Block diagram and frequency response of alternating current machines

Block Diagram and Frequency Response of Alternating Current Machines L. HANNAKAM Machine of Double Non-Symmetry

The basis for plotting the block diagrams and calculating the frequency response curves is given by the electrical and mechanical equations of the confined system representing the a.c. machine. These consist of the circuit equations covering each winding plus the equation of motion, and they form a set of simultaneous non-linear differential equations capable of describing the transient phenomena following any disturbance. Let a given electrical machine developing a torque dM * be coupled to a driving or braking machine maintaining a torque do, then the system of equations can be denoted in the dimensionless form u

=

r .i

. h' + 'I. f' Wit 'If' =

d'lf'

d

d T an 'If' = 'If'a

winding f3 which will be connected to a voltage up = 3t . Urn • cos T and conduct a current if! = 2ib/3t = -2ic/3t.

+ 'If'h

= Xa. i

+ x,,(y) . i

'q

Figure I . Structure of salient pole synchronous machine

(1)t

_ d2y

-

initial conditions for

T

dT2

an

dd M

_ -

*

1 2" • i

.

dxh(y) . ----ctY .I

(2)*

= 0:

(i)o = 10 or ('If')o = 'P' 0' (y)o =

Using the definition of main flux components

r 0 and (y)o

=

r0

wherein the notation uses T ( = DN . t) for the time variable I multiplied by the rated angular frequency DN , TA for the inertia constant, u, i, 'If'a, 'If'h and 'If' for the vectors of voltages, currents, leakage fluxes, main fluxes and total flux magnitudes, r for the diagonal matrices of ohmic resistors (including variable seriesconnected resistors, if such exist), Xa and Xh for the symmetrical matrices of leakage and main reactances, the latter being a function of the rotor position-angle y, in the conventional per-unit system. The currents i, flux 'If' and the angular position y of the rotor represent the variables of the system, whereas the voltages u and the counteracting torque do characterize disturbance functions imposed from without the system. The generally valid form 1 of the electrical and mechanical system equations is applied to machines of so-called double non-symmetry, stator and rotor of which are built nonsymmetrically. As an example a machine of this type is represented in Figure 1 which shows a salient-pole machine with symmetrical stator windings (Ra = Rb = Re = Rs and Xaa = Xab = Xac = Xas). The stator windings band c in series are connected to an alternating voltage source providing Ube = U m • cos T, while phase winding a remains open. Now we substitute the series of phase windings band c by a single

* Small letters imply variables, while capital letters suggest constant quantities. t The star index (*) characterizes the transpose of a matrix.

and 'If'hq = Xhq . ihq = X hq . (iq • cos y + iQ) where Xhd and X hq are the main reactances of direct and quadrature axis, and applying the per-unit system, the system equations assume the dimensionless form

w;,h

[~;] 0 ] 'If'af!] _ [xas 'If'aD 0 0X aD 0 [ 'If'aQ 0 0 XaQ 0 . 'If'aF 0 0 0 XaF

f 1f l l 'lf'hfJ

and 'If'hD

Sin y

cos

0 1

1 0

=

'lf'hQ 'If'hF

dG = DN T

= 0:

· TA •

('!f'rJ)o

Y-

=

ti.lf

1

0

[~fJ] ID

(3)

iQ

iF

Yl . ['If'hdJ

J

'If'hQ

with dM = if!. ('If'hd' cos Y - 'If'hd' sin y)

'P'po; (IPD)O

=

('If'F)O

'P'DO; ('If'Q)o

= lYFO ;

(y)o

= 'P'Qo; = 1"0; (y)o =

ro

t In order not to neglect saturation effects the respective magnetizing characteristics 'Pha
1707

BLOCK DIAGRAM AND FREQUENCY RESPONSE OF ALTERNATING CURRENT MACHINES

which serves to plot the block diagram of the single-phase generator shown in Figure 2. The trigonometrical functions Zl = cos y and Z2 = sin y are produced from the relative angular speed y by solving the differential equations dz1 dT T ~

u,

=

= 0:

• Zl

•

(Zl)O

dZ 2 dT =

and

= -y. Z2

= cos ro

and

(zJo

=

• Z2

•

= Y . Zl (4)

sin ro

from the main reactances X h the dependence of the rotor angle y by introducing a transformed current system i' = v . i* using a transformation matrix v = v(y). We will use the d-q transformation as derived from the cross field theory, i.e. we work with quantities of the direct axis current id and quadrature-axis current iq • Suppose the salient pole machine in Y connection is linked to a symmetrical three-phase supply of voltage amplitude Urn and constant angular frequency DiY' then the electrical and

