1. ELEC 3105 Basic EM and Power Engineering
Rotating DC Motor
PART 2 Electrical

2. Motor / Generator Action
Loop
Equivalent circuit
Expression of Vemf
Slide extracted from linear motor and modified for loop motor.

3. Motor / Generator Action
Slide extracted from linear motor and modified for loop motor.
Linear relation between
speed and torque
Current flows in a direction to charge the battery.
Stall torque
Generator
Motor
Link

4. Electrical Equivalent
𝐿 𝑎
𝑅 𝑎
𝑅 𝑓 
𝐿 𝑓
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
E = motor voltage
Ra = armature resistance
La = armature inductance
V = Applied motor voltage
Ia = armature current
 = magnetic flux
Rf = field resistance
Lf = field inductance
Vf = Field voltage
If = field current

5. Electrical Equivalent
𝐿 𝑎
𝑅 𝑎
𝑅 𝑓 
𝐿 𝑓
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
In steady state operation: FIELD SIDE
𝐼 𝑓 = 𝑉 𝑓 𝑅 𝑓
Φ= 2 𝑛𝐼 𝑓 𝔑
Magnetic circuit
reluctance

6. Electrical Equivalent
𝐿 𝑎
𝑅 𝑎
𝑅 𝑓 
𝐿 𝑓
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
In steady state operation: ARMATURE SIDE
𝐸= 𝑘 𝐸 Φ 𝜔 𝑚
Back emf
Motor constant

7. Electrical Equivalent
𝐿 𝑎
𝑅 𝑎
𝑅 𝑓 
𝐿 𝑓
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
In steady state operation: ARMATURE SIDE
𝑇 𝑑𝑒𝑣 = 𝑘 𝑇 Φ 𝐼 𝑎
Developed torque
Motor constant
𝑘 𝐸 = 𝑘 𝑇
Same motor constant in emf and developed torque

8. Electrical Equivalent
𝐿 𝑎
𝑅 𝑎
𝑅 𝑓 
𝐿 𝑓
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
Power flow: ARMATURE SIDE; Conservation of energy
KVL
𝑉= 𝐼 𝑎 𝑅 𝑎 +𝐸
Armature copper loss
POWER
𝑉 𝐼 𝑎 = 𝐼 𝑎 2 𝑅 𝑎 + 𝐼 𝑎 𝐸
Power developed
Power in

9. Electrical Equivalent
𝐿 𝑎
𝑅 𝑎
𝑅 𝑓 
𝐿 𝑓
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
Power flow: ARMATURE SIDE; Conservation of energy
𝑃 𝑑𝑒𝑣
Power developed
𝑃 𝑖𝑛
𝑃 𝑙𝑜𝑠𝑠
𝑃 𝑑𝑒𝑣 = 𝐼 𝑎 𝐸= 𝜔 𝑚 𝑇 𝑑𝑒𝑣
Electrical Mechanical
copper

10. Electrical Equivalent
𝐿 𝑎
𝑅 𝑎
𝑅 𝑓 
𝐿 𝑓
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
Power flow: ARMATURE SIDE
𝑃 𝑜𝑢𝑡 = 𝑃 𝑑𝑒𝑣 − 𝑃 𝑟𝑜𝑡
𝑃 𝑜𝑢𝑡
𝑃 𝑑𝑒𝑣
𝑃 𝑟𝑜𝑡
𝑃 𝑖𝑛
𝑃 𝑙𝑜𝑠𝑠
Rotational loss
Copper
Electrical Mechanical

11. Electrical Equivalent
𝐿 𝑎
𝑅 𝑎
𝑅 𝑓 
𝐿 𝑓
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
Motor sequence
𝑃 𝑜𝑢𝑡 = 𝑃 𝑑𝑒𝑣 − 𝑃 𝑟𝑜𝑡 𝑉
𝐼 𝑎
𝑇 𝑑𝑒𝑣
𝜔 𝑚 𝐸
𝑃 𝑑𝑒𝑣
Speed of rotation limiting loop

12. Shunt Connected Field 𝐿 𝑎 𝑅 𝑎 𝑉= 𝐼 𝑎 𝑅 𝑎 +𝐸 𝑇 𝑑𝑒𝑣 = 𝑘 𝑇 Φ 𝐼 𝑎 R 𝑅 𝑓
𝑉= 𝐼 𝑎 𝑅 𝑎 +𝐸
𝑇 𝑑𝑒𝑣 = 𝑘 𝑇 Φ 𝐼 𝑎 R
𝑅 𝑓
𝐸= 𝑘 𝐸 Φ 𝜔 𝑚
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
𝐿 𝑓 
𝑇 𝑑𝑒𝑣 = 𝑘Φ𝑉 𝑅 𝑎 − 𝑘 2 Φ 2 𝑅 𝑎 𝜔 𝑚
Rotation rate
Developed torque

