Harmonic modelling of synchronous machines
P.M. Hart, M.EngSc
W.J. Bonwick, PhD, FlEE
Indexing terms: Power systems and plant, Synchronous motors
Abstract: The connection of power converter and
other nonlinear loads to the electrical power
system results in other system components being
subject to voltage waveform distortion. In order
to predict the resultant heating and harmonic
current flows, it is necessary to have fairly accurate models for components such as synchronous
machinery. The paper presents a theoretical and
experimental study into the electrical behaviour of
synchronous machines when subjected to harmonic voltage disturbance at the terminals. It is
shown that the time-varying nature of the
machine inductances results in current flow at
both the applied and at an associated harmonic
frequency. Consequently, it is inadequate to model
the machinery by a pure reactance at each harmonic frequency.
List of symbols
phase indices
index
= harmonic number (always positive)
= current and voltage at harmonic frequency,
no
= current and voltage at harmonic frequency,
(n T 2)o
= angle of field winding at t = 0 (see eqn. 3)
a, b, c
=
f
= field
n
' n y Vn
-
. -
1n 9 u n
a
i
La, L f f
L&, Lb
L&',Lb'
zn
z
Zn
TI
*XY
*x
N
9: 0
PPS
nPs
1
= machine
= appaient
inductances
impedance at frequency, n o
= apparent impedance at frequency, (n T 2)w
= supply impedance at frequency, n o
= supply impedance at frequency, ( n f 2 ) o
= flux linking winding x due to current
in
winding, y
= total flux linking winding, x
= denotes phasor
= denotes phase angle
= positive phase sequence
= negative phase sequence
small to medium sized distorting loads if they are within
preset size limits, for large loads a detailed harmonic penetration study is necessary [l]. The study will usually
extend back to the zone substation for strong systems,
but may need to extend to generators for small systems.
Power converter loads act as harmonic sources of
current which propagate through the distribution system.
Shunting loads within the distribution system provide
harmonic current paths, and thus affect the overall
current flow in the system. At supply frequency it is usual
for the impedance of shunting loads to greatly exceed
that of the system impedance as seen at the load. This is
not necessarily true at harmonic frequencies because resonances in the power system may occur, expecially if
capacitors are connected nearby. If an accurate harmonic
penetration study is to be achieved it will be necessary
that system components including shunting loads are
accurately modelled. The study is made more difficult
because of changing loads and changing circuit arrangements. Consequently, the study should include likely
extremes of network loading variation and anticipated
network changes.
The paper considers the harmonic modelling of the
synchronous machine, and presents an analysis of the
harmonic currents and voltages that result at the
machine terminals when the machine is subjected to harmonic disturbance of the supply. The effects on shaft
torques are not considered here, but they have been
treated by Williamson 161. Basic plant harmonic item
models are given in References 2, 3,4 and 5.
A second major concern is whether a machine will
overheat when subjected to disturbance. This can be predicted if the machine impedance at harmonic frequencies
is known. The losses in machines drawing nonsinusoidal
current wneforms have been studied by Williamson et
al. [6, 71.
It is usual to model synchronous machinery by a reactance which is based on either the negative sequence or
the transient reactances [2, 31, and induction machines
by an inductance value based on the direct on-line
current. Losses may be included by adding resistances
[5] as shown in Fig. 1. In the paper, the authors report
Introduction
The increased use of power converters in industry raises
concern about harmonic disturbance levels in the power
system. Whilst the supply authority generally accepts
b
Paper 5797B (Pl), first received 5th January and in revised form 26th
June 1987
The authors are with Monash University, Department of Electrical
Engineering, Clayton, Victoria 3168, Australia
52
Fig. 1 Harmonic equivalent circuits for synchronous and induction
machines
a Circuit based on negative sequence impedance
b Formal equivalent circuits
IEE PROCEEDINGS, Vol. 135, P I . B, N o . 2, M A R C H 1988
on their own theoretical work and discuss supporting
evidence. It is shown that the model of Fig. l a is generally inappropriate for synchronous machine modelling.