------------------~ ~~

_____

~------------~----~ ,------------------------------------------~

.----------------------------1. !----.:~====i__------- sin ro ~ ~----------------~ ~:!:.:...{:)--

Ufl

L~~~=====:t-----C05t;,

~~---------------------

Figure 2. Block diagram of single-phase generator

To exemplify the characteristic transient behaviour of the single-phase generator, the torque curve dM caused by a short circuit of a heavy generator for railway service is shown in Figure 3. Assuming that the single-phase generator is connected to the voltage Ubc with interconnection of a resistance (R,A) and an inductance (X,A), the quantities Rs and X"s are to be replaced

mechanical equations in dimensionless per-unit notation can be written as Fp. u' = R' . i'

and do T

=

-dM

+ X'.

dj'

dT

+ (1 + Y'P). K.

+ DN . T,A . Y with dlJ1

=

= 0: (no = 1'0; (y)o = ro and

[

~dJ

Iq

iD

~

X'. i'

t' . K. (Y)o

ruml [Sin0yp cos0 yp = lO J; Fp = 0 0 0 0 0 uF

iF'

0

= 1\

o0 0 0 0 0J

Urn

; U'

X' . i'

1 0

0 ;

010

o

0

0

1

[RS 0 0 o Rs R' =

0

o o

0 Xq 0 X hq Xhd 0

~ ~" [ X,

X'

Figure 3. Torque dM(t) following a short-circuiting of the single-phase generator

by (R."; + R,A) and (X"s concerning the voltage

+ X,A) respectively. The block diagram

. u,A = R:.4. lp

+

X

dip

,A. dT

0 0 J 0 RD 0 0 ; 0 0 RQ 0 0 0 0 RF X hd 0 0 Xhq XD 0 0 XQ X hd 0

-1o 0 0 0 0 000 000 000

(5)

across the terminals of the series R,A, X,A appears in Figure 2 in dashed lines.

Xq = Xhq

Machine of Single Non-Symmetry

(6)

~d

X"'J ;

XF

0J 0 0 0 0

+ X"s;

XQ = Xhq

+ X"Q

* Transformed magnitudes are characterized by the prime mark.

When connecting all three phases of the salient pole machine above (Figure J) to a three-phase supply, it is possible to remove

195

1708

L. HANNAKAM

where yp indicates the conventional load angle. The block diagram of the salient pole machine can be easily plotted following the system equations 6; it is omitted here. Instead, the method of setting up the electromechanical frequency response of the synchronous machine will be discussed. The equations 6 for small deviations of the variables from their constant stationary quantities provide the basis for expressing the frequency response. Using the operator p = d/dT we get for the machine rotating at its synchronous speed

+ Fp . D.u' - K. Fp . U' . D.yp = R' . (I' + D.i') + X'pD.i' + (1 + P . D.y,,) . K . X ' . (I' + D.i') + D.da = -(I' + D.i')* . K. X' . (I' + D.i') + p2 . ON . TA . D.y"

variable amplitude, ofvoItage U", and constant angular frequency ON' the stator of which carries the symmetrical windings (SI, S2' S.J) while the windings of the rotor (R!> R 2 , R3 ) are short-circuited by their variable starting resistors r v. The

0·15

Fp. U'

DG

Z

(7)

to

H

H

"ooc '

After subtracting the stationary balanced relation F" . U' = R' . !' DG =

+ K.

and

X' . !'

* -!'.K.X'.!'

jvD

(8) =

"

-DM

.

the system equations for small deviations from a stationary zero position become F" . D.u' = (R'

D. . do

=

+ pX + K. X) . D.i' + K. (pX' . !' +

* . (K. X' -I'

F". U'). D.y"

(9)

+ X*' . K) . D.i' + p20,v . TA . D.y"

..

Hence the several frequency response relations may be established. In order to select a frequency response relation of general interest which especially characterizes the salient pole machine an oscillation of the rotor according to D.y" = D.r" . cos (v . T) with D.r p = constant at a constant armature voltage Urn = U m = 1 and field voltage UF = UF , proceeding from a load angle of r" = 0 is considered. The generated oscillating torque

* . K) + X' X')-I . K. (iv . X' . !' + F" . U')]

-

Rs '0·1

jvD

*

D.dJJf = Re['I9.11(v) . ejvT ] = Re{[ -!' . (K. X'

. (R'

+ jv . X + K.

