# EnergyConversion short

```Outcome of this lecture
At the end of this lecture you will be able to:
• Calculate the field energy and co-energy
• Calculate magnetic forces and force densities
• Calculate the torque of an electrical machine
• Understand how the torque is produced in different machines
You will enhance your understanding of the energy conversion process in electrical machines.
Classification of Electric Motors
Electric Motors
Direct Current (DC)
Alternating Current (AC)
Separately excited
Series
Asynchronous Motors
Cage Induction
Shunt
Self excited
Compound
Synchronous Motors
Wound Rotor
PM Motors
Field Excitation
Reluctance
Energy Conversion Process
Conservation of energy
• Continuous and discrete energy conversion
• Motors
dWe < dWm ∗ dWf
• Generators
• Actuators
• Losses RI2, Pc
Electrical power
Electrical
Losses
Electrical
System (windings)
Magnetic
Losses
Coupling field
(iron and air)
Mechanical
Losses
Mechanical system
(bearings and fan)
Mechanical power
Energy conversion and losses
Typical values for induction motors P < 100 kW
Type of loss
Percentage of
total loss (100%)
Fixed loss or core
25
Variable loss: stator RI2
34
Variable loss: rotor RI2
21
Friction & rewinding loss
15
5
Pin < 3 UI cos π
Electrical power
Electrical
Losses
Electrical
System (windings)
Magnetic
Losses
Coupling field
(iron and air)
Mechanical
Losses
Mechanical system
(bearings and fan)
Mechanical power
Pout < Tmech ϖ mech
Pout
Eff <
Pin
Example: losses in pump drives
Fuel in
925 kW
Pump out
100 kW
Condensate
power station
556 kW (eff 40 %)
Motor and
Distribution
Pump losses
losses 37 kW chocking valve
33 kW (eff 75 %)
200 kW (eff 40 %)
(eff 90 %)
Fuel in
434 kW
Pump out
100 kW
Condensate
power station
260 kW (eff 40 %)
Motor and
Distribution
losses 18 kW inverter 31 kW
(eff 80 %)
(eff 90 %)
Pump losses
25 kW (eff 80 %)
Field Energy
No motion
dW m < 0
dWe < dWf
dWe < ei dt
e<
What is the expression for
spring potential energy?
dκ
dt
dWf < i dκ
κ
Wf < 〉 i dκ
0
Field Energy
Remember
Ni < H c lc ∗ H g lg
κ < NΕ < NAB
Wf < Vc wfc ∗ Vg wfg
Core energy
B
wfc < 〉 H c dB
0
Air gap energy
wfg
B2
<
2 λ0
linear material
Bc
Hc <
λc
Bc2
wfc <
2 λc
Energy – Co energy
Energy
Co energy
κ
Wf < 〉 i dκ
0
Wfϒ ∗ Wf < κ i
Characteristic depends on the air gap length
i
Wfϒ < 〉 κ di
0
Mechanical Force – Scenario 1
Movable part moves slowly
from x1 to x2
Constant current
Why ?
Mechanical Force – Scenario 1
ΧWe < 〉 ei dt
ΧWf <
κ2
〉 i dκ
κ1
ΧWm < ΧWe , ΧWf
Differential displacement dx
f m dx < dWm < dWfϒ
fm
∝Wfϒ ∋ i , x (
<
∝x
i <constant
Mechanical Force – Scenario 2
Movable part moves
quickly from x1 to x2
dWe < 0
f m dx < dWm < , dWf
fm
∝Wf ∋ κ , x (
<,
∝x
κ <constant
Force in a Linear System
Force from field energy
fm
κ < L ∋x (i
1
Wf < L ∋ x ( i 2
2
∝ ∑ κ2 ⌡
<, 

∝x  2L ∋ x ( 
1 2 dL ∋ x (
fm < i
2
dx
κ < constant
Force from co energy
Wf < Wfϒ <
1
L ∋x (i 2
2
fm
∝ ∑1
1 2 dL ∋ x (
2⌡
<
< i
 L ∋x (i 
∝x  2
dx
 i < constant 2
Linear System (Rc << Rg)
Ni < Hg 2 g <
Wf <
fm
Bg2
2 λ0
λ0
Vg <
2g
Bg2
λ0
Ag g
2
2
∑
⌡
B
B
∝
g
g
< 
Ag g  <
Ag
 λ0
∝g  λ0


Force pressure
Fm <
Bg
Bg2
2 λ0
Energy confined
in the air gap
λc < ⁄
Hc < 0
Rotating Machines
Stored field energy
dWf < es is dt ∗ er ir dt
< is dκs ∗ ir dκr
κs < Lss is ∗ Lsr ir
κr < Lrsis ∗ Lrr ir
dWf < Lss is dis ∗
Lrr ir dir ∗
Lsr d ∋ is ir (
1
2 1
Wf < Lss is ∗ Lrr ir2 ∗ Lsr is ir
2
2
Torque
∝Wfϒ ∋ i ,π (
T<
∝π
i <constant
1
2 1
Wf < Lss is ∗ Lrr ir2 ∗ Lsr isir
2
2
dLsr
1 2 dLss
1 2 dLrr
T < is
∗ ir
∗ is ir
2
dπ
2
dπ
dπ
For linear systems the energy
is equal to the co-energy
First two terms represent reluctance torque; variation of self-inductance
Third term represents the torque produced due to the variation of mutual inductance
Cylindrical Machines
• No reluctance torque
dLsr
T < is ir
dπ
• Mutual inductance
Lsr < M cosπ
• Currents
is < I sm cos ϖ st
ir < I rm cos ∋ϖ r t ∗  (
• Rotor position
π < ϖmt ∗ χ
Basis for Synchronous and Asynchronous Machines
 sin

 sin
I sm I rm M 
T<,
 sin
4

 sin

ζ∋ϖm ∗ ∋ϖs ∗ ϖr ( ( t ∗  ∗ χ | ∗ 
ζ∋ϖm , ∋ϖs ∗ ϖr ( ( t ,  ∗ χ | ∗ 
ζ∋ϖm ∗ ∋ϖs , ϖr ( ( t ,  ∗ χ | ∗ 
ζ∋ϖm , ∋ϖs , ϖr ( ( t ∗  ∗ χ | 
• Torque in general varies sinusoidally with time
• Average value of each term is zero unless the coefficient of t is zero
• Nonzero average torque exists only if ϖ m < ° ∋ϖ s ° ϖ r (
ϖm < ϖs ° ϖr
Synchronous Machines
synchronous machine
ϖr < 0
ϖm < ϖs
 <0
I sm I R M
T <,
sin ∋ 2ϖst ∗ χ ( ∗ sin χ |
ζ
2
Tavg
I sm IR M
<,
sin χ
2
• Single-phase machines, 1 winding at the stator
• Pulsating Torque : NOT OK for larger machines!
• Poly-phase machines to minimize pulsating torque
• ωm=0  Tavg=0 ⇓ Not self starting
Asynchronous Machines
asynchronous machine
ϖ m < ϖ s , ϖr
ϖm ÷ ϖr
ϖm ÷ ϖs
I sm I rm M sin ∋ 2ϖ st ∗  ∗ χ ( ∗ sin ∋ ,2ϖr t ,  ∗ χ ( ∗ 
T<,


4
sin ∋ 2ϖ st , 2ϖ rt ,  ∗ χ ( ∗ sin ∋ ∗ χ ( 
Tavg < ,
I sm I rm M
sin ∋ ∗ χ (
4
•
Single-phase machines
• Pulsating Instantaneous Torque
• Not self-starting
•
Poly-phase machines minimize pulsating torque and self starting
```