Solution manual of electric machines 2n edition charles i hubert PDF

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University of Karbala

DC Machine

Electrical & Electronics Eng. Dep.

CH. 3- D.C. GENERATORS 3.1. Generator Principle: An electrical generator is a machine which converts mechanical energy (or power) into electrical energy (or power). The energy conversion is based on the principle of the production of dynamically (or motion ally) induced e.m.f. Whenever a conductor cuts magnetic flux, dynamically induced e.m.f. is produced in it according to Faraday’s Laws of Electromagnetic Induction. This e.m.f. causes a current to flow if the conductor circuit is closed. Hence, two basic essential parts of an electrical generator are (i) a magnetic field and (ii) a conductor or conductors which can so move as to cut the flux. 3.2. Types of Generators Generators are usually classified according to the way in which their fields are excited. Generators may be divided into (a) separately-excited generators and (b) self-excited DC generators. (a) Separately-excited DC generator: The field winding are energized from an independent external d.c. current source, as shown in Fig. 3.2. If

+ DC Source

-

Ia = IL

IL

A1

F1

Ra Vt

Ea

Rf F2

A2

Fig. 3.2 Separately-excited d.c generator.

(b) Self-excited DC Generators: The field windings are energized by the current produced by the generators themselves. Due to residual magnetism, there is always present some flux in the poles. When the armature is rotated, some e.m.f. and hence some induced current is produced which is partly or fully passed through the field coils thereby strengthening the residual pole flux. There are three types of self-excited generators named according to the manner in which their field coils (or windings) are connected to the armature. 33 Ms.c. Haider M. Umran

University of Karbala

(i)

DC Machine

Electrical & Electronics Eng. Dep.

DC Shunt Generator: The field windings are connected across or in parallel with the armature conductors and have the full voltage of the generator applied across them, as shown in Fig. 3.3. IL

Ia F1

A1

Rsh

Ea

F2

A2

Vt

Fig. 3.3: DC shunt generator.

Current and voltage relations can be expressed as: Ia = IL + Ish Ish =

Vt

Rsh

The induced e.m.f is: Ea= Vt + Ia. Ra + Vbrush

And

Ea =

ФpNZ 60 a

In practice, Vbrush is neglected. (ii)

DC Series Generator: In this case, the field windings are joined in series with the armature conductors, as shown in Fig. 3.4. As they carry full load current, they consist of relatively few turns of thick wire or strips. Such generators are rarely used except for special purposes i.e. as boosters etc. Ia S1

S2

Ise

IL

A1

Ea

Vt

A2

Fig. 3.4: DC series generator. 34 Ms.c. Haider M. Umran

University of Karbala

DC Machine

Electrical & Electronics Eng. Dep.

The relations between voltage and current can expressed as: Ia = Ise= IL Ise: Current through series winding. E.m.f equation is: Ea = Vt +Ia. Ra + Ia. Rse +Vbrush Ea = Vt +Ia (Ra +Rse) +Vbrush Where; (iii)

Ea =

ФpNZ 60 a

DC Compound Generator: It is a combination of a few series and a few shunt windings and can be either short-shunt or long-shunt, as shown in Fig. 3.5 and 3.6. In a compound generator, the shunt field is stronger than the series field. When series field aids the shunt field, generator is said to be commutativelycompounded.

a) Long Shunt Compound Generator: Ish Ia= Ise And

Se. Sh.

sh

Ea

Ea = Vt + Ia . Ra + Ia. Rse +Vbrush

IL

Ia

A1

Ia= Ish+ IL V Ish = R t

Where;

Ise

Vt

A2

Rsh: Resistance of shunt field winding. Fig. 3.5: Long shunt compound generator

b) Short Shunt Compound Generator: And

∴ ∴

Ia = Ise+ Ish Ise = IL

IL

S Ish

Ia = IL+ Ish E− I .R Ish = R a a

A1

sh

Sh.

Voltage equation is; Ea = Vt+ Ia Ra +Ise Rse +Vbrush Ise= IL Ea = Vt + Ia Ra + IL Rse+ Vbrush Neglecting Vbrush, Ea = Vt + Ia Ra + IL Rse Ea - Ia Ra = Vt + IL Rse

Ise Ia Vt

E

A2 Fig. 3.6: Short shunt compound generators.

35 Ms.c. Haider M. Umran

University of Karbala

DC Machine

Electrical & Electronics Eng. Dep.

V + I .R

∴

Ish = t RLsh se a) Cumulative and Differential Compound Generator: When the field excitation is produced by a combination of shunt field winding and series field winding as shown in Fig. 3.7, the shunt and series fields help each other, the compound generator is termed cumulative compound. A1

ΦT = Φsh + Φse Where:

F1

Φsh = Flux produced by shunt. Φse = Flux produced by series.

