3	21.6	2.40	1.00	0.17	11	30	2.72	0.66
3	” 4″	26	28.0	15.8	4.0	6.1	0.96	0.5	0.14	9	20	2.2	0.51
4	” 3″	20.5	31.2	14.7	2.2	8.1	1.33	0.66	0.09	4	10	2.5	0.51
5	” 2.5″	14.5	16.6	8.7	2.0	4.7	1.0	0.53	0.12	3	9	3.0	0.47
6	” 1.5″	7.4	11.9	6.1	1.0	4.3	1.35	0.69	0.11	4	4	1.0	0.54
7	” 1.5″	7.4	12.0	5.7	1.0	3.4	1.36	0.65	0.11	6	12	2.0	0.46
8	Bulldozer “Geeveland”	7.5	9.1	3.63	0.7	3.7	1.03	0.43	0.08	4	4	1.0	0.51
9	Hammer “Erie” 2500 f	14.8	38	10.3	3.0	14.6	2.28	0.62	0.19	10	15	1.50	0.51
10	”	14.8	37	21.5	1.8	13.3	2.22	1.28	0.11	20	35	1.75	0.72
11	”	18.9	30	20.7	1.8	12.1	1.46	1.0	0.10	12	27	2.25	0.65
12	”	14.0	40	19.8	1.2	12.4	2.40	1.15	0.07	12	19	1.58	0.75
13	” 1600 f	15	30.3	17.8	1.8	11.4	1.77	1.04	0.11	9	13.5	1.5	0.71
14	”	15	33.0	7.5	1.8	4.4	1.94	0.44	0.11	9	20	2.2	0.44
15	”	15	33.0	19.5	1.8	12.4	1.94	1.14	0.11	14	21	1.5	0.63
16	” 1200 f	7.4	15.8	10.0	1.5	5.5	1.80	1.14	0.20	8	20	2.5	0.70
17	”	7.4	14.8	11.6	1.5	7.7	1.68	1.32	0.20	15	20	1.3	0.72
18	”	7.4	16.2	10.4	1.5	6.6	1.85	1.18	0.20	7.5	11	1.5	0.70
19	”	10	19.4	9.3	0.7	6.3	1.65	0.79	0.06	10	15	1.5	0.65
20	Press “Kiserling”	6.8	8.0	5.8	1.2	6.2	1.0	0.72	0.15	12	36	3.0	0.43

It follows from Table No. 4 that, with a relatively large maximum working-load value ($P_{\max}$ and $\eta_{\text{v max}}$), the average value of the power factor, calculated from equation (9), is below the normally accepted 0.8.

The chief influence on the value of $\cos \varphi$ of forging machines is exerted by $\gamma=\dfrac{t_1}{t_2}$, i.e., the duration of the motor idling, and by the relative magnitude of the load, characterized by the coefficient $\alpha=\dfrac{\eta_{\text{v2}}}{\eta_{\text{v1}}}$. A reduction of these ratios improves the average $\cos \varphi$ of the motor*). As a final conclusion it may be considered that forging shops lower the plant-wide $\cos \varphi$.

5) Influence of the operating regime of the enterprise on the daily power factor.

Usually the shift-by-shift operation of metalworking plants varies, with the maximum lighting load occurring in the second shift. This determines the corresponding values of the power factor by shift; and since projects rarely provide for the possibility of disconnecting and switching transformers on the basis of their maximum load, the underloading of the second and third shifts leads to a decrease in the average daily and consequently monthly power factor due to a reduction in $\cos \varphi$

) For more detail see: Vestnik El-nosti*, No. 3, 1934, article by Eng. L. B. Geiler.

transformer. Let us analyze the design of one locomotive-repair plant from the point of view of the power factor, taking into account the operating schedule of the enterprise, using as a basis the average values adopted in design (Table 5).