2
Synchronous machine
and
2.1 Application of a single current harmonic
The stator voltage components which result when balanced harmonic currents of harmonic number n are
applied to a 3-phase synchronous machine are determined. The synchronous machine has a single field
winding on a round rotor and no subtransient properties.
Subtransient affects are considered in Section 2.3.
In complex exponential notation the applied currents
are
+ 3: in]
i, = in exp j [ n o t + 3: in T 27c/3]
i, = in exp j [ n o t + 3: in f 2n/3]
i,
=
in expj[nwt
(1)
The upper signs in eqn. 1 apply for a positive-phase
sequence applied-voltage set, and the lower for negative
phase sequence. Expressions for voltage, current and flux
may be obtained by taking the real part of each relevant
complex equation.
The flux linking the rotor field is
$y =
in L,,(expj[nwt
where the inductances
L & - Lh -
(7)
4 Lrr
This equation is valid for a round rotor machine only as
saliency effects on the inter-phase inductances Lab and La,
are ignored.
Because there is a single field winding, the rotor flux
pulsates along the field axis. Such a flux can be represented as the resultant of two counter rotating waves (relative
to the field winding) of constant amplitude, one at rotational frequency n o and the other at (n f 2)w with
respect to the stator. This is the interpretation of eqn. 6.
The a-phase stator voltage is obtained by differentiating eqn. 6 and is
2
v, = injno( L d
\
+ 3: in] COS (wt + a)
+ exp j[nwt + 3: in T 2n/3]
x COS (wt +
2~/3)
+ exp j[nwt + 3: in & 2n/3]
x cos ( o f + a + 2x/3)}
= $ L a f i nexp j [ ( n
1)wt T u + 3: in]
ub = injnw(
La, COS (ut + a).
(3)
Inductance symbols are defined in Reference 8.
The field winding is assumed to be rotating at synchronous speed. Thus the field current induced is at frequency ( n T l ) w and is
+
x exp j [ ( n f 1)wt T a
3: in + A] (4)
where i.= arctan (rr/[n & l ) o L r r ] ) .
This equation is not valid for the fundamental
( n = 1 ) pps supply as there is no relative motion of field
and flux wave under this condition, and hence no induction.
Considerable simplification results if
(n T l)wLj, % r,
(5)
This will be assumed to apply, and is well founded provided losses do not rise too rapidly with frequency. For
laminated rotor machines, eddy current losses will be
small, whereas in solid pole machines, eddy currents may
dramatically increase losses, and decrease the effective
inductance L,, and La,.
The flux linking the stator a-phase has contributions
from the field and the b and c phase windings and is
IEE PROCEEDINGS, Vol, 135. Pt. B, No. 2, M A R C H 1988
'
exp j[nwt
+ 3: in]
/
(6)
.)
L&+ L'
+ inj ( n T 2)w
(L'
(2)
where the mutual inductance between the field winding
and the a-phase winding is
+ 3: in]
lL')
Using a similar analysis the b-phase voltage is
-
x exp j [ ( n T 2)wt T 2u
3 L:,
exp jCnwt T 2x13
2"9
+ 3: in]
+
x exp j [ ( n f 2)ot & 2a T 2x13
3: in]
(9)
Comparing Eqns. 8 and 9, it is seen that the additional
harmonic voltage term has the opposite phase sequence
to that which is applied. For example, if n = 7 with
positive-phase sequence, the additional harmonic component has harmonic number 5 with negative-phase
sequence. This is the usual situation for a balanced
system. If, however, n = 7 with negative-phase sequence,
the additional component has harmonic number 9 and is
positive-phase sequence : not zero-phase sequence as is
usually expected. Consequently, the phase voltage set will
not be balanced. The analysis is also valid for the application of negative-phase sequence current at fundamental
frequency in which case the additional component has
harmonic number 3, and is positive phase sequence [ l o ] .
2.2 Application of a single voltage harmonic
If a balanced set of harmonic voltages is applied to a
synchronous machine, an additional harmonic current
with harmonic number n f 2 is anticipated, based on the
previous results. The proposed solution is
i,
=
in exp j[nwt + 3: in]
+ i, exp j [ ( n T 2)wt + 3: i,]
+ 3: in T 2x131
+ exp j [ ( n f 2)wt + 3: i, i 2+]
i, = in exp j[nwt + 3: in k 271131
+ < exp j [ ( n T 2)wt + 3: k 2n/3]
i,
=
in exp j[nwt
(10)
The barred quantities are the current components at frequency ( n f 2)w. The phase sequence of these com53
ponents is the reverse of that at the applied frequency,
nw.