F2 A3

A4 A2

Fig.3.7: Cumulative compound.

When the shunt and series field oppose each other, then the generator is differential compound as shown in Fig. 3.8. As a result, the terminal voltage falls drastically with increasing load.

ΦT = Φsh - Φse

A1 F1

F2 A3

A4 A2

Fig. 3.8: Differential compound

Ex.: A DC shunt generator has shunt field winding resistance of 100 Ω. It is supplying a load of 5 Kw at a voltage of 250 V. if its armature resistance is 0.22 Ω. Calculate the induced e.m.f of generator. Sol.: Ia = IL + Ish Ish = Vt / Rsh = 250 v / 100 Ω = 2.5 A IL= PL / Vt = 5×103 / 250 = 20 A Ia = IL + Ish = 20 + 2.5 = 22.5 A Ea = Vt + Ra.Ia = 250 + 0.22 × 22.5 = 254.95 V.

Ish

Ia

IL

F1 G

Vt

F2

36 Ms.c. Haider M. Umran

University of Karbala

DC Machine

Electrical & Electronics Eng. Dep.

Ex.: A 4 pole, compound DC generator long-shunt type, supply’s 100 A at a terminal voltage of 500 V. If armature resistance is 0.02Ω, series field resistance 0.04 Ω and shunt field resistance 100Ω, find the generated EMF. Take drop per brush as 1 V, Neglect armature reaction. Sol: Ish = Vt / Rsh = 500 / 100 = 5 A Ia = IL + Ish = 100 + 5 = 105 A Voltage drop on series field windings =105×0.04= 4.2V Armature voltage drop = 105 × 0.02=2.1 volt Drop at brushes = 2 × 1= 2 V Now, E.m.f = V+ Ia. Ra +series drop+ brush drop = 500 + 2.1 + 4.2 + 2 = 508.3 V. Ex.: A 20 kW compound generator works on full load with a terminal voltage of 250 V. The armature, series and shunt windings have resistances of 0.05Ω, 0.025Ω and 100 Ω respectively. Calculate the total E.M.F generated in the armature when the machine is connected as short shunt. Sol.: Load current =P / V =20000/ 250 = 80 A Voltage drop in the series windings = 80 × 0.025 = 2V Voltage across shunt winding =Vt+ Voltage drop in the series windings IL = 250 + 2 = 252 V. Ish = Vt +IL Rse /Rsh = 250 +80 ×0.025 / 100 =2.52 A Ia = IL+Ish = 80 + 2.52 = 82.52A Ia.Ra = 82.52 × 0.05 = 4.13V The generated E.m.f = Vt+ Ia. Ra +Voltage drop in the series windings = 250 + 4.13 + 2 = 256.13 V.

37 Ms.c. Haider M. Umran

University of Karbala

DC Machine

Electrical & Electronics Eng. Dep.

3.3. Characteristics of D.C. Generators: Following are the three most important characteristics or curves of a d.c. generator: 1. No-load saturation Characteristic (Eo/If): It is also known as Magnetic Characteristic or Open-circuit Characteristic (O.C.C.). It shows the relation between the no-load generated e.m.f. in armature, Eo and the field or exciting current If at a given fixed speed. It is just the magnetization curve for the material of the electromagnets. Its shape is practically the same for all generators whether separately-excited or self-excited. 2. Internal or Total Characteristic (Eo/Ia): It gives the relation between the e.m.f. Eo actually induces in the armature (after allowing for the demagnetizing effect of armature reaction) and the armature current Ia. 3. External Characteristic (Vt/IL): It is also referred to as performance characteristic or sometimes voltage-regulating curve. It gives relation between that terminal voltage Vt and the load current IL. This curve lies below the internal characteristic because it takes into account the voltage drop over the armature circuit resistance. The values of Vt are obtained by subtracting Ia .Ra from corresponding values of Eo. This characteristic is of great importance in judging the suitability of a generator for a particular purpose. 3.2.1 Characteristics of Separately Excited Generator:1. No-load Saturation or Open Circuit Characteristic (Eo/If): The arrangement for obtaining the necessary data to plot this curve is shown in Fig. 3.9. The exciting or field current If is obtained from an external independent d.c. source. It can be varied from zero upwards by a potentiometer and its value read by an ammeter A connected in the field circuit as shown. Ф𝐩𝐍𝐙 Now, the voltage equation of a d.c. generator is, 𝐄𝐨 = volt 𝟔𝟎 𝐚 A

F1

A1

D.c supply

V Rheosta F2 A2

Fig. 3.9: Separately excited gen. with no-load.

