Lecture notes on synchronous machine (ac generator) transients

1. AC TRANSIENTS
SYNCHRONOUS MACHINE TRANSIENTS
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2. Synchronous machine Review
• Per phase equivalent circuit is a constant generated e.m.f in series
with a simple impedance.
• For salient pole rotor, generator is modeled with a direct axis
reactance Xd and a quadrature axis reactance Xq.
Synchronous machine equivalent circuit
Synchronous machine connected to an infinite bus
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3. Synchronous machine Review
Synchronous generator phasor diagram
phasor diagram of the generator showing excitations corresponding to lagging, unity and leading power
factors
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4. Transient Phenomena
To better understand the synchronous machine transient phenomena, we look first at the transient
behavior of a simple RL circuit. Consider a sinusoidal voltage v(t) applied to the circuit at time t=0
𝑣 𝑡 = 𝑉
𝑚 sin(𝜔𝑡 + 𝛼)
• oooo
A simple series circuit with constant R and L
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5. R L Circuit Transients
𝑅𝑖1 + 𝐿
𝑑𝑖1
𝑑𝑡
= 2𝐸𝑠sin(𝜔𝑡+𝛼)
The solution is of the form
𝑖1(𝑡) =
2𝐸𝑠
𝑍
sin(𝜔𝑡 − 𝜃) + 𝐴1𝑒
−
R
L
t
• Where
Load impedance 𝑍 = 𝑅2
+ 𝜔𝐿 2
1
2
Load angle 𝜃 = tan−1 𝜔𝐿
𝑅
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6. R L Circuit Transients
Constant A1 can be determined from the initial condition; at 𝜔𝑡 = 0, 𝑖1 = 0. A1 is found as;
𝐴1 = −
2𝐸𝑠
𝑍
sin(𝛼 − 𝜃) 𝑒
R
L
𝑡
𝑖1(𝑡) =
2𝐸𝑠
𝑍
sin(𝜔𝑡 + 𝛼 − 𝜃 − sin(𝛼 − 𝜃) 𝑒
R
L
𝑡
]
𝜃 = tan−1 𝜔𝐿
𝑅
𝑍 = 𝑅2 + 𝑋2
𝑖1 𝑡 = 𝑖1𝑎𝑐 𝑡 + 𝑖1𝑑𝑐(𝑡)
𝑖1𝑎𝑐 𝑡 = symmetrical current (steady state sinusoidal component)
𝑖1𝑑𝑐 𝑡 = DC off set current (dc transient component - decays with time)
Both components are equal and opposite at 𝑡 = 0
𝑖1 𝑡 = Asymmetrical current
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7. R L Circuit Transients
• Magnitude of the dc component depends on the instant of
application of voltage as defined by the angle 𝛼. It is zero when
𝛼 = 𝜃.
• It will have a maximum initial value of
2𝐸𝑠
𝑍
if the circuit is closed
when 𝛼 = 𝜃 −
𝜋
2
radians
• Current Wave form with maximum dc offset is shown in (b)*
• If 𝜔𝐿 ≫ 𝑅 then 𝜃 ≈
𝜋
2
. Circuit closure at voltage maximum will give
no dc component and closure at voltage zero will cause the
maximum dc transient current to flow.
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8. Example
R=0.15𝞨 L=10mH v t = 151sin(377𝑡+𝛼)
Determine the current response after closing the switch for the
following cases (1) no dc offset (2) maximum dc offset
• Solution
𝑍 = 0.125 + 𝑗3.77 = 3.772∠88.10 Im =
2𝐸𝑠
𝑍
= 40𝐴 τ =
L
R
= 0.08𝑠
𝑖(𝑡) = 40 sin(𝜔𝑡 + 𝛼 − 88.10
− sin(𝛼 − 88.10
) 𝑒
−
𝑡
0.08 ]
• No dc offset when switch is closed at 𝛼 = 88.10
• Has maximum dc offset when 𝛼 = 88.10 − 900 = −1.90
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9. • Current waveform (a) with no dc offset (b) with maximum dc offset
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10. 10

11. Transient conditions
• Sudden change of load
• Sudden change in speed such as under starting and braking
conditions
• Under short circuit conditions
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12. Balanced three phase short circuit
• Consider a three phase synchronous generator operating at
synchronous speed with constant excitation. We will explore the
nature of the three armature currents and the field current
following a three phase short circuit at the armature terminals.
The machine is assumed to be initially unloaded.
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13. Short circuit transients in synchronous
generator
• By far the severest transient condition that can occur in a
synchronous generator is a situation where the three
terminals of the generator are suddenly shorted out
• There are several components of current present in a
shorted synchronous generator
• When a fault occurs, the resulting current flow in the phases
of the generator can appear as shown in oscillogram*
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14. Short circuit transients in synchronous generator
It helps to determine
• winding and bus bar stresses
• Protective relay settings
• Circuit breaker ratings
To examine the nature of currents resulting from a sudden
three phase short circuit at the synchronous generator
terminals
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15. Transient process in a three phase synchronous
generator under sudden three phase short circuit at
its terminals.
• Assumption
• Machine is unloaded prior to short circuit and its speed remains
constant at rated value.
• Such condition provides a method of determining principal reactances
and time constants of the machine.
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16. Armature current in the various phases vary with time in a complicated way. The current waveforms
consist of the following components.
1. An alternating component (a fundamental frequency component and a
double frequency component)
2. A dc component which depends on the instant of switching. It is given by the
midpoint curve of the oscillogram.
When steady short circuit occurs at the machine terminals, resistances of the
armature coils are very small in comparison with the reactances so that the
resulting the armature m.m.f is nearly along the direct axis and the direct axis
parameters are mainly involved.
Three phase sudden short circuit of a
synchronous machine
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17. Typical Oscillogram
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