Using the results of the previous section, the a-phase
voltage is
u, = in jnw(
+
L& L'
q)
exp jrnwt
+ 9: i n ]
+ in j ( n T 2)w( L b 2 " i ) exp j [ ( n T 2)
x wt T 2a + 9: in]
Voltage equations for the general case are obtained by
applying a voltage mesh equation at each frequency nw
and ( n f 2 ) o . The voltage components applying across
the machine are those given by eqn. 11 in response to the
currents given by eqn. 10.
Hence at frequency n o
u, exp j 9: u,
=
-
i n z , exp j 9: z ,
in jnm(L'
')
exp j 9: in
and at frequency (n T 2 ) o
u, exp j 9: ir;; - i n z , exp j 9: U;;
-
For an applied voltage
v,
= v,
= i,,j(n T
2)-(
L:, + L'
q)
expjxi,
exp jnwt
(12)
comparing eqns. 11 and 12 gives the following equations
u, = i,jnw('
2
' i ) + i,jnw(L:, 'i)
+ i , , j ( r 1 k 2 ) o ( ~ ; ';)expj[T2a+
<in]
(16)
These equations may be manipulated to give equations
for inand inas follows
in exp j 3: in =
u , c exp j 9: u, - F, exp
9: i j ~ j n (L'
w
2' i )
exp +2aj
Eqns. 13a and b may be rearranged to give
u, = j n w in
2L&L;
L& L;
+
Comparing eqns. 9 and 14, it is seen that the effective
inductance at the applied frequency is different for the
voltage and current sourced cases. If harmonic voltages
are applied at both frequencies n o and (n f 2 ) o , then the
eqn. 13 will be modified by the inclusion of and 3: G.
The currents at both frequencies will consequently be
affected.
and
-
-
in exp j 9: in =
2.3 The general case
In the general case, a machine will be subjected to
applied voltages v, and y;; via system impedances z, and
z;; as shown in Fig. 2. The system model may be viewed
In these equations
5,
= z , exp j 9: z ,
+j n w
(Ld
and
-
5,
Fig. 2
System in general case; balanced representation
as a Thevenin model and consequently linearity is being
assumed.
It will be assumed at this stage that the applied harmonic voltages have opposite phase sequences, one positive and the other negative.
This is the condition that would apply if the distorting
load responsible for the harmonic disturbance draws balanced currents. Consequently, a single phase model may
be used.
54
=
exp j
< + j ( n f 2)w
Eqns. 17 and 18 show that the effective voltages at each
frequency involve the voltage sources at both frequencies,
there being a phase term which depends on angle a. The
resonance condition for the equation is
under which condition currents at both frequencies
become large.
Eqns. 15 and 16 suggest the equivalent circuit shown
in Fig. 3. The interaction between the two sides of the
circuit is shown as a mutual inductance, but this is an
over simplification. The mutual inductance is nonIEE PROCEEDINGS, Vol. 135, P t . B, N o . 2, M A R C H 1988
reciprocal, and the frequency difference between sides has
been ignored. The model may be used in schemes to calculate harmonic current flow. The results presented here
Fig. 3
Equivalent circuit,forgeneral case
show that the machine cannot be modelled by an impedance (Fig. l(a)) if significant voltage sources exist at each
frequency.
In the case where one harmonic voltage source is large
and the other small so that q N 0, the apparent machine
impedance to current flow is
f
The eqns. 17 and 18 are not valid for the application of
positive phase sequence voltage at fundamental frequency
( n = 1) as there is no induced field current, and V, and
are zero. For negative sequence voltages with n = 1, the
equations are valid and the additional frequency harmonic number is 3. If there is no background voltage at this
frequency then the machine impedance offered to negative sequence fundamental current is given by eqn. 21 and
contains terms dependent on the system impedance at the
third harmonic.
