6. Three-Phase Systems. Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna

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1 6. Three-Phase Systems J B

2 G Three-Phase Systems v v v i i i The generation and the distribution of electrical energy is usually done by three- phase systems. There are three wire system connected to a generator consisting of three AC sources having the same amplitude and frequency (mostly 50 Hz, ω = 4 rad/s, V e - 0/400 (effective phase tension/line tension) in Europe, most of Asia and Australia, 60 Hz, ω = 77 rad/s, V e - 0/0 and 0/40 in Nord America, Canada, and others) but out of phase with each other by 0. Three are the main motivations:. Usually the electric power is generated by three phase electrical machines (alternators synchronous generators).. In balanced three phase systems the total instantaneous power is constant (not pulsating). This results in an uniform transmission and less vibrations.. For the same amount of transported power, a three phase system is more economical than a single phase. U

3 Three-Phase Systems The Set of voltages and currents in three phase systems are the following. The line currents i (t), i (t), and i (t) are the currents flowing in each of the three lines. From the KCL it follows: i (t) + i (t) + i (t) = 0 I + I + I! =!0 The line tensions ( line to line tensions or line voltages tensioni concatenate) v (t), v (t), and v (t) are also the po- tential differences between the termi- nals and, and, and and. From the KTL it follows: v (t) + v (t) + v (t) = 0 E! E! n E! V! V! V! I I I V + V + V! =!0

4 The#three#phase#system#is#isofrequential.#The#three#phase#currents#and# the#three#phase#voltages#must#fulfill#the#kirchhoff's#laws.#in#the#phasor# plane#it#follows: ################################### I + I + I # = #0 ################################## V + V + V # = #0 Therefore#the#three#phase#line#currents#are#represented#by#the#tringle(of( the(line(currents#and#the#the#three#phase#line#tensions#are#represented#by# the#tringle(of(the(line(tensions. I! Three-Phase Systems I! V! V! I! Triangle of the line currents V! Triangle of the line tensions

5 Three-Phase Systems Balanced line tension system (sistema simmetrico) p a! = e j(p /) p V! V! p p V! V! p V! V! p Positive sequence (Sequenza (Clockwise rotation from V diretta) "to" V " and from" V "to V ) V ;"" V " = " V e &j(π /) ;"" V " = " V &j(4π /) "e V ;"" V " = " α V ;"" V " = "" α V " Negative sequence (Sequenza indiretta) (Counterclockwise rotation from V "to " V " and from" V "to V ) V ;"" V " = " V e j(π /) ;"" V " = " V j(4π /) "e V ;"" V " = " α V ;"" V " = " α V "

6 Three-Phase Systems Balanced line current system (sisatema equilibrato) p a! = e j(p /) p I! I! I! I! p p p I! Positive sequence (Clockwise rotation from I "to" I " and from" I "to I ) I ;"" I " = " I e &j(π /) ;"" I " = " I &j(4π /) "e I ;"" I " = " α I ;"" I " = "" α I " I! p Negative sequence (Counterlockwise rotation from I " "to" I " and from" I "to I ) I ;"" I " = " I e j(π /) ;"" I " = " I j(4π /) "e I ;"" I " = " α I ;"" I " = " α I "

7 Three Phase System Supply The phase tensions E," E," E (or phase voltages) are the potential difference between the center n of the Y connected tension system and the terminals,, and. V! V! E! E!n Triangles of line and phase tensions E! E! E! n E! n center of the tension system: V! V! V! The n-node can be any point of the plain of the phasor plain for which a three phase tensions correspond to V! the same three line tensions. Therefore there are are sets of three impedances corresponding to the same set of line tensions

8 Three Phase System Supply Balanced Phase Tension Balanced line tension system (positive sequence):" """"" V ;"" V " = " α V ;"" V " = "" α V """ Balanced phase tension system (positive sequence):"!a V! E! n!a E!!a V! a! E! V! """"" E ;"" E " = " α E ;"" E " = "" α E """""""""""

9 Three Phase System Supply Balanced line tension system (positive sequence):" """"""""" V " = "V ( ) " (% + j" ) """"""""" V " = " V "e %jπ / " = "V"e %jπ / " = "% V + j" """"""""" V " = "" V "e %j4π / " = "V"e %j4π / " = " V """ Balanced phase tension system (positive sequence):" """""""" E " = "E"e %j"π /6 """" where ""E" = " V """"""""" E " = " E""e %j"π /6 "e %jπ / %j5π /6 " = "E"e """"""""" E " = " E""e %j"π /6 "e %j4π / %j9π /6 " = "E"e """"""""""" α V!a E! n a! E! E! V!a V!

