Download Motors: Equivalent Model

Motors: Equivalent
Model
Filippas 2016
Equivalent model
Node A
Node A
2
Back_EMF
+
Te
Node B
3
MotorResistor
R
Equivalen
t to
V_emf
θ
Ea
Motor
Node B
Motor in a circuit
Going around the loop:
The circuit is comprised of the
voltage source and the motor.
DC voltage source: rise : +V
Set up KVL in the one loop – taking into
account the equivalent circuit of the motor.
Ammeter: zero voltage drop
Ammeter
+
0.000
A
3
MotorResistor
R
1
DC_Source
V
Resistor: voltage drop consistent
with the direction of the current : VR
+
+ VR
+
V_emf
KVL
-
0
2
Back_EMF
Back-emf voltage: drop : -Vemf
𝑉 − 𝑉𝑅 − 𝑉𝑒𝑚𝑓 = 0
Motor in a circuit
From previous:
𝑉 − 𝑉𝑅 − 𝑉𝑒𝑚𝑓 = 0
Ammeter
+
0.000
A
3
MotorResistor
R
+ VR
1
DC_Source
V
I
0
-
2
Back_EMF
Ohm’s law
𝑉𝑅 = 𝐼 ∙ 𝑅
V_emf
𝑉 − 𝐼 ∙ 𝑅 − 𝑉𝑒𝑚𝑓 = 0
In lab, you use the ammeter to
measure the current.
You also know the value of the
source voltage, V.
Solve for Vemf: 𝑉𝑒𝑚𝑓 = 𝑉 − 𝐼 ∙ 𝑅
Motor in a circuit
Ammeter
+
0.000
A
3
+ VR
1
DC_Source
V
Why is this important?
MotorResistor
R
I
0
-
2
Back_EMF
V_emf
We cannot directly measure
the back-emf Vemf.
How can I use the Vemf?
I know from
electromagnetism that:
𝑉𝑒𝑚𝑓 = 𝑁 ∙
𝑉𝑒𝑚𝑓 = 𝑉 − 𝐼 ∙ 𝑅
𝑑𝜓
,
𝑑𝑡
where:
𝜓: magnetic flux
N: Number of coils
Varying magnetic flux
F
I
B
Fperp
F
F B
I F
perp
𝜑
F
Side view
Side view
No field lines go through
surface area of coil.
As the coils rotate, more lines
go through the surface area of
𝑑𝜓
𝑑𝜑
the coil. So, 𝑑𝑡 ∝ 𝑑𝑡
Induced voltage
𝑉𝑒𝑚𝑓 = 𝑁 ∙
𝑑𝜓
𝑑𝑡
𝑚2 ∙𝑘𝑔
𝑉𝑒𝑚𝑓 : Voltage induced in motor in 𝑉 = 𝑠3 𝐴
𝑚2 ∙𝑘𝑔
𝜓: Time-varying magnetic flux in 𝑊𝑏 = 2
𝑠 ∙𝐴
𝑑𝜓
𝑊𝑏
𝑚2 ∙𝑘𝑔
: Rate of change of magnetic flux in 𝑠 = 𝑠3 ∙𝐴
𝑑𝑡
So:
𝑉𝑒𝑚𝑓 = 𝑁 ∙ 𝜓0 ∙ 𝑓
You know:
Vemf, N, and 𝜓0 .
So:
you can calculate 𝑓
𝑑𝜓
𝑑𝜙
= 𝜓0
= 𝜓0 ∙ 𝑓
𝑑𝑡
𝑑𝑡
𝜓0 : Magnetic flux from magnets in 𝑊𝑏 =
≡𝑉
𝑑𝜙
𝑑𝑡
= 𝑓: Frequency of rotation in s-1
𝑚2 ∙𝑘𝑔
𝑠 2 ∙𝐴
Power and efficiency
Ammeter
+
0.000
A
3
MotorResistor
R
Power dissipator
2
Back_EMF
1
DC_Source
V
V_emf
0
“Useful” power
Power generator
How do we calculate power in electrical systems?
Power
I
I
+
V
-
This is the picture of a power dissipator.
+
V
-
This is the picture of a power generator.
𝑃𝑑𝑖𝑠 = 𝑉 ∙ 𝐼
𝑃𝑔𝑒𝑛 = 𝑉 ∙ 𝐼
𝑃𝑔𝑒𝑛 = −𝑉 ∙ 𝐼
𝑃𝑑𝑖𝑠 = −𝑉 ∙ 𝐼
Power and efficiency
Ammeter
+
0.000
A
3
MotorResistor
R
Power dissipator
2
Back_EMF
1
DC_Source
V
𝑃𝑅 = 𝑉𝑅 ∙ 𝐼 – dissipated
V_emf
0
“Useful” power – power dissipated
𝑃𝑒𝑚𝑓 = 𝑉𝑒𝑚𝑓 ∙ 𝐼 – dissipated
Power generator
𝑃𝑠𝑜𝑢𝑟𝑐𝑒 = 𝑉 ∙ 𝐼 – generated
Add all powers generated:
𝑃𝑖 = 0
Power and efficiency
Ammeter
+
0.000
A
3
MotorResistor
R
Power dissipator
2
Back_EMF
1
DC_Source
V
𝑃𝑅 = 𝑉𝑅 ∙ 𝐼 – dissipated
V_emf
0
“Useful” power – power dissipated
𝑃𝑒𝑚𝑓 = 𝑉𝑒𝑚𝑓 ∙ 𝐼 – dissipated
Power generator
𝑃𝑠𝑜𝑢𝑟𝑐𝑒 = 𝑉 ∙ 𝐼 – generated
Efficiency:
𝜂=
𝑃𝑒𝑚𝑓
𝑃𝑠𝑜𝑢𝑟𝑐𝑒