Table 5

Substation No.	Operating time	Installed transformer capacity, \(P_{\text{inst}}\)	Data without taking transformer \(\eta_{\text{vt}}\) into account: \(P_a\)	Data without taking transformer \(\eta_{\text{vt}}\) into account: \(P_p\)	Data without taking transformer \(\eta_{\text{vt}}\) into account: \(\cos \varphi\)	Data taking transformer \(\eta_{\text{vt}}\) into account: \(\eta_{\text{v}}\)	Data taking transformer \(\eta_{\text{vt}}\) into account: \(\cos \varphi\)	Data taking transformer \(\eta_{\text{vt}}\) into account: \(P_a\)	Data taking transformer \(\eta_{\text{vt}}\) into account: \(P_p\)
1	8—16	\(8 \times 320\)	648	655	0.70	0.965	0.665	648	738
2	”	\(2 \times 560\)	830	745	0.76	0.975	0.73	830	772
3	”	\(2 \times 320\)	476	440	0.73	0.975	0.70	476	450
		By plant . . .	1954	1840	0.727	—	0.700	1954	1960
1	16—24	\(3 \times 320\)	745	559	0.8	0.97	0.765	745	630
2	”	\(2 \times 560\)	950	667	0.82	1.035	0.79	950	738
3	”	\(2 \times 320\)	524	339	0.84	0.975	0.81	524	374
		By plant . . .	2219	1565	0.82	—	0.786	2219	1742
1	24—8	\(2 \times 320\)	466	212	0.91	0.80	0.87	466	274
2	”	\(2 \times 560\)	695	416	0.86	0.72	0.81	695	502
3	”	\(2 \times 320\)	406	252	0.85	0.75	0.81	406	293
		Average daily values . . . .	1560	880	0.872	—	0.824	1567	1069
		Average daily values . . . .	5740	4285	0.802	—	0.768	5740	4770

It is evident that if, in the design, the influence of the power factor of underloaded transformers had been taken into account, then for the third shift a different combination of transformers would have had to be adopted. Usually the load curve fluctuates even more sharply between the first and third shifts, and yet operating personnel do not always proceed to disconnect transformers operating with underload. Let us give the calculation of \(\cos \varphi\) for the same plant only for the summer load schedule (Table 7).

Table 6

Substation No.	Operating time	Installed transformer \(P\), kVA	Without taking \(\eta_{\text{vt}}\) into account: \(P_a\)	Without taking \(\eta_{\text{vt}}\) into account: \(P_p\)	Without taking \(\eta_{\text{vt}}\) into account: \(\cos \varphi\)	Taking \(\eta_{\text{vt}}\) into account: \(\eta_{\text{vt}}\)	Taking \(\eta_{\text{vt}}\) into account: \(\cos \varphi\)	Taking \(\eta_{\text{vt}}\) into account: \(P_a\)	Taking \(\eta_{\text{vt}}\) into account: \(P_p\)
1, 2, 3	8—16	By plant	1954	1840	0.727	—	0.719	1954	1960
1	16—20	\(3 \times 320\)	470	533	0.66	0.74	0.61	470	610
2	”	\(2 \times 560\)	733	605	0.77	0.85	0.73	733	682
3	”	\(2 \times 320\)	357	329	0.72	0.765	0.675	357	390
		By plant . . .	1560	1467	0.727	—	0.68	1560	1682
1, 2, 3	20—24	—	2219	1565	0.817	—	0.786	2219	1742
1, 2, 3	24—4	By plant	1567	880	0.872	—	0.824	1567	1069
1	4—8	\(1 \times 320\)	199	202	0.7	0.89	0.66	199	235
2	”	\(2 \times 560\)	484	396	0.77	0.56	0.71	484	481
3	”	\(2 \times 320\)	244	241	0.70	0.545	0.64	244	293
		By plant . . .	927	839	0.741	[[unclear: parenthesized dash]]	0.676	927	1009
		Daily . . .	—	—	0.77	—	0.730	—	—

In design practice it is customary to take, for the whole year, the calculated value \(\cos\varphi = 0.802\), whereas even with the most cautious calculation—without allowance for lunch breaks, with a 7-hour working day, and with starting time before shifts—the value obtained for the summer period is \(\cos\varphi = 0.730\), i.e., a difference of 0.072.

Insufficient allowance for the plant load schedule and an excessive fondness for “average” values lead to the fact that the power factor of metalworking plants differs very sharply from the design value. Table 8 gives values of \(\cos\varphi\) for several plants according to survey results, confirming the above:

Table 7

No.	Plant	I shift	II shift	III shift	Daily
1	Elektroapparat	0.51	0.44	0.33	0.44
2	Elektrosila (OHM)	0.599	0.652	0.595	0.61
3	Elektropribor	0.669	0.62	0.578	0.643
4	Telegraph Plant	0.53	0.48	0.50	0.508
5	Copper-rolling plant	—	—	—	0.47
6	Average for 26 metalworking plants of Moscow	—	—	—	0.62

b) Influence of the type of electric-power receiver on the plant-wide power factor.

It is very difficult to make any generalizations, since everything depends on the relative weight of each type of electric-power receiver in the overall balance of the plant’s electrical economy.

We shall indicate only the value of \(\cos\varphi\) of welding transformers for several plants and their relative influence on the plant’s \(\cos\varphi\) (Table 9).

Table 8

No.	Character of plant	\(\sum P_{\text{inst}}\) for the plant, kW	Load \(P_a\), kW	\(\sum P'_{\text{inst}}\), kW	Load \(P'_a\), kW	\(\dfrac{\sum P'_{\text{inst}}}{\sum P_{\text{inst}}}\), %	\(\dfrac{P'_a}{P_a}\), %	Design \(\cos\varphi\)
1	Locomotive-repair	2787	1954	97	68	3.5	3.5	0.35–0.8
2	Agricultural-machinery parts	856	539.6	15	1.70	1.70	0.31	0.30
3	Freight-lifting structures	1992	1140	155	32.3	7.8	7.2	0.5–0.8
4	Machine-building	3136	500	1429	90	1.6	6.3	0.6

The table was compiled from Gipromash design data. It follows from it that the relative weight of welding units and transformers is insignificant and therefore does not exert any special influence on the plant’s \(\cos\varphi\).

Large electric furnaces usually have an independent transformer and therefore do not affect the plant’s power factor on the low-voltage side. They improve the plant-wide \(\cos\varphi\), since they have their own \(\cos\varphi = 0.8–0.9\).

The same may be said of large motors, for example compressor installations, which are connected on the high-voltage side. While improving the plant-wide \(\cos\varphi\), they do not solve the problem as a whole, since the low-voltage network is not freed from reactive currents.

7) Regulation of speed by electrical means is most economical when direct current is available at the plant. This in turn necessitates the installation of equipment converting alternating current into direct current, and, depending on the method of conversion, affects the overall $\cos \varphi$ of the plant. High-power converters have a high $\cos \varphi$ and therefore improve the plant-wide power factor.

At metalworking plants without their own metallurgical base and with small motor installations, the role of direct current in the overall system is very insignificant, and therefore it need not be discussed separately.

8) The correctness of the choice of installed capacity ultimately determines the $\eta_{\mathrm{в}}$ of the motors and, consequently, their $\cos \varphi$. An excess of installed motor capacity increases the required capacity of transformers, which in this case also have incomplete loading.

How unsuccessful the approach to this question sometimes is can be seen from Table 10, which gives the ratios of the capacities of various machine tools at one of the machine-building plants.

Table 9

No.	Name of machine tool	$P_{\text{inst}}$ of machine tool, kW	$P_{\text{inst}}$ of motor, kW	No.	Name of machine tool	$P_{\text{inst}}$ of machine tool, kW	$P_{\text{inst}}$ of motor, kW
1	Horizontal milling No. 1½ Cincinnati	2.61	5.5	15	Horizontal milling Barikand	1.62	1.75
2	〃	3.02	3.7	16	〃 Sonderman	2.43	4.5
3	〃	2.90	5.5	17	Vertical milling Sonderman	3.17	4.5
4	〃	2.9	3.7	18	〃	7.4	5.5
5	Horizontal milling No. 2 Cincinnati	6.25	3.7	19	Universal milling Ingersoll	7.4	10
6	〃	4.20	3.7	20	Center lathe Preytus	5.5	7.4
7	〃	4.0	3.7	21	〃	4.8	7.4
8	〃	5.9	2.6	22	〃	5.96	7.4
9	Horizontal milling No. 3 Cincinnati	4.1	3.7	23	〃	8.6	7.4
10	〃	5.0	3.7	24	〃	8.85	7.4
11	〃	5.4	3.7	25	〃	1.89	5.5
12	Horizontal milling Reinecker	3.02	4.5	26	Revolver lathe Werner	2.36	3.7
13	〃	4.8	7.4	27	〃	2.36	3.7
14	Horizontal milling Wanderer	2.95	4.5	28	〃	2.36	3.7
				29	Planer, double-column, Revol.	2.94	4.5
				30	Center lathe Lange	8.85	11.05
				31	〃	11.8	14.7

The table is sufficiently indicative both from the standpoint of the unsystematic selection of motors and from the standpoint of excessive motor capacity. Incidentally, we note that the machine tools taken for seasoning constitute about 15% of their total number at the plant. The situation is still worse with the loading of machine tools, i.e., with their actual utilization, as can be seen from Table 2.

