Electric Drives Q
An electric drive is a system that converts electrical energy to mechanical energy • Parts: – electric motor (or several) – control system (including software)
• Constant-speed drives – only a start/stop and protection system in addition to the electric motor
• Variable-speed drives (VSDs) – include an electronic power converter
1
Electric Drive and the Surrounding System Energy Power supply Electric drive
Display and control panel
Drive control
Converter
Process Effective work
Measurements Motor
Gear
Process control
Load
Field bus
2
Acceleration of Inertial Mass • Torque needed for accelerating the moment of inertia J:
TJ = J
dΩ m dt
• Moment of inertia of a thin-walled cylinder
J = mr 2 • Moment of inertia of a solid cylinder
J=
1 2 mr 2
J=
π 2
ρ lr 4
3
Mechanical Transmissions • Gear ratio
i=
n1 Ω m1 = n2 Ω m2
i=
Z2 r2 = Z1 r1
• Torque reduced to the motor shaft (power is preserved):
Ω m1T1 = Ω m2T2 Ω T T1 = m 2 T2 = 2 i Ω m1
Number of teeth Z1
Number of teeth Z2
• Moment of inertia reduced to the motor shaft (kinetic energy is preserved): 1 1 J1Ω m2 1 = J 2 Ω m2 2 2 2
Ω m2 2 J J 1 = 2 J 2 = 22 i Ω m1 4
Reduction of Linear Motion Ω m1T1 = v2F2 T1 =
v2
Ω m1
F2
1 1 J 1Ω m2 1 = m2 v22 2 2 J1 =
v22
Ω m2 1
m2
5
Equation of Motion Te = J
d Ωm + TL dt
Acceleration of a constant-torque drive
d Ω m Te − TL = dt J
Acceleration of a constant-torque/ constant-power drive 6
Field-Weakening (Constant Power) • The nominal speed is obtained at nominal armature voltage. • Higher speeds can be achieved by decreasing the magnetic flux. • The torque must be restricted (constant power operation).
ua ≈ kφ Ω m
Te = kφ ia
kφ = K aΦ
Pm = Ω mTe
7
Four-Quadrant Operation • Operation in all four quadrants of the Wm-T plane can be achieved: motor and generator (braking) operation in both rotational directions. • The direction of the armature current is changed for reversing the torque direction. T
Generator
Motor
Speed reversal
Operating point
Ωm
Motor
Generator 8
DC Motor Drives
Electromechanics and Electric Drives
9
Dynamic Model of a DC Motor • Armature circuit
u a = Ra i a + La
dia + kφ Ω m dt
kφ = K aΦ = f(i f )
• Equation of motion and torque
dΩ m Te = J + TL dt
f( i f )
Te = kφ ia
if
• Field circuit
uf = Rf if +
dψ f dt
ψ f = f(i f )
10
Dynamic Model of a DC Motor u a = Ra i a + La
dia + kφ Ω m dt
Te = kφ ia
Te = J
dΩ m + TL dt
After Laplace transform:
ua ( s) = Ra ia ( s) + sLa ia ( s) + kφ Ω m( s)
Te ( s) = kφ ia ( s)
Te ( s) = sJ Ω m( s) + TL ( s) 11
Block Diagram of a DC Motor 1/ Ra ⎡ u (s ) − kφ Ω m (s )⎤⎦ ia ( s ) = La ⎣ a 1+s Ra Te (s ) = kφ ia (s )
Ω m (s) =
Armature time constant L τa = a Ra Electromechanical time constant
Te (s ) − TL (s ) sJ
τm =
JRa kφ2
Block Diagram
12
Derivation of Transfer Functions ua + −
uai
1/ Ra 1 + sτ a
ia
TL
Te
kφ
−
+
1 sJ
Ωm
Ω m ( s) ua ( s)
=?