10 Three Phase System Loading Δ-Connected Load I! I! I! I! I! I! Y-Connected load I! n I! Z I! V = Z I V = Z I V = Z I I = I - I I = I - I I = I - I V = Z I - Z I V = Z I - Z I I + I + I = 0 E = Z I E = Z I E = Z I There are sets of three Y connected impedances which induce sets of line tensions when connected to the same set of line tensions.

11 Three Phase System Loading Y-Connected load I! I! Z I! n n n ) n ) n n n ) n n As it is shown AT the right figure, there are n-points (which correspond to the n-node in the left figure) that correspond to sets of line to the n-node tensions n, n, n. and to sets of Y connected impedances Z, Z, Z which induce sets of line tensions when connected to the same set of line tensions,,.

12 Three Phase System Supply Balanced Y Connected System In a balanced line to line system with a positive sequence it is = V, = V e -jp /, = V e jp / For a balanced Y connected impedance system where Z = Z = Z = Z Y The phase tensions (line to the n-node tensions) become: with: E, E = E e -jp /, E = E e jp / E = V/!a V! n n!a V! I! I! I! Z E α E V! α E

13 Three Phase System Supply Balanced D Connected System In a balanced line to line system with a positive sequence it is I = I - I, I = I - I, I = I - I For a balanced D connected impedance system where Z = Z = Z = Z the currents flowing through the three impedances are: I, I = I e -jp /, I = I e jp / with: I = I / I I! I! I! I I! I! α I α I I = α I I! I =α I

14 Y-Connected load Three Phase System Loading I! O I! I! D -Connected Load I! I! I! I! I! I! V! V! V! V! I! O I! I! I! I! I! I! I! I!

15 Wye and Delta Load Connections For impedances as for resistors each impedance of the wye connection is the product of the two impedances of the delta connection connected to the same node, divided by the sum of the three impedances of the delta connection. Z Δ Z Δ Z Δ Each impedance of the delta connection is the sum of all the products of the impedances of the wye connection two by two, divided by the impedance in the opposite branch of the wye connection. Z Y Z Y Z Y!!!!!!!!!!!!!!!!!!!!!!!!! Z Y = Z Δ Z Δ Z Δ + Z Δ + Z Δ ;"" Z Y = Z Δ Z Δ Z Δ + Z Δ + Z Δ ;!! Z Y = Z Δ Z Δ Z Δ + Z Δ + Z Δ Z Δ = Z Y Z Y + Z Y Z Y + Z Y Z Y Z Y ;## Z Δ = Z Y Z Y + Z Y Z Y + Z Y Z Y Z Y ;## Z Δ = Z Y Z Y + Z Y Z Y + Z Y Z Y Z Y When Ż a = Ż b = Ż c = Ż Δ and Ż = Ż = Ż = Ż Y the load is said a balanced load. In this case It is: Z Y = Z Δ!;!!! Z Δ = Z Y

16 Three Phase System Loading n n n n neutral Z Z n E n E E E n E ' Z E As it is shown in the right figure, there are n-points that correspond to sets of line to the n-node (phase) tensions E, E, E. and to sets of Y or D connected impedances Z, Z, Z which induce sets of currents when connected to the same set of line tensions,,. In order to have always a balanced system of phase tensions (sistema simmetrico), a connection between the center of the Y connected generator system with the center of the Y load system (neutral) is used.

17 Three Phase System Loading A n n neutral Z n n n n n. B C Z n Z n In a system with neutral, as in balanced three phase system (sistemi simmetrici) each y connected impedance Z Y is supplied by a phase voltage where n Z Y E = 4. e -jp /6 n The same relation holds also for E, n,, and.

18 Three Phase System with Neutral E! E! O E! I n I I I n A three phase generator is equivalent to three y-connected mono-phase generators with a symmetric tension system. The node of center of the y-connection is linked to a fourth line. The relation between the phase tensions E and the line tensions V in a balanced system is: V E = The tension with an effective (rms) voltage E e = 0 V is obtained by means of a three phase symmetrical system with an effective line tension V e = 400 V. In this case it is: I + I + I + I n! =!0 ü The line tension system with neutral can afford the same utilization of the system without neutral. ü Both a balanced line tension set (,, with V =V =V = V) and a balanced phase tension set (E,E,E with E =E =E = E= V/ ) are available.