II. The Most Economical Methods of Compensating Phase Shift at Metalworking Plants

Let us briefly summarize the conditions under which compensation of phase shift should take place. These conditions are as follows:

1) the presence of a large number of small motors as a result of the wide spread of individual drive, which makes it possible to subdivide capacity not only as applied to a given unit, but also, within the unit, to break down individual operations. The average installed capacity of motors does not exceed 10 kW;

2) a non-stationary operating regime of the motors driving machine tools, as a consequence of changes in operations on the machine tools;

3) a relatively large and branched low-voltage distribution network;

4) low values of the average load factors of machine tools, which correspondingly cause low load factors of electric motors and reduce their power factors;

5) insignificant use of direct current, since for the auxiliary drives of metalworking machine tools (the plates of grinding machines, electromagnetic clutches of planing machines, etc.) small converters may be used. Crane motors are used, as a rule, on alternating current, since great accuracy in regulating their speed is not required, as, for example, in large steel-casting shops, where direct current is often preferred to alternating current precisely for reasons of speed regulation;

6) large electric-motor installations above 250 kW, which may include compressors; electric furnaces, if there is an in-house foundry shop, are installed either directly on high voltage (compressors) or have an independent supply transformer (electric furnaces), and therefore their high power factor affects the plant-wide $\cos \varphi$;

7) the variable operating regime of metalworking plants by shifts.

Let us limit the problem of phase-shift compensation only to the incoming feeders of the factory substation, taking a typical metalworking plant supplied from the district network.

Of the existing methods of phase-shift compensation within a plant, three basic ones are distinguished.

1) Centralized compensation, understood as the installation on the high-voltage side for improving the plant-wide power factor without affecting the magnitude of the reactive lagging power in the low-voltage network.

2) Group compensation—the installation of low-voltage reactive-power generators at shop substations or directly in shops for a group of receivers of electric energy.

3) Individual compensation of phase shift, understood as compensation of the phase shift to a definite value for each receiver of electric energy.

Centralized compensation of phase shift, applied in high-voltage networks for reasons of unloading generators for regulation of the network voltage, under the conditions of metalworking plants obviously does not solve the question as a whole.

Centralized improvement of the power factor solves two problems:

a) relieving the high-voltage network and the generators of the central station from the lagging reactive power, which in one way or another increases the available capacity of the station;

b) reducing payment for electric energy, whereby the second question is essentially only a question of tariff policy—“control by the ruble”; it does not solve the basic problem of freeing the plant from reactive power.

The most favorable solution of the question will obviously be such a solution in which the entire distribution network and transformer substations are freed from reactive power, since under these conditions the equipment will be utilized to the maximum.

Let us determine the most advantageous conditions for phase-shift compensation.

1) If the active power of the transformers is equal to $P_a$, then at $\cos \varphi_1$ their required apparent power is equal to:

$$ P_1 = \frac{P_a}{\cos \varphi_1} $$

Having raised the power factor to a value equal to \(\cos \varphi_2\) with \(P_a\) constant, we shall evidently obtain the required apparent power:

\[ P_2=\frac{P_a}{\cos \varphi_2} \]

Whence the excess of the installed power of the transformers will be equal to:

\[ \Delta P=\frac{P_1-P_2}{P_1}\,100 = \left(1-\frac{\cos \varphi_1}{\cos \varphi_2}\right)100\% \ . . . . . . . (1) \]

Since \(\cos \varphi_1<\cos \varphi_2\), the difference will be positive. Comparing with \(\cos \varphi_2=1\), we obtain equation (1-a)

\[ \Delta P=(1-\cos \varphi_1)\,100\% \ . . . . . . . . . . . (1\text{-a}) \]

i.e. the required excess apparent power of transformers at a power factor less than unity is directly proportional to the existing power factor.

On the other hand, the difference \(\Delta P\) is equal to zero at \(\cos \varphi_1=1\), whence it follows that compensation is advantageous up to \(\cos \varphi=1\), since in this case the minimum transformer power is required.

2) The efficiency of a transformer at reduced \(\cos \varphi_1\) may be expressed as:

\[ \eta_{\mathrm{tr}1} = \frac{P_n\cos \varphi_1} {P_n\cos \varphi_1+\dfrac{\Delta U\,P_n}{100}+P_0} \ . . . . . . . . . . . (2) \]

and at \(\cos \varphi_2=1\)

\[ \eta_{\mathrm{tr}2} = \frac{P_n} {P_n+\dfrac{\Delta U\,P_n}{100}+P_0} \ . . . . . . . . . . . (3) \]

The relative decrease in the transformer efficiency will be equal to

\[ \Delta \eta_{\mathrm{tr}2} = \frac{\eta_{\mathrm{tr}2}-\eta_{\mathrm{tr}1}}{\eta_{\mathrm{tr}1}}\,100\%, \]

which, after substitution and transformations, gives:

\[ \Delta \eta_{\mathrm{tr}} = \frac{0.01\,\Delta U\,P_n+P_0} {P_n+0.01\,\Delta U\,P_n+P_0} \cdot \left[\frac{1}{\cos \varphi_1}-1\right] = C_1 \left[\frac{1}{\cos \varphi_1}-1\right] \ . . . (3\text{-a}) \]

whence it follows that the transformer efficiency has a maximum at \(\cos \varphi_1=1\).