38 Ms.c. Haider M. Umran

University of Karbala

DC Machine

Electrical & Electronics Eng. Dep.

When current If is increased from its initial small value, the flux Φ changed and hence induced e.m.f. Eo increase directly along the poles are unsaturated. This is represented by the straight portion OA in Fig. 3.10. But as the flux density increases, the poles become saturated, so a greater increase in If is required to produce a given small increase in voltage than on the lower part of the curve. E0

Saturation Increasing

A Constant Open circuit Characteristic

O

If

Fig. 3.10: Magnetization ch.cs. for constant speed.

2. Load Saturation Curve (Vt/If): The curve showing relation between the terminal voltage Vt and field current If when the generator is loaded, is known as Load Saturation Curve. The curve can be deduced from the no-load saturation curve provided the values of armature reaction and armature resistance are known. While considering this curve, account is taken of the demagnetizing effect of armature reaction and the voltage drop in armature which are practically absent under no-load conditions.

Fig. 3.11: Load saturation curve.

The no-load saturation curve of Fig. 3.10 has been repeated in Fig. 3.11 on a base of field ampturns (and not current) and it is seen that at no-load, the field amp-turns required for rated noload voltage are given by Oa. Under load conditions, the voltage will decrease due to 39 Ms.c. Haider M. Umran

University of Karbala

DC Machine

Electrical & Electronics Eng. Dep.

demagnetizing effect of armature reaction. This decrease can be made up by suitably increasing the field amp-turns. Let ac represent the equivalent demagnetizing amp-turns per pole. Then, it means that in order to generate the same e.m.f. on load as at no-load, the field amp-turns/pole must be increased by an amount ac = bd. The point d lies on the curve LS which shows relation between the voltage E generated under load conditions and the field amp-turns. The curve LS is practically parallel to curve Ob. The terminal voltage V will be less than this generated voltage E by an amount (Ia Ra) where Ra is the resistance of the armature circuit. From point d, a vertical line de = Ia Ra is drawn. The point e lies on the full-load saturation curve for the generator. Similarly, other points are obtained in the same manner and the full-load saturation curve Mp is drawn. The right-angled triangle bde is known as drop reaction triangle. Load saturation curve for half-load can be obtained by joining the mid-points of such lines as mn and bd etc. In the case of self-excited generators, load saturation curves are obtained in a similar way. 3. Internal and External Characteristics: Let us consider a separately-excited generator giving its rated no-load voltage of Eo for a certain constant field current. If there were no armature reaction and armature voltage drop, then this voltage would have remained constant as shown in Fig. 3.11. By the dotted horizontal line I. But when the generator is loaded, the voltage falls due to these two causes, thereby giving slightly dropping characteristics. If we subtract from Eo the values of voltage drops due to armature reaction for different loads, then we get the value of E, the e.m.f. actually induced in the armature under load conditions. Curve II is plotted in this way and is known as the internal characteristic. The straight line Oa represents the Ia Ra drops corresponding to different armature currents. If we subtract from E o the armature drop Ia Ra, we get terminal voltage Vt. Curve III represents the external characteristic and is obtained by subtracting ordinates the line Oa from those of curve II.

Fig. 3.11: Load characteristics curve.

40 Ms.c. Haider M. Umran

University of Karbala

DC Machine

Electrical & Electronics Eng. Dep.

3.2.2: Characteristics of Self- Excited DC Shunt Generator:1. No-load Curve Characteristic (Eo / If): This curve shows the relation between the generated e.m.f. at no-load (Eo) and the field current (If). The field or exciting current If is varied by rheostat and its value read on the ammeter (A). The generator is derived at constant speed by the prime mover and the generated e.m.f. on no-load is measured by the voltmeter connected across the armature as shown Fig. 3.12. Due to residual magnetism in the poles, some e.m.f. (OA) is generated even when I f = 0. Hence, the curve shown in Fig. 3.13; starts a little way up. The generated e.m.f. is directly proportional to the exciting current. However, at high flux densities, where μ is small, iron path reluctance becomes appreciable and straight relation between E and If after point B, saturation of poles starts. Ish

A

Ia

Eo

IL

B

F1 F2

G

Vt

V A If

O

Fig.3.13. Load characteristics curve

Fig.3.12. DC shunt generator

2. Load Characteristics of DC Shunt Generator:A. Internal Characteristic (E/Ia): Ideally the induced e.m.f. is not dependent on the load current IL or armature current Ia. but as load current increased, the armature current I a increases to supply load demand. As Ia increased armature flux increases. The effect of flux produced by armature on main flux produced by the field winding is called armature reaction. Due to the armature reaction, main flux distorted. Hence lesser flux gets linked with the armature conductor and this reduces the induced e.m.f as shown Fig. 3.14. E Eo

Drop due to Armature reaction

IL

O

Fig.3.14: Internal characteristics curve. 41 Ms.c. Haider M. Umran

University of Karbala

DC Machine

Electrical & Electronics Eng. Dep.