kφ
D( s) R( s) + −
G1( s)
+ +
H( s)
G2 ( s )
Y ( s)
Y ( s) G1( s)G2 ( s) = R( s) 1 + G1( s)G2 ( s)H( s) Y ( s) G2 ( s) = D( s) 1 + G1( s)G2 ( s)H( s)
13
Transfer Functions of a DC Motor Ω m ( s) ua ( s)
=
1 1 kφ 1 + sτ m + s2τ mτ a
ia ( s) 1 sτ m = ua ( s) Ra 1 + sτ m + s2τ mτ a
Ω m ( s) TL ( s)
=−
Ra 1 + sτ a kφ2 1 + sτ m + s2τ mτ a
14
Open-Loop Speed Control
• The angular speed is adjusted by means of the armature voltage ua. • The speed will change if the load or the supply voltage is varying. • The method can be used only for coarse speed adjustment in lowpower drives.
15
Closed-Loop Speed Control
• The method is rarely used (only in very small servo motors).
16
Cascade Speed Control
• Voltage disturbances are corrected by the inner control loop. • Load changes are corrected by the outer control loop. • It is easy to limit the current by ia,ref. 17
Feedback Control Objectives xref (s )
+
e(s )
−
Controller
Plant
Gc (s )
Gp ( s )
Q
Feedback control makes the system insensitive to disturbances and parameter variation.
Q
Control objectives:
x(s )
• zero steady-state error • good dynamic response (fast response, small overshoot and short settling time) 18
Definitions xref (s )
+
Controller
Plant
Gc (s )
Gp ( s )
e(s )
−
x(s )
• Open-loop transfer function
GOL ( s) = Gc ( s)Gp ( s)
• Closed-loop transfer function
x( s) GOL ( s) GCL ( s) = = xref ( s) 1 + GOL ( s)
• 0 dB crossover frequency fc ,ωc
GOL ( jωc ) = 0 dB
19
Closed-loop Step Response x(s ) = 1 xref (t )
0.9 0.8 0.7 0.632
0.6
1 1 s 1+ s kOL
x(t )
τ
x(t ) = 1 − e −t /τ
0.5
τ=
0.4 0.3
1 kOL
= 0, 5 ms
0.2 0.1 0 0
0.5
1
1.5 t (ms)
2
2.5
3
20
Steps in Designing the Controller 1. Assume that the system is linear about the steady-state operating point and design the controller using linear control theory. 2. Simulate the design under large-signal conditions and "tweak" the controller as necessary. •
For small-signal analysis, it is assumed that the steady-state operating point = 0.
•
If the highest bandwidth is at least an order of magnitude lower than the switching frequency, the switching-frequency components can be ignored.
21
Cascade Control reference speed reference position
+ −
Position controller
+ −
Speed controller
reference torque +
−
Torque controller
torque Electrical system
Mech. system
speed 1 s
position
torque (current) speed position
• Torque loop: fastest • Speed loop: slower • Position loop: slowest
22
Modeling of DC Machines
TL ( s) ua ( s )
+ −
1 / Ra 1 + sτ a uai ( s)
ia (s )
kφ
Te (s ) +
−
1 sJ
Ω m (s )
kφ
23
Controller Design Q
Procedure • Design the torque loop (fastest) first. • Design the speed loop assuming the torque loop to be ideal. • Design the position loop (slowest) assuming the speed loop to be ideal.