The machine model leading to eqns. 17 and 18 is
linear and time varying. If several harmonic frequency
sources exist, the total current flow will be the superposition of those given by the theory, when applied to
each harmonic n and its associated harmonic ( n T 2).
If the rotor carries two windings, one on the d-axis
and one of the q-axis, then the analysis is altered. If the
two windings are electrically identical, the net gap flux
wave resulting from induced current in both windings
will rotate at frequency no. The previously discussed
wave at frequency ( n T 2)w will be suppressed, and consequently the machine impedance at frequency nw is
Z, =jwn
L&+ L;
~
f
However, if the windings are not electrically identical, a
counter rotating flux wave will occur with rotational frequency ( n T 2)w and a magnitude dependent on the
extent of the electrical difference between windings.
In the case of two rotor windings the substitution
is invalid. However, the equations presented previously
which are expressed in terms of L&and Lb are valid.
I E E PROCEEDINGS, Vol. 135, Pt. E , No. 2, M A R C H 1988
The results of this section are clearly associated with
the transient response of the machine. If the machine has
subtransient properties, application of harmonic disturbances will induce currents in the damper windings or
pole face which will partly shield the field windings from
the applied flux wave. The machine response will be
given by eqns. 17 and 18 with the following replacements
L:,+ L: and Lb+ Li
(24)
Because of the variation of flux penetration depth with
frequency, these machine impedances will also be a function of frequency.
Subtransient saliency will occur if the damper windings are incomplete (i.e. confined to the pole faces), if the
dampers have a different construction between poles to
that used on the pole axis [9] or if eddy currents in the
pole faces of salient pole machines contribute to the subtransient response. Incomplete shielding of the field windings of a machine fitted with dampers subjected to
harmonic disturbance has been demonstrated by Meyer
et al. [9] for a 4.6 MW synchronous motor.
For large to medium sized machines it is expected that
the subtransient response will dominate. The additional
frequency generated from within the machine, as demonstrated for example in eqn. 8, will then depend on the
level of subtransient saliency. The results of Meyer indicate that the subtransient saliency can be significant.
2.4 Non -sinusoidal MMF effects
Non-sinusoidal gap flux wave effects can be accounted
for by representing the mutual inductances between the
field and stator, and between the stator windings by a
trigonometric series in the rotor angle 8. The resulting
mathematics is extremely complex. It is found that the
inclusion of non-sinusoidal inductance variation causes a
spreading of the spectrum such that injection of a single
harmonic current results in voltage arising at all odd harmonic numbers, including the fundamental. The extent to
which the energy is spread across the spectrum depends
upon the pole shape, the numbers of stator and rotor
slots, and the damper design. The spectral spread is a
second order effect compared to the generation of the
associated voltage at frequency n f 2.
3
Measurements on synchronous machines
To test the preceding theory a series of measurements
was taken on two 5 kVA, 415 V, 4-pole synchronous
machines (A and B). Full details of the machines used for
measurement are given in Appendix 7. Machine A has an
incomplete pole face damper winding built in and so
exhibits subtransient saliency. Machine B exhibits transient saliency only and is known to have a voltage waveform with very little harmonic corruption.
3.1 Current spectrum for an applied 7th harmonic
voltage, machine A
A 350 Hz ( n = 7) 3-phase voltage was applied to machine
A and the current spectrum was measured using a spectrum analyser applied across a resistive shunt. Measurements were taken for both negative and positive-phase
sequence supply. Measured values are given in Table 1.
These results are in accord with the theory presented in
Section 2 where it was concluded that for n = 7 pps, a
strong signal is expected at the 5th harmonic and for
n = 7 nps a strong signal occurs at the 9th harmonic.
The spectrum shows the spreading which is associated
with non-sinusoidal inductance variations. The flux
55
Table 1 : Measured current spectrum f o r 350 Hz voltage
aodication. Machine A.
Positive-phase sequence
Frequency, Hz Harmonic No.