In equations (2), (3) and (3-a) the following notation is adopted:

\(P_0\)—no-load power of the transformer,
\(P_n\)—rated power of the transformer in kVA,
\(\Delta U\)—voltage drop at the secondary terminals of the transformer, in %%.

3) The relative increase in power losses in the network may be expressed as follows:

\[ P_1=\frac{P_a L}{KqU^2\cos^2\varphi_1} \qquad P_2=\frac{P_a L}{KqU^2\cos^2\varphi_2} \]

The following notation is adopted:

\[ K=\frac{1}{S} \]
—specific conductivity,

\(q\)—cross-section of the cable,
\(U\)—line voltage,
\(L\)—length of the distribution network or of an individual cable, feeder, etc.

For $\cos \varphi_1 < \cos \varphi_2$

\[ \Delta P=\frac{P_1-P_2}{P_2}\,100\%=\left[\frac{\cos^2\varphi_2}{\cos^2\varphi_1}-1\right]100\% \ldots \ldots \ldots (4) \]

For $\cos \varphi_2 = 1$, i.e., in comparison with the most favorable case:

\[ \Delta P=\operatorname{tg}^2 \varphi_1\,100\% \ldots \ldots \ldots (4\text{-a}) \]

4) The reduction in the efficiency coefficient of the network can be obtained from the preceding equations, taking into account the following:

\[ \eta_{c1}=\frac{P_a}{P_a+P_1}, \qquad \eta_{c2}=\frac{P_a}{P_a+P_2} \]

\[ \Delta \eta_c=\frac{\eta_{c2}-\eta_{c1}}{\eta_{c2}} =\frac{P_1-P_2}{P_a+P_2}\ldots \ldots \ldots \ldots (5) \]

The relative reduction of the network efficiency after the transformations will be:

\[ \Delta \eta_c= \frac{L\left(\cos^2\varphi_2-\cos^2\varphi_1\right)} {\cos\varphi_1\left(a\cos^2\varphi_2+L\right)}, \]

where $a=KqU^2$.

For $\cos\varphi_2=1\ldots \varphi_2=0$ we obtain:

\[ \Delta \eta_c=\frac{L}{a+L} \left[\frac{1}{\cos^2\varphi_1}-1\right] =C_2\operatorname{tg}^2\varphi_1 \ldots \ldots \ldots (5\text{-b}) \]

From equation (5) it follows that the efficiency of the network will also have a maximum at $\cos\varphi_1=1$, which is quite understandable, since the Joule losses determining $\eta_c$ are proportional to the square of the current, which in turn is inversely proportional to $\cos\varphi$.

5) The relative reduction of the cross-section of the distribution network for a given and constant active power is equal to:

\[ \Delta q=\frac{q_1-q_2}{q_1}\,100 =\left[1-\frac{\cos^2\varphi_1}{\cos^2\varphi_1}\right]100\% \ldots \ldots \ldots (6) \]

If we are interested in the relative saving at $\cos\varphi_2=1$, i.e., the relative increase of the cross-section when $\cos\varphi_2$ is reduced from 1 to $\cos\varphi_1$, then the equation will take the form:

\[ \Delta q=\left[1-\cos^2\varphi_1\right]100\% \ldots \ldots \ldots (6\text{-a}) \]

which again shows that the most advantageous case will be $\cos\varphi=1$.

6) The increase in the weight of the cable is obviously equal to:

\[ \Delta G=\frac{G_1-G_2}{G_1}\,100\% =\frac{\gamma L}{\gamma L}\left[\frac{q_1-q_2}{q_1}\right] =\left[1-\frac{\cos^2\varphi_1}{\cos^2\varphi_1}\right]100\% \]

or, for $\cos\varphi_2=1$,

\[ \Delta G=\left[1-\cos^2\varphi_1\right]100\% \ldots \ldots \ldots (7) \]

I. Thus, the technically most advantageous case is compensation of the phase shift on the low-voltage side to an angle equal to zero, or to a power factor equal to unity. In this same case, the required cross-section of the distribution network and the power of the transformers are reduced to the greatest extent, and their efficiency has a maximum.

As stated above, by group compensation is meant the installation of reactive-power generators:

a) on the low-voltage busbars of the substation,

b) directly in the shop for a group of electric-power receivers.

The first method, more convenient from the standpoint of servicing the compensating installation and of selecting large units, is technically disadvantageous in that the distribution network is not relieved of reactive power.

The second method, while relieving, in addition to the transformers, the distribution network, has the drawback that servicing conditions are worsened and, moreover, generators of smaller capacities must be installed; this is, of course, provided that the entire distribution network of the plant is to be relieved of reactive power (from the group panels).