B. External Characteristic (Vt/IL): For D.C generator (E = Vt - Ia Ra) neglected other drops. So as load current IL increases, Ia increases. Thus will increase drop Ia Ra and terminal voltage Vt = E - Ia Ra decreases. The value of armature resistance Ra is very small; the drop in terminal voltage as I L changes from no load to full load is very small. Hence shunt generator is called constant voltage generator. If the load resistance is reduced beyond point a break down point i.e. load IL is increased beyond P then it increases momentarily. This is very large current and generator gets overloaded. Due to a large current the armature reaction is high and drop Ia Ra decreases from P to Q, rather than increasing. Thus on curve aqr, the voltage goes on reducing rapidly and point r becomes zero as shown Fig. 3.15. Vt Eo Small drop Ia.Ra

a Drooping nature

Break down q r

O

IL Q

P

Fig.3.15: External characteristics curve.

3.2.2.1 Critical Field Resistance in DC Shunt Generator: Now, connect the field windings back to the armature and run the machine as a shunt generator. Due to residual magnetism in the poles, some e.m.f. and hence current, would be generated. This current while passing through the field coils will strengthen the magnetism of the poles (provided field coils are properly connected as regards polarity). This will increase the pole flux which will further increase the generated e.m.f. Increased e.m.f. means more current which further increases the flux and so on. This mutual reinforcement of e.m.f. and flux proceeds on till equilibrium is reached at some point like P (Fig. 3.16). The point lies on the resistance line OA of the field winding. Let R be the resistance of the field winding. Line OA is drawn such that its slope equals the field winding resistance i.e. every point on this curve is such that volt/ampere = R. The voltage OL corresponding to point P represents the maximum voltage to which the machine will build up with R as field resistance. OB represents smaller resistance and the corresponding voltage OM is slightly greater than OL. If field resistance is increased, then slope of the resistance line increased, and hence the maximum voltage to which the generator will build up at a given speed, decreases. If R is increased so much that the resistance line does not cut the O.C.C. at all (like OT), then obviously the machine will fail to excite i.e. there will be no ‘build up’ of the voltage. If the resistance line just lies along the slope, then with that value of field resistance, the machine will just excite. The value of the 42 Ms.c. Haider M. Umran

University of Karbala

DC Machine

Electrical & Electronics Eng. Dep.

resistance represented by the tangent to the curve, is known as critical resistance Rc for a given speed. (Rsh) increased, slope increases, Max; generate voltage decreases. Line of Rc

E.m.f

A Normal max; excitation voltage

B

E

C

Normal field resistance line (Rsh)

O.C.C at Normal speed

D

Critical Voltage EC If

O Fig.3.16: No-load saturation curve.

Steps to Find Critical Resistance Rc:1. O.C.C. is plotted from the given data. As shown in Fig. 3.16. 2. To draw the line of field resistance Rf, consider an equation of line (y = m.x). Where y = Eo, x = If and m= slope = Rf. one point of the line is (0, 0), second point (Eo, If) If (If) is any value from the given data by the above eq. we determine the new value of Eo. The second point on the line is (Eo, If) and draw the line passing through (0, 0) and (Eo , If), OA line. 3. Draw a tangent line to (O.C.C) i.e. tan θ, is the critical resistance of the field resistance. 𝐃𝐄 ∆𝐄 Slope of tangent line is 𝐑𝐜 = ∆𝐈 = 𝐂𝐃 = 𝐭𝐚𝐧 𝛉. 𝐟

We know that E α N. As speed decreased the induced e.m.f. decreases, we gate (O.C.C) below the (O.C.C) just tangential to normal field resistance line. As shown in Fig.3.17. Critical Speed (Nc) is the speed at which machine just excites for the given field circuit resistance. 𝐄𝐨𝟏 𝐍𝟏 = 𝐄𝐨𝟐 𝐍𝟐 ∴

𝐄𝐨𝟐 = 𝐍 𝟐 . 𝐄𝐨𝟏 𝐍

𝟏

43 Ms.c. Haider M. Umran

University of Karbala

DC Machine

Electrical & Electronics Eng. Dep.

Steps to Find Critical S...