RMS current, mA
~~
50
250
350
450
550
650
850
1150
1750
1850
1
5
7 (applied)
9
11
13
17
23
35
37
19.7
143.1
324.0
0.8
8.2
3.3
0.3
0.2
1.o
4.9
machine
cored
N
I
v
Negative-ohase seauence
50
150
250
350
450
550
650
750
850
1650
1750
1
3
5
7 (applied)
9
11
13
15
17
33
35
26.3
6.6
0.6
329.2
141.4
0.1
3.3
1.o
0.3
0.3
2.3
waveforms arising from induced damper currents exhibit
significant harmonic corruption and so the spreading of
the spectrum is more extensive than would have been
expected based on the non-sinusoidal inductance variation measureable under static conditions.
It is seen in Table 1 that the ratio of currents i,,/< at
frequencies no and (n f 2)o is close to equal for both
positive and negative sequence cases. This is expected
from eqn. 13 since
-
i,,/i, =
L:, + L;
L:, - L;
~
which is independent of frequency or phase sequence.
3.2 Measurement of apparent impedance, machine A
The apparent impedance at harmonic frequencies of
machine A was measured using the arrangement shown
in Fig. 4. The bridge rectifier draws odd, non-triplen harmonic currents, some of which flow into the machine
because the supply impedance has been artificially
increased by adding X , . Using a special purpose, dual
channel frequency unit, the voltage and current at each
harmonic frequency can be displayed, and the apparent
The measured apparent normalised impedance against
frequency is given in Fig. 5. The values are seen to oscillate in a saw-tooth manner with frequency. The voltage
component at each harmonic frequency has two components, one being the first term in eqn. 8 and the other
associated with current flow at the associated frequency
(i.e. second term in eqn. 8) which has phase angle dependence on a. These two components may either reinforce
or cancel resulting in larger or smaller normalised impedances as observed.
The results of Table 2 are generally in accord with
theory, although some significant discrepancies d o occur.
Factors which are not accounted for in the theory are
machine losses (the variation with frequency which is not
known), accurate modelling of the source machine, and
incomplete maintenance of field flux linkages. These
results do confirm that the apparent machine impedance
at the applied harmonic frequency varies significantly
when the system impedance is varied.
Oscillographs of the current waveform drawn by the
machine during test 1, table 2 are shown in Fig. 7a. The
waveforms exhibit beating affects as expected from the
two associated frequency terms. Machine B was also
machine
60V
350Hz
0
+d
Fig 6
Measured circuitfor machine B
Quiet reactances are at 350 Hz
mochiine
50HZ
hormonic number, n
56
Fig. 5 Measured normalised appurrnt impedance against
frequency, machine A
1 Noload
2 Reluctance m/c, field s.c.. 2.9 A. 193 Ci$
3 Reluctance m/s, field o.c., 2.9 A, 193 V+
4 Synchronous m/c. 6.0 A. 200 V+
IEE PROCEEDINGS, Vol. 135, Pt. B, N o . 2, M A R C H 1988
Table 2: Measured and theoretical apparent impedance
Machine B. Circuit connection is shown in Figure 6.
for
Circuit
description
Sequence
Measured, fl
Theoretid, fl
1
n = 7 pps
(nT2)=5
29 3 @ 78"
3 1 @82"
27 9 (a 90"
6 1 (g 90"
n = 7 nps
(n T 2) = 9
29 5 (a 80"
6 3 @ 77'
27 9 (a 90"
11 0 (a 90"
n=7pps
(nT2)=5
4 5 7 (a 79"
635@81"
465
558
2
-n=7nps
'L
3
4 5 3 (& 792"
(nT2)=9 1059@846"
n = 7 pps
69 7 6 72 5"
(n T 2) = 5 203 1 @ 63'
4 6 5 @ 90"
1005@90"
60 5 (a 90"
296 0 @ 90"
21 7 (a 71 8'
1 3 2 ( a -785"
19 7 (a 90"
5 8 6 -90
n = 7 nps
(nT2)=9
II
25pF
4
Conclusions
v
I
v
Fig. 7
Oscillographs
Current (i) and applied voltage (V) waveforms. (i) and (ii) positive sequence
applied harmonic. (iii) and (iv)negative sequence applied harmonic.
b Waveforms with the quadrature field short-circuited, machine B. (V) armature
voltage, (i) current, (i,) field current waveforms for machine B with 350 Hz voltage
applied
a
fitted with a rotor winding on the quadrature axis which
was left open during all previous tests. When this winding
is short-circuited, the machine becomes zero transient
salient with X &= X b . The current waveform is a pure 7th
harmonic as shown in Fig. 7b. The measured machine
impedance in this case is 13.3 R which is close to the calculated value of noL& (14.3 R). The measured results
illustrate that a synchronous machine which exhibits
transient saliency cannot be modelled by a simple equivalent circuit like those given in Fig. la.