The relative disadvantage of group compensation also consists in the fact that, owing to the very uneven operating schedule of metalworking plants, with compensation up to a $\cos \varphi$ close to unity, overcompensation is possible, i.e., loading of the network with reactive leading power, which will lead to the consequences indicated above.

The variable operating schedule of an enterprise by shifts, as well as the relative decrease in reactive (lagging) power during the evening and night shifts, both because of a reduction in the number of operating motors and because of an increase in $\cos \varphi$ due to the switching-on of the lighting load, requires special supervision of the installations for compensating the phase shift. All these three circumstances do not permit recommending group compensation under the conditions of metalworking enterprises.

The principal consumers of reactive (lagging) energy at metalworking plants are small motors with the operating conditions enumerated above; therefore, if one takes into account, in addition, conclusion I, then the most applicable is the 3rd method, i.e., individual compensation.

II. Consequently, under the conditions of metalworking plants, individual compensation of the phase shift is the most expedient.

Let us turn to the question of selecting reactive-power generators for metalworking plants.

1) Synchronous and asynchronous compensators obviously cannot at all be used at metalworking plants as the principal reactive-power generator for centralized improvement of $\cos \varphi$, especially since they are built for large capacities (from 1000 kVA and above) and predominantly for high voltage.

2) Synchronous motors will find very limited application for driving compressors.

The minimum power of the ST series of the VEO plants is 82 kVA, or $P_m \simeq 60$ kW.

Moreover, they do not solve the problem, since they are a means of centralized or group compensation.

3) Phase compensators are applicable only for motor capacities of the order of not less than 200–250 kW and higher, and therefore are in general not applicable under the conditions of metalworking plants, since for driving large machines it is more expedient to take a normal synchronous motor.

4) Synchronized asynchronous motors at full load have low stability and a low efficiency. When there are several large motors located at a sufficient distance, it is necessary either to use a low-voltage exciter of the order of 10–15 volts and below, or else to create a special low-voltage network of the same order, since the magnitude of the generator voltage in this case is determined only by the voltage drop in the rotor winding of the motor. If several large motors, but of different…

capacities, an additional expenditure of energy appears because motors of lower capacity, as requiring a smaller excitation current, must be shunted (the usual arrangement is the series connection of the rotor circuits of several motors in order to increase the generator voltage). If, however, one takes into account the magnitude of the average installed capacity of the motors (Table 1), it becomes clear that this method of improving the power factor cannot be recommended as a mass method under the conditions of metalworking enterprises. It is possible that in certain special cases of a motor operating underload, when its capacity is large, this method too may be applicable as a special exception; although, with a large already existing motor, it is more advantageous to use a phase compensator, and, when installing a new one, to take a normal synchronous motor.

5) Compensated asynchronous motors with $\cos \varphi = 1$ are not manufactured at all in this country because of their high cost and the complication of operating conditions. Abroad they find very little application, and therefore they, too, may be left out of further consideration.

6) AC commutator motors, while having a high $\cos \varphi$ and wide speed regulation, have a relatively low efficiency, are expensive and inconvenient to operate, and therefore have found no distribution under the conditions of metalworking plants.

7) Static capacitors are the only means that can be recommended, without special reservations, for improving the $\cos \varphi$ of metalworking plants.

III. Thus, for improving the $\cos \varphi$ of metalworking plants, the technically most correct procedure will be the use of static capacitors at low voltage, directly at the consumers of reactive energy.

Let us consider in more detail the advantages and disadvantages of static capacitors.

1) Economic comparison of static capacitors with other types of compensation.

The economic comparison has been made for those types of compensation which can be applied under the conditions of metalworking enterprises, i.e., not above 400 kVA per unit. Since synchronous condensers are a means of centralized compensation of phase shift, naturally they are not considered here.

The annual cost of operating any installation for the production of reactive power consists of:

a) fixed costs, which must include depreciation deductions for the equipment and interest charges on the initial expenditures, and

b) variable costs, consisting in turn of the cost of the active-energy losses necessary for the production of reactive power, and the cost of maintenance.

Let us introduce the following notation:

$K$ — cost of annual operation of the installation,

$A$ — cost of the initial expenditures,

$P$ — interest charges,

$V$ — losses in the installation,

$h$ — number of hours of use,

$b$ — cost of one kilowatt-hour,

$W$ — cost of maintenance of the installation.

