SME
Power Systems · Problem type
DC motor speed and torque
Use this for “A 240 V DC motor draws 20 A, with Rₐ = 0.5 Ω and kΦ = 2.3 V·s/rad. Find its speed and torque.”
Diagram
KVL: 240 = 10 + 230 V (rises = drops)
This is an example. Type your own values to start a new problem.
Eₐ (V)
=
Vₜ (V)
−
Iₐ (A)
×
Rₐ (Ω)
ω (rad/s)
=
Eₐ (V)
÷
kΦ (V·s/rad)
T (N·m)
=
kΦ (V·s/rad)
×
Iₐ (A)
IₐRₐ
Armature drop
worked out (not typed)
P
Converted power
calculated
Formulas
IₐRₐ = Iₐ × Rₐ
10 = 20 × 0.5
Eₐ = Vₜ − IