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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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