Then the total annual operating costs of the compensation installation may be expressed in the form

\[ K = pA + Vhb + W \ldots \ldots \ldots \ldots \ldots \ldots (8) \]

In order to make it possible to compare different methods of compensation, let us introduce the concept of the specific cost of operation, understanding by this the cost of the annual operating expenses referred to one reactive kVA. In this case, obviously, we shall have:

\[ K = pa + vhb + w \ldots \ldots \ldots \ldots \ldots \ldots (9) \]

All the notation is as before, only referred to one reactive kVA. In Fig. 4 a comparison is given of this specific cost of operation for synchronized motors at different numbers of hours of utilization, according to data of the German firm Siemens-Schuckert. From a comparison of the curves it follows that, despite the high initial cost of static capacitors \((h = 0)\), already at \(h = 2000\) hours per year static capacitors become more economical (at small powers, less than 50 kVA). The unconditional economic advantage of static capacitors will occur at \(h = 4000\) hours per year.

Fig. 4.

In Fig. 5 a comparison is given of static capacitors and compensated motors of small powers according to the same indices, with static capacitors being more advantageous to the left of the dotted line, and compensated asynchronous motors to the right. It is characteristic that within the range of small powers the magnitude of the specific costs does not depend on the power of the capacitor. This is quite explainable, since the 1st and 3rd terms of equation (9) remain—

remain unchanged, while the 2nd changes only in proportion to the number of hours of use.

Fig. 5.

2) Processes of switching capacitors on and off.
a) Switching on a capacitor.
The instantaneous value of the current and voltage may be expressed by the following known equations:

\[ i_c = J_{\max}\left(-\sin \omega t+\frac{e^{-\frac{t}{2T}}}{\omega}\sin \nu t\right) \tag{10} \]

\[ U_c = U_{\max}\cos \omega t - e^{-\frac{t}{2T}}\cos \nu t) \tag{11} \]

where \(J_{\max}\), \(U_{\max}\) are the amplitude values of the charging current and voltage in the steady state,

\[ T=\frac{L}{R} \]

is the time constant of the oscillatory circuit.

\[ \nu=\sqrt{\frac{1}{LC}-\left(\frac{R}{2L}\right)^2} \tag{12} \]

is the natural number of oscillations of the circuit.

It follows from equation (10) that the switching-on current is the sum of the steady-state and equalizing currents, the equalizing current being equal to the product of the steady-state current by the ratio of frequencies: the natural frequency of oscillation of the given circuit to the frequency of the network. Usually the frequency of natural oscillations lies within the limits of 250–350 periods/sec, and therefore one should expect a current surge at the first moment of switching on equal to 5–7 times the normal-

... nominal current. This is under the condition of an insignificant active resistance of the circuit. If, however, \(R\) is large, then, as follows from equation (10), the natural frequency of the circuit is lowered and, correspondingly, the switching-in current must decrease. The duration of the switching-in process, the magnitudes of the current surges thereby obtained, and the influence of the additional resistance during switching-in are shown in Fig. 6, where two instants of switching-in are taken, namely

Fig. 6.

at the voltage maximum and when the voltage curve passes through zero. A comparison of the curves shows how strongly the additional resistance affects the magnitude of the current surge and, consequently, that to reduce it, it is expedient to switch in an additional resistance operating only at the instant of switching in the capacitor.

If it is taken into account that squirrel-cage motors connected directly to the network have a current surge of the same order, it is clear that switching in capacitors without an additional resistance is no more dangerous than switching in squirrel-cage motors. The duration of the process, measured in hundredths and thousandths of a second, evidently ensures its normal operation with respect to the thermal regime, even with relatively frequent switching-in, since the amount of energy released in it should not be especially large \((I^2 R_c t)\), because \(t\) is very small.

Figures 7—13 present oscillograms of switching-in at various instants of passage of the voltage curve. Table 11 (see p. 232) gives the data relating to the listed oscillograms; moreover, for clarity, some of the experiments were carried out with a single-phase circuit.

Fig. 7

\(C = 13\,\mu\mathrm{F}\)
\(U = U_{\max}\)

Visible annotations in the oscillogram: \(T = 0\), \(U = 0\), \(8\,\mathrm{A}\), \(600\,\mathrm{V}\), \(8\,\mathrm{A}\), \(600\,\mathrm{V}\).

Fig. 8

\(C = 13\,\mu\mathrm{F}\)
\(U = 0\)

Visible annotations in the oscillogram: \(T = 0\), \(U = 0\), \(8\,\mathrm{A}\), \(600\,\mathrm{V}\), \(8\,\mathrm{A}\), \(600\,\mathrm{V}\).

Fig. 9

\(C = 82\,\mu\mathrm{F}\)
\(U = U_{\max}\)

Visible annotations in the oscillogram: \(T = 0\), \(U = 0\), \(600\,\mathrm{V}\), \(600\,\mathrm{V}\), \(100\,\mathrm{A}\), \(100\,\mathrm{A}\).

b) Switching off the capacitor.