. D.r" .

(10)

ejVT }

in the representation -'I9M (v) = qv) + jvD(v) with v running from 0 to 00 is plotted in F(fJure 4 for several armature resistance quantities*. The frequency response -{}M(v) indicates the spring constant C(v) and the damping constant D(v) varying with the relative frequency v. The influence of the ohmic stator resistance, hitherto neglected in almost all calculations, is obvious. If Rs assumes larger values, e.g. if the connection of the machine to the supply contains a long line of small crosssection, self-excited oscillations and falling out of step may occur.

l'

"

"

Rs=O·Z

Figure 4. Frequency response curves -{fM(I') of the salient pole machine for r = 0°

trigonometrical dependence of the system equations may be totally removed owing to the transformation of currents

Machine of Full Symmetry

The system equations of the single-phase gener2.tor contain trigonometrical functions of the position angle y in the main reactances; when dealing with the stator-symmetrical salient pole machine the dependence of the angle y was transferred to the transformed voltage variables. Now we consider the slipring motor connected to a symmetrical three-phase system of

(11)

rCOS(T-Y)

[~~~] = I cos (T -

• The frequency response curves were calculated by the magnetic drum computer IBM 650 in the AEG Computing Centre in Berlin. The time required for computing one point was 2·5 sec.

IR3

196

1709

l (

Y -

cos T - Y

J Y - 2;) .[;oJ

-sin(T-Y)

2;) -

27T) +3

sin (T -

sin (T - Y

27T) +3

D

BLOCK DIAGRAM AND FREQUENCY RESPONSE OF ALTERNATING CURRENT MACHINES

By use of the power-invariant transfonnation 11 the slip-ring motor is transfigured into a d.c. machine carrying two rotating annatures, each with two sets of brushes positioned rectangular to each other, as shown in Figure 5. The exterior armature

Regarding small deviations Ll of the single variables from their constant stationary quantities the system equations 12 become

+ Llu'

+ Llr') . (/' + Lli') + X' . pLli' + [Ks + (S + Lls) . K R ] . X' . (/' + Lli') + Llda = -(/' + Lli')* . KR . X' . (/' + Lli')

U'

Da

= (R'

(13)

- QSTA' pLls

where p stands for the operator d/dT. Subtracting the stationary balance condition U'

= R' . I' + (Ks + S. K R ) . X' . I'

and

*

(14)

Da = - / ' . K R

.

X'. I' = -DM

we get the system equations for small deviations from a given zero position Llu' - Llr' . /' = [R' + (Ks + S. K R) . x' + pX'] . Lli'

+ K R . X' . /' . Lls Llda

* (KR . X' + X' * . K'A) . Lli' = -I'.

which provides the basis for establishing the frequency response relations of the slip-ring motor considered. Suppose the magnitude of the starting resistance is subject to periodic changes according to Llr = LlRR . cos (v . T) with LlRR = constant on a slip-ring motor connected to an infinite bus (voltage u' = U') by a relatively long cable and working at a constant load torque d a = D a , then the deviation of slip Lls = R.[O'(v) . e jv7 ] will follow the frequency response equation

Figure 5. Transformed arrangement of the three-phase slip-ring motor

rotating at synchronous speed (~QN) is connected to the variable voltage Urn by the pair of brushes AA while the second pair of brushes BB is short-circuited. The interior annature runs at slip-speed (~s . QN), both pairs of brushes CC and DD being short-circuited by the variable rotor-resistors rv' Using the notation

* ( ) _ /' . (KH . X' 0'1'

-

for the total reactances Rs and rR for the ohmic resistances (including r v) XI! for the main reactance

= r' . i' +

di'

X'. dT

+ (Ks + s. K R ).

d a = -c/.lf - QN' TA . s with

}

= 0: (i')o = 1'0;

T

i'=

x'

r;~l.

U~J IX~

_,0

-l~h

X' . i'

d.:.lf = (s)o

19

0

XI!

Xs

0

0

XR

Xh

0

KR

XJ.

Ks

0j'

XR

=

rg 19

0 0 0 0

0 0 0 1

*

j' .

~ [~

(16)

LlR R

_~~_ _-,,0'=005 _ _ _-,O{)10

0020 real

0{)15

and KR . X' . j'

/.