IEE PROCEEDINGS, Vol. 135, P t . B, N o . 2, MARCH I988
When a synchronous machine IS subjected to a harmonic
voltage disturbance at frequency n o , harmonic current
components are drawn at n o and at the associated frequency ( n & 2)o. The upper sign applies for positive and
the lower sign for negative-phase sequence. Current flow
at the associated frequency occurs because the machine is
a time-varying electrical system.
In a similar way, a synchronous machine fed by harmonic current at frequency n o develops a voltage across
its terminal at both the applied and the associate frequency ( n f 2)w.
In consequence of this behaviour, the machine cannot
be modelled by an impedance applying at each harmonic
frequency. The harmonic behaviour is governed by a set
of coupled equations, eqns. 17 and 18, which involve
machine and system parameters at both frequencies.
Harmonic behaviour is also affected by non-sinusoidal
inductance variation with rotor position and by the
extent to which induced damper winding currents shield
the field winding from the harmonic gap flux waves.
5
Acknowledgment
The authors gratefully
the
of the Electrical Research Board of Australia.
6
References
1 'Disturbances in mains supply networks', AS227Y, Pts. 1 & 2
1979, Standard Association of Australia
2 PILEGGI, D.J., H A R K , H., CHANDRA, N.. and EMANUEL,
A.E. : 'Prediction of harmonic voltages in distribution systems',
Trans. PAS., 1981,100, pp. 1307-1315
3 C E D D E Y , D.K.: 'Computer calculation of harmonic voltage levels
in high voltage transmission systems'. Seminar on miniqg and industrial loads, Sydney, August 1983
4 IEEE Working group on power systems harmonics: 'The effects of
power system harmonics on power system equipment', IEEE PAS.,
1985, 104, pp. 2555-2563
~
51
5 CIGRE Working group 36-05 : ‘Harmonics, characteristic parameters. methods of study, estimates of existing values in the
network’, Electra, 1981, pp. 35-54
6 WILLIAMSON, A.C.: ‘The effects of system harmonics upon
machines’, I J E E E , 1982,19, pp. 145-155
7 WILLIAMSON, A.C., and URQUHART, E.B.: ‘Analysis of the
losses in a turbine-generator rotor caused by unbalanced loading’,
IEE Proc. B, 1976, 123, pp. 1325-1332
8 FITZGERALD, A.E., and KINGSLEY, C.: ‘Electric Machinery’,
Chapter 5 , McGraw-Hill, 1961
9 MEYER, A., and ROHRER, H.J.: ‘Calculations and comparative
measurements for converter-fed synchronous motors’, Brown Boveri
Review, 1985,72, pp. 71-77
10 KIMBARK, E.W., ‘Direct Current Transmission - Vol. l’, WileyInterscience, N.Y., 1971
7
Appendix
(a) Machine A Salient pole synchronous machine
5 kVA, 4 15 V, 4 pole, 50 Hz, 1500 rev/min
X,
=
45.5 Q, R
1.83 R (at d.c.), X ,
=
15.0 0.
( b ) Machine B Round rotor synchronous machine,
5 kVA, 415 V, 4 pole, 50 Hz, 9 A, 1500 rev/min
(c) Data for Machine B
Single-field mode
ZTS mode
5x
=
54 mH
6.5 mH
L,, = 4.9 H
La, = 0.40 H
Ld = 54 mH
Lh = 6.5 mH
L,, = 3.6 H
La, = 0.34 H
Ld
=
Li
=
I E E P R O C E E D I N G S , Vol. 135, P t . E , N o . 2, M A R C H 1988