The discharge of a capacitor through an external resistance, in mathematical form, may be expressed by the following equations:

\[ u_c = U e^{-\frac{t}{T_1}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots (13) \]

\[ I_c = \frac{U}{R} e^{-\frac{t}{T_1}} \ldots \ldots \ldots \]

where \(U\) is the voltage at the moment of switching off, and \(R_1\) is the discharge resistance (external resistance); \(T_1 = RC\) is the time constant of the discharge, if the inductive resistance is neglected. Consequently, the greater the capacitance and the resistance, the longer the discharge process will be. If the capacitor is discharged only through its own resistance, which is usually very large,

Table 10

Fig. Nos.	\(C\), \(mF\)	\(R_{\mathrm{add}}\), \(\Omega\)	Switching instant \(e=\)	\(U_{\mathrm{eff}}\)	\(v\)	\(t\)	\(U'/U_{\mathrm{eff}}\)	\(J'/J_{\mathrm{eff}}\)	Capacitor connected to
7	13	0	max.	505	350	0.017	1.8	7	Induct. reg. 35 kva
8	13	0	0	495	350	0	1.0	1	Tr-r 45 kva
9	82	0	max.	540	650	0.014	1.75	12	—
10	\(3 \times 89\)	choke		504	300	0.02	1.56	4.7	Transf. 50 kva
11	2	0	any	520	3000	0.008	—	—	280 ″
12	134	1	″	520	0	—	—	—	″ 280 ″
13	134	2	″	520	…	—	—	—	″ ″

Fig. 12
\(C = 134\,mF\)
\(R_1 = 1\,\Omega\)

Fig. 13
\(C = 134\,mF\)
\(R_1 = 2\,\Omega\)

naturally, the discharge process will last a considerable time. Obviously, during discharge one need not fear a rise in voltage even without a protective resistance, and the discharge is mainly needed for quickly removing the voltage from the capacitor in order to avoid a possible case of a worker touching the capacitor. The duration of the discharge process, obviously,

can be reduced by connecting an additional ohmic resistance in parallel with it. This effect of the added resistance is shown in Figures 14–15: in the first case with the capacitor short-circuited (Fig. 14), in the second with \(R_1 = 2\) ohms.

Connecting an inductive resistance also has a beneficial effect on the process of capacitor discharge. Figure 16 shows two curves: the first for the case of discharge through a high-ohmic resistance, and the second curve—discharge through a choke. As is seen from the comparison, in the second case the damping proceeds very rapidly.

As a choke for individual compensation, the winding of the motor to which the capacitor is connected may serve.

\(C = 134\,\mu\mathrm{F}\)  \(R_1 = 1\,\Omega\)

Fig. 14

Consideration of the nonstationary operating conditions of capacitors leads us to the conclusion that, with appropriate measures (protective discharge resistance), these conditions will not affect the normal operation of the plant electrical system; therefore, from this point of view as well, capacitors are entirely applicable.

\(C = 134\,\mu\mathrm{F}\)  \(R_1 = 2\,\Omega\)

Fig. 15

Among the other advantages of static capacitors, the following should be noted:

1) the simplicity of capacitor construction, requiring minimal maintenance, or rather periodic inspection;

2) the divisibility of static capacitors, which can be made from a few tenths to hundreds of kVA in a single unit, which makes it possible—

their installation at each receiver of electrical energy in accordance with the magnitude of the reactive power being compensated.

The divisibility and simplicity of the design permit their installation under any conditions, which in turn makes it possible to switch them on and off together with the receiver of electrical energy, thereby achieving simple regulation of the power factor within the specified limits.

Fig. 16.

3) The high efficiency of capacitors, which accounts for their low own consumption of active energy \((\eta_k = 0.99—0.995)\) and its independence of the size of the capacitor.

Thus, as a final conclusion one may say that, under the conditions of metalworking plants, the most effective, technically and economically, will be the application of phase-shift compensation by static capacitors on the low-voltage side directly at the consumer, up to a power-factor value equal to unity.

REFERENCES USED

1) Scharowsky. Der Leistungsfaktor. 1930. Berlin.

2) K. Baudisch and H. Kann. Die Kondensatoren in Industrieanlagen und Verteilungsnetzen (Simens-Zeitschrift, 1932, H. 10).

3) Max Hürbin. Statische Kondensatoren zur Verbesserung des Leistungs-faktor in ihrer Rückwirkung auf das Netz. Schweiz. Elektrotechnischer Verein Bulletin, No. 19, 1929.

4) Fig. 2 is taken from the technical report of a study of the metal plants of the city of Leningrad by the LOVEO brigade under the direction of Prof. S. A. Rinkevich.