O{}3 /.

MS

0

~l

0 0

Rs

0 0

**

+ X'KR). [R + (Ks + S

The corresponding curve is plotted in Figure 6*. In the region

= So

I"m] r' = r~~ u' = l ~ ;

*

+ jvX']-l . I' . K R) . X' + jvX,]-l . KR . X' . /"

of both stator and rotor windings

the electrical and mechanical equations of the slip-ring motor in dimensionless per-unit form are given by /I'

* . K*R) . [R + (Ks + S. KR) . X' + X'

jv. QSTA - 1'. (KRX'

s = 1 - dy/dT for the slip and s = ds/dT

x s and X R

(IS)

- pQNTA . Lls

rH

0 -1 0 0 0

gl. gJ

()O]

-OOt0i' 007

dr.~~ RR_ TAD P S

-J"

rR

0 0 0 0

"

Figure 6. Frequency response curves a(v) of the three-phase slip ring motor

(12)

-rl

The block diagram is omitted but could be established easily.

of very small:values of v the frequency response approximated by the dashed circle described by O'a(V)

=

G

G

SA' . T . -R . ~RR With

+ JV

G

=

0'(1')

may be

aD -a"ll

(17)

S * Computing time per point 1·2 sec using the magnetic drum computer IBM 650. 197

1710

A

R

L. HANNAKAM

The fonnula G in 17 indicates the derivative of the stationary. torque curve with respect to the stationary slip S which is here considered as a variable. The approximation 17, however, does not cover oscillations of the slip-ring motor at higher frequencies, and therefore an approximation valid for a broader frequency spectrum is required. In most cases the stationary starting position allows approaching the system equations 12 for a range of small quantities of slip, neglecting the ohmic resistance

(Rs = 0) of the stator, by the system

die dT ds dT

-

I rR. SK RR I TAQ' X h

--'-'Ie

= -

Xh • U SK .-·S X h + X"s RR

(X+ X"s U . ie + do )

(18)

h •

with initial conditions for T = 0: (ie)o = leo and (s)o = So, where SK is the breakdown slip of the three-phase machine corresponding to the rotor resistance R R'