19 Power in Three Phase Systems In a balanced three phase system with a positive sequence the phase rms effective voltages are v p = Ecosωt;"""v p = Ecos( ωt 0 );"""v p = Ecos ωt +0 ( ) where E is the rms effective value of the phase voltage: E = V where V is the rms value of the line tension. In Y-connected balanced load with Z Y = Ze jϕ the currents lag behind their corresponding phase voltages by ϕ. Thus i p = Icos( ωt ϕ);"""i p = Icos( ωt ϕ 0 );" """"""""i p = Icos ωt ϕ +0 ( ) where I is the rms effective value of the current. The total instantaneous power absorbed by Y-connected load is p( t) = p + p + p = v p i p + v p i p + v p i p!!!!!!!!!!!!!!!!!!! = EI cosωt cos( ωt ϕ) +!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!+cos( ωt 0 )cos( ωt ϕ 0 ) +!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!+cos( ωt +0 )cos( ωt ϕ +0 )

20 Power in Three Phase Systems From the trigonometric identities cos(a+b) = cosa cosb - sina sinb and cos(a-b) = cosa cosb + sina sinb, and setting α =ωt-j, from the expression of the power: p(t) = EI [cos w t cos(w t- j ) + cos(w t-0 ) cos(w t- j -0 ) + + cos (w t+0 ) cos(w t- j +0 )] it results: ( ) = EI cosϕ + cos( ωt ϕ) ( ) = p t ( ) + +cos ωt ϕ 40!!!!!!!!!!!!!!!!!+cos ωt ϕ + 40!!!!!!!!!=EI cosϕ + cosα +cosα cos40 +sinα sin40 +!!!!!!!!!!!!!!!!!+cosα cos40!sinα sin40 =!!!!!!!!!=EI cosϕ + cosα +cosα cos40 =!!!!!!!!!=EI cosϕ + cosα+ cosα = EIcosϕ Ø Thus the total instantaneous power in a balanced three phase system does not change with time as the instantaneous power of each phase does. This result is true for a balanced system with balanced loads weather the loads are Y- connected or Δ-connected. This is one of the motivations to use three phase systems to generate and distribute AC power.

21 and Power in Three Phase Systems The total average power and the reactive power in a three phase balanced system are (E and V are the rms effective values of the phase and line tensions respectively): P = EIcosϕ = VIcosϕ Q = EI sinϕ = VI sinϕ The power factor is Q cosϕ = cos tan - P () The power factor is defined by eq. for every three phase system independently from the connection and whether it is balanced or not. For unbalanced system the power factor is given by eq. where the average and the reactive power can be calculated from: l P!! =!! R k!i k!!!and!!!!q!! =!! X k!i k k= where R k and X k are the resistances and the reactances of the circuit. l k=

22 Power in Three Phase Systems In a unbalanced system the phase shift angle between phase voltage and line current usually is different for each phase. In this case the shift angle correction has to be done for each phase. For unbalanced system the power factor is given by cosϕ = cos tan - Q P = where the average and the reactive power can be calculated from the resistances and the reactances of the circuit. P P + Q = P N ( N=P+jQ and N= P + Q ) For unbalanced systems, the rotation of an angle φ of the current system pruduces the maximal value of the average power. İ Ė İ Ė Ė φ φ İ φ

23 Power in Three Phase Systems In a unbalanced system the phase shift angles between the three phase voltages and the corresponding line currents are φ, φ, and φ. After the rotation they are: φ - φ, φ - φ, and φ - φ. The average power for a rotation of φ is P = E k I k cos( ϕ k ϕ) ; Q = E k I k sen( ϕ k ϕ) k= The maximum of P corresponds to ϕ for which: ( ) dp dϕ = E I sen ϕ ϕ = 0 k k k k= - cosϕ E k I k senϕ k + senϕ E k I k cosϕ k = 0 k= tan ϕ = senϕ cosϕ = k= k= E k I k senϕ k E k I k cosϕ k k= k= = Q P The power factor is thus defined as Q cosϕ = cos tan - P = P P + Q = P N İ Ė Ė Ė İ φ İ φ φ

24 Example (a)

25 Example

26

27 Example

28 N Q =N N N N P =N N N

29 The total complex power N absorbed by the two loads is N = N + N N I L = I D = > A N = V L I D and the phase to phase current, I D flowing through each impedence of the connected laod, is I D = I L / (I L line current). I L I L

30 (Q C =Q C /). The the figure

31 Terminology Balanced current system Balanced load Balanced tension system Line tension Negative sequence Phase tension Positive sequence Three- phase electric system Sistema di correnti equilibrato Carico equilibrato Sistema simmetrico Tensioni concatenate Sequenza negativa (Sequenza indiretta) Tensioni d fase Sequenza positiva (Sequenza diretta) Sistema elettrico trifase