24
Torque (Current) Loop • Simplifying assumption: The moment of inertia J is assumed to be high enough, and the feedback from the angular speed can be ignored. TL ( s) ia ,ref (s) +
kT
PI
−
ua ( s ) +
−
1 / Ra 1 + sτ a
ia (s )
kφ
uai ( s)
−
Te (s ) +
1 sJ
Ω m (s )
kφ
ia (s )
PI ia ,ref (s) + −
kiI s
k pI ⎞ ⎛ s + 1 ⎜ ⎟ kiI ⎠ ⎝
kT
ua ( s )
1/ Ra 1 + sτ a
i a (s )
i a (s ) 25
Design of the Torque (Current) Loop PI ia ,ref (s) +
kiI s
−
k pI ⎞ ⎛ s + 1 ⎜ ⎟ kiI ⎠ ⎝
kT
ua ( s )
1/ Ra 1 + sτ a
i a (s )
i a (s )
k GI ,OL (s ) = iI s
⎛ kPI 1 + s ⎜ kiI ⎝
⎞ 1/ Ra k ⎟ T ⎠ (1 + sτ a )
• The zero of the PI controller is selected to cancel the motor pole:
k GI ,OL ( s) = iI s
⎛ k ⎜⎜ 1 + s PI kiI ⎝
⎞ 1/ Ra ⎟⎟ kT ⎠ ( 1 + sτ a )
k pI kiI
= τa
• kiI is chosen to achieve the desired 0 dB crossover frequency ωcI:
GI ,OL (j ωcI ) =
kiI kT =1 ωcI Ra
kiI =
RaωcI kT 26
Speed Loop Ω m ,ref (s) +
ia ,ref (s)
PI
−
1
ia (s )
kφ
Te (s )
1 sJ
Ω m (s )
Ω m (s )
• The current loop is assumed to be ideal (represented by unity). • The open-loop transfer function is GΩ ,OL ( s) =
k iΩ s
k pΩ ⎛ 1 + s ⎜ k iΩ ⎝
kφ ⎞ 1 ⋅ ⋅ ⎟ sJ ⎠
or GΩ ,OL ( s) =
kiΩ kφ J
1+ s
k pΩ k iΩ
s2
27
Design of the Speed Loop GΩ ,OL ( s) =
kiΩ kφ
1+ s
J
k pΩ k iΩ
s2
• The 0 dB crossover frequency ωcΩ is chosen an order of magnitude lower than ωcI with a reasonable phase margin φpm,Ω (e.g. 60°): GΩ ,OL (s )
kiΩ kφ J
s = jωcΩ
1 + jω c Ω −ωc2Ω
∠ ⎡⎣GΩ ,OL (s)⎤⎦ s= jω
=1 k pΩ k iΩ
=
k pΩ ⎛ ⎜ k k 1 + jωcΩ kiΩ ∠ ⎜ iΩ φ ⎜ J −ωc2Ω ⎜ ⎝
kiΩ kφ J
k pΩ ⎞ ⎛ 1 + ⎜ ω cΩ ⎟ ki Ω ⎠ ⎝
ωc2Ω
⎞ ⎟ k pΩ ⎟ = arctan ⎜⎛ ω cΩ ⎟ k iΩ ⎝ ⎟ ⎠
cΩ
= −180° + φ pm ,Ω
2
=1
⎞ ⎟ − 180° = −180° + φ pm ,Ω ⎠ 28
Design of the Position Loop θm ,ref (s ) + −
Ω m ,ref (s)
kθ
1
Ω m (s )
1 s
θm (s)
θm (s)
Gθ ,OL ( s) =
kθ s
• The speed loop is assumed to be ideal. (corresponds to unity). • Proportional gain kθ alone is adequate due to the presence of a pure integrator. • The 0 dB crossover frequency ωcθ is chosen
Gθ ,OL (j ωcθ ) =
kθ
ωcθ
=1
kθ = ωcθ
29
Further Issues Q
Feedforward: to improve dynamic response
30
Further Issues Q
Effect of limits: nonlinearity, further delay
Q
Antiwindup integration • Integration is suspended when the output saturates.
31
Effects of Measurements and Converter Unidealities
32
Rectifier-Fed Drives
• Transfer function of a rectifier: gain KT and delay τT
GT ( s ) = KT e− sτ T
e
− sτ
1
=
1 + sτ + • Approximate model:
KT GT ( s ) ≈ 1 + sτ T
( sτ )2 2
+…
• DC-DC chopper with a high switching frequency:
GT ( s ) ≈ KT 33