Summary While the block diagram of d.c. machines is of simple structure due As an example for the synchronous salient pole machine the to the axis of the individual windings being stationary in space the frequency response of the angular displacement of the rotor resulting dynamic theory of a.c. machines is considerably more difficult since from periodic fluctuations of the driving or load torque is derived the mutual inductances of stator and rotor windings vary in magnitude from the block diagram. Such frequency response curves are mainly depending on the instantaneous angular position of the rotor within being used when investigating diesel power plants. The existing the stator bore. The system of equations covering any transient resonant frequencies are given special consideration. From the wide condition involves trigonometric functions of the angle indicating field of applications of the induction motor the frequency response the rotor position. By a suitable transformation these functions may of a slip-ring rotor is discussed for a periodical variation of the be eliminated, and in the new form, independent of the rotor angle, secondary resistance. This frequency response reveals the magnitude the equations present the basis for designing the block diagrams. and phase relation of the machine speed with respect to any change The block diagrams thus developed are given for the most important of the secondary resistance, and is, for instance, successfully used in types of a.c. machinery, i.e. the synchronous and the induction speed control systems. The given electromechanical frequency machines. The synchronous machine and the induction motor will response curves take into full account the electrical transients and be regarded as examples of machines with unsymmetrical and also--contrary to steady-state considerations of electrical machines symmetrical rotor, respectively. The block diagrams devised may then as practised hitherto- fast changes of the individual system variables, be incorporated in overall block diagrams of complete control i.e. those which occur with higher frequencies. It may be noted that systems. Some permissible omissions are also being discussed which the given frequency response curves have been calculated with the allow considerable simplifications of the exact block diagrams under aid of a digital computer. The block diagrams are the basis for the typical load conditions. simulation of electrical a.c. machines on the analogue computer. Sommaire Alors que le schema fonctionnel des machines it courant continu a On donne ces schemas pour les types les plus importants de une structure simple, du fait que les axes des enroulements sont fixes machines it courant aiternatif, c'est it dire les machines synchrones et dans l'espace, la theorie dynamique des machines it courant aiter- les machines d'induction. Ces schemas peuvent etre incorpores dans natif est beaucoup plus difficile, car les inductances mutue11es des les schemas globaux des systemes de controle. On discute quelques enroulements statoriques et rotoriques varient en valeur selon la omissions tolerables qui permettent des simplifications considerables. position angulaire instantanee du rotor par rapport au stator. Le A titre d'exemple, dans le cas d'une machine synchrone it poles systeme d'equations qui englobe n'importe quelle condition dynami- saillants, on peut tirer du schema fonctionnella reponse en frequence que, comporte des fonctions trigonometriques de l'angle du rotor. du mouvement angulaire du rotor, provenant de fluctuations periodiPar une transformation convenable on peut eliminer ces fonctions, ques des couples moteur et resistant. et sous cette nouvelle forme, independante de l'angle du rotor, les Les schemas fonctionnels sont it la base de la simulation des equations fournissent une base pour etablir les schemas fonctionnels. machines eJectriques it courant alternatif sur calculatrices analogiques. Zusammenfassung Wahrend die Strukturbilder von Gleichstrommaschinen infolge der Als Beispiel ftir die synchrone Schenkelpolmaschine wird der sich raumlich ruhenden Achsen der einzelnen vorhandenen Wicklungen aus dem Strukturbild ergebende Frequenzgang des Polradwinkels dieser Maschinen einfache und leicht erfal3bare Formen aufweisen, dieser Maschine angegeben, der sich bei periodischer Anderung des treten bei den elektrischen Wechselstrommaschinen infolge der Antriebs- bzw. Belastungsmomentes einstellt. Derartige FrequenzAbhangigkeit der Beeinflussung der Stator- und Rotorwicklungen gange werden vornehmlich bei der Untersuchung von Dieselkraftvon der jeweiligen Stellung des Rotors in der Maschinenbohrung anlagen verwendet. Die auftretenden Resonanzstellen werden wesentliche Erschwerungen der dynamischen Theorie auf. Die besonders hervorgehoben. Aus dem Anwendungsgebiet der Asynelektromechanischen Systemgleichungen dieser Maschinen , die jeden chronmaschine solI der Frequenzgang eines Regelschleifringlaufers beliebigen Obergangszustand beschreiben, enthalten trigonometrische bei periodischer Anderung des Rotorwiderstandes besprochen werden . Funktionen des die jeweilige Rotorste11ung charakterisierenden Ein derartiger Frequenzgang gibt an, mit welcher Grol3e und Phase Rotorpositionswinkels; diese lassen sich durch eine zweckmal3ig die Maschinendrehzahl der vorgenommenen Anderung des Rotorgewahlte Transformation beseitigen und in der sich somit ergebenden, widerstandes folgt, und wird mit Erfolg z.B. bei der Regelung der vom Rotorpositionswinkel freien transformierten Form werden die Maschine auf eine bestimmte vorgegebene konstante Drehzahl elektromechanischen Systemgleichungen der Aufste11ung der Struk- verwendet. turbilder zugrunde gelegt. Die angegebenen elektromechanischen Frequenzgange beriicksichDie nach diesen Gesichtspunkten entwickelten Strukturbilder tigen im vollen Mal3 die inneren elektrischen Ausgleichsvorgange und werden fur die beiden wichtigsten Wechselstrommaschinen, die erfassen somit- im Gegensatz zu den bisherigen stationaren BetrachSynchronmaschine und die Asynchronmaschine, angegeben. Die tungen elektrischer Maschinen-auch schne11e, also mit hoheren Synchronmaschine solI dabei als Beispiel ftir Maschinen mit unsym- Frequenzen erfolgende Anderungen der einzelnen SystemgroBen. metrischem Rotor, die Asynchronmaschine ftir Maschinen mit Erganzend soli noch erwahnt werden, dal3 die angegebenen Fresymmetrischem Rotor angesehen werden. Die entwickelten Struktur- quenzgange mittels einer digitalen Rechenmaschine ermittelt wurden' bilder konnen dann in die Gesamtstrukturbilder ganzer Regelkreise die angegebenen Strukturbilder bilden den Ausgangspunkt ftir di~ eingebaut werden. Aul3erdem werden noch die zulassigen Vernach- Nachbildung elektrischer Wechselstrommaschinen auf dem eleklassigungen besprochen, die bei charakteristischen Belastungszustan- tronischen Analogrechner. den der beiden Maschinentypen wesentliche Vereinfachungen der angegebenen exakten Strukturbilder ermoglichen. 198

1711