Boise State University
Department of Electrical and Computer Engineering
ECE 570 – Electric Machines
Spring 2021
Simulation Project: Synchronous Machine Dynamics
1 Objective
The objectives of this experiment are to further understand machine dynamics through the math-
ematical simulation of synchronous machine step loading.
2 Background
The instantaneous rotor position of a synchronous machine may be written in mechanical radians
as,
θm(t) =
∫ t
o
ωm(t)dt+ θm(0) (1)
where ωm(t) is the instantaneous rotor speed in mechanical radians per second. For a 6-pole 60-Hz
synchronous machine running at 1200 rpm, the steady-state shaft speed is equal to the mechanical
synchronous speed
ωm(t) = ωms =
(
2
p
)
ωs =
(
2
6
)
(2pi60) (2)
and
θm(t) = 40pit+ θm(0) (3)
Defining θ(t) in electrical radians by
θ(t) =
(
p
2
)
θm(t) (4)
and ω(t) in electrical radians per second by
ω(t) =
(
p
2
)
ωm(t) (5)
In steady state,
ω(t) = ωs = 2pifs = 2pi60 (6)
θ(t) = 2pi60t+ θ(0) (7)
where θ(0) = (p/2) θm(0). For a fixed voltage source, an electrical angular displacement may be
written as
va(t) =
√
2Va cos(θs(t)) =
√
2Va cos(ωst+ θv) (8)
where for a 60-Hz source, ωs = 2pi60 electrical radians per second. Define the power (torque)
angle delta,
δ(t)
∆
= θ(t)− θs(t) + pi
2
(9)
1
where, θ(t), θs(t) and thus δ(t) are all in electrical radians. Newton’s second law for rotating bodies
requires that
J
d2θm
dt2
=
∑
T+ (10)
where
∑
T+ is the sum of torques acting in the positive +θm-direction. Since
θm =
(
2
p
)
θ (11)
then
d2θm
dt2
=
(
2
p
)
d2θ
dt2
(12)
Also note that
d2θ
dt2
=
d
dt
(
dθ
dt
)
=
d
dt
(
dδ
dt
+ ωs
)
=
d2δ
dt2
(13)
and
d2θ
dt2
=
d
dt
(
dθ
dt
)
=
dω
dt
(14)
It can be shown that the steady-state electrical torque into a balanced 3φ round-rotor p-pole
synchronous motor may be expressed as
Te =
Pe
ωms
=
Pe
(2/p)ωs
=
3|V˜a||E˜a|
(2/p)ωsXs
sin(−δ) (15)
where |V˜a| is the line-to-neutral rms stator phase-a voltage, Xs is the synchronous reactance, and
|E˜a| = ωsMsfIf√
2
(16)
where ωs is in electrical radians per second, and δ is in electrical radians. The damper (amortis-
seur) windings of a synchronous machine provide an asynchronous torque through induction which
contributes only when ωm 6= ωms so that
TD = D(ωm − ωms) = D
(
2
p
)
(ω − ωs) (17)
The steady-state mechanical shaft torque is
Tm =
Pm
ωms
=
Pm
(2/p)ωs
(18)
where Pm is the shaft load in watts.
Assuming that the steady-state electrical and mechanical torque expressions are valid for the rela-
tively slow mechanical dynamics being investigated in this experiment, the equations which describe
the synchronous machine dynamics are as follows:
dδ
dt
= ω − ωs (D.E.1)
dω
dt
=
1
J
(
p
2
)
(Te − TD − Tm) (D.E.2)
2
2.1 The Problem
Consider a balanced 3φ 6-pole 60-Hz round-rotor synchronous motor with
Ra = 0 Ω |V˜a| = 230/
√
3 V
Xs = 9.0 Ω D = 2 N-m/s
Msf = 0.4 H p = 6 poles/phase
If = 1.2 A J = 0.4 kg-m
2
Let Pm = 0 at t = 0 so that the machine is unloaded, with δ(0) = 0 and, in steady state,
ω(0) = ωs. The maximum load power Pm,max (W) that the motor could handle theoretically is
found by setting δ = − 90o in Equation (15) and solving for power instead of torque:
Pm,max =
3|V˜a||E˜a|
Xs
=
3× (230/√3)× (120pi ∗ 0.4 ∗ 1.2/√2)
9
∼= 5664 W
a) Determine the stability of the synchronous motor (i.e., stable or unstable) by examining the
dynamic responses δ(t) and ω(t) to sudden shaft loads of Pm = 3000, 4000, 5000 Watts,
each applied at t = 1.0 s using a “switch” block. Perform these simulations for D = 2 N-m
and D = 0 N-m and write down your answers in the following table:
Pm (W) 3000 4000 5000
D = 2 (N-m) Stable/Unstable Stable/Unstable Stable/Unstable
D = 0 (N-m) Stable/Unstable Stable/Unstable Stable/Unstable
Comment on these results.
b) Use your Simulink model to find the critical Pm = Pcr and Tm = Tcr which can be added
suddenly from no load and such that the machine will not lose synchronism with the network
for D = 2 N-m. Repeat this simulation for the case when D = 0 N-m.
c) Verify that the critical Tcr satisfies the equal-area criterion discussed in class for the case of
zero damping, i.e., when D = 0 N-m. In other words, verify numerically (using MATLAB)
that the steady-state angle |δcr| corresponding to Tm = Tcr (or Pm = Pcr) satisfies the
criterion that the decelerating area A1 equals the accelerating area A2 when D = 0.
d) Submit a copy of your Simulink model and three plots of δ(t) for Pm = 3000, 4000, 5000
Watts when = 0 on separate graphs. Comment on each graph.
3
AA
1
2
pi |δ|0 cr cr|δ | pi−|δ |
Te
0
max
Tm
T ,Te m
Tcr
T
4

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