13. Shunt Connected Field 𝐿 𝑎 𝑅 𝑎 R 𝑅 𝑓 𝜔 𝑚 , 𝑇 𝑑𝑒𝑣 𝐿 𝑓 
𝑇 𝑑𝑒𝑣 = 𝑘Φ𝑉 𝑅 𝑎 − 𝑘 2 Φ 2 𝑅 𝑎 𝜔 𝑚
Similar type of graph

14. Shunt Connected Field 𝐿 𝑎 𝑅 𝑎 𝑇 𝑑𝑒𝑣 = 𝑘Φ𝑉 𝑅 𝑎 − 𝑘 2 Φ 2 𝑅 𝑎 𝜔 𝑚 R 𝑅 𝑓
𝑇 𝑑𝑒𝑣 = 𝑘Φ𝑉 𝑅 𝑎 − 𝑘 2 Φ 2 𝑅 𝑎 𝜔 𝑚 R
𝑅 𝑓
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
𝐿 𝑓 
𝑇 𝑑𝑒𝑣
𝐾Φ𝑉 𝑅 𝑎
Force motor to spin backwards
Force motor to spin to fast
𝑉 𝑘Φ
Generator
Motor
Generator
𝜔 𝑚

15. Series Connected Field
𝑅 𝑓
𝐿 𝑎
𝐿 𝑓
𝑅 𝑎 
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
Universal motor design: works for D.C. and for A.C.
𝑉= 𝐼 𝑎 (𝑅 𝑎 + 𝑅 𝑓 )+𝐸
𝑇 𝑑𝑒𝑣 = 𝑘 𝑇 Φ 𝐼 𝑎
𝑇 𝑑𝑒𝑣 = 𝑘 ′ 𝐼 𝑎 2
Since
Φ∝ 𝐼 𝑎
𝐸= 𝑘 𝐸 Φ 𝜔 𝑚

16. Series Connected Field
𝑅 𝑓
𝐿 𝑎
𝐿 𝑓
𝑅 𝑎 
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
Universal motor design: works for D.C. and for A.C.
𝑇 𝑑𝑒𝑣
1 𝜔 𝑚 2
𝑇 𝑑𝑒𝑣 = 𝑘 ′ 𝑉 𝑅 𝑎 + 𝑅 𝑓 + 𝑘 ′ 𝜔 𝑚 2
𝜔 𝑚

17. Maximum Power Transfer
𝐿 𝑎
𝑉= 𝐼 𝑎 𝑅 𝑎 +𝐸
𝑅 𝑎
𝑅 𝑓 
𝐿 𝑓
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
Power developed in the motor
𝑃 𝑑𝑒𝑣 =𝐸 𝐼 𝑎
𝑃 𝑑𝑒𝑣 =𝐸 𝑉−𝐸 /𝑅 𝑎
Find maximum with respect to the motor voltage
𝑃 𝑑𝑒𝑣 = 𝐸𝑉− 𝐸 2 𝑅 𝑎

18. Maximum Power Transfer
𝐿 𝑎
𝑉= 𝐼 𝑎 𝑅 𝑎 +𝐸
𝑅 𝑎
𝑅 𝑓 
𝐿 𝑓
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
For extremes of a function, take derivatives and set to zero
𝑃 𝑑𝑒𝑣 =𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑤ℎ𝑒𝑛 𝐸= 𝑉 2
𝑃 𝑑𝑒𝑣 {𝑚𝑎𝑥 = 𝑣 2 4𝑅 𝑎

19. Calculation example Solution provided in class 𝑉= 𝐼 𝑎 𝑅 𝑎 +𝐸 𝐿 𝑎 𝑅 𝑎
𝑉= 𝐼 𝑎 𝑅 𝑎 +𝐸
𝐿 𝑎
𝑅 𝑎
𝑅 𝑓
𝑇 𝑑𝑒𝑣 = 𝑘 𝑇 Φ 𝐼 𝑎 
𝐿 𝑓
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
𝐸= 𝑘 𝐸 Φ 𝜔 𝑚
A 120 volt dc motor has an armature resistance of 0.70 Ω. At no-load, it requires 1.1 A armature current and runs at 1000 rpm. Find the output power and torque at 952 rpm output speed. Assume constant flux.
Solution provided in class

20. Calculation example Solution provided in class 𝑉= 𝐼 𝑎 𝑅 𝑎 +𝐸 𝐿 𝑎 𝑅 𝑎
𝑉= 𝐼 𝑎 𝑅 𝑎 +𝐸
𝐿 𝑎
𝑅 𝑎
𝑅 𝑓
𝑇 𝑑𝑒𝑣 = 𝑘 𝑇 Φ 𝐼 𝑎 
𝐿 𝑓
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
𝐸= 𝑘 𝐸 Φ 𝜔 𝑚
A permanent magnet dc motor has the following information: 50 hp, 200 V, 200 A, 1200 rpm and armature resistance of 0.05 Ω. Determine the output power if the voltage is lowered to 150 V and the current is 200 A. Assume rotational losses are proportional to speed. Determine the rotational loss, armature resistance, no-load rpm, machine constant, efficiency?
Solution provided in class

21. Calculation example Solution provided in class 𝑉= 𝐼 𝑎 𝑅 𝑎 +𝐸 𝐿 𝑎 𝑅 𝑎
𝑉= 𝐼 𝑎 𝑅 𝑎 +𝐸
𝐿 𝑎
𝑅 𝑎
𝑅 𝑓
𝑇 𝑑𝑒𝑣 = 𝑘 𝑇 Φ 𝐼 𝑎 
𝐿 𝑓
𝜔 𝑚 , 𝑇 𝑑𝑒𝑣
𝐸= 𝑘 𝐸 Φ 𝜔 𝑚
An 80 V dc motor has constant field flux, separately excited, and a nameplate speed of 1150 rpm with 710 W output power. The nameplate armature current is 10 A and the no-load current is 0.5 A. Assume constant rotational losses.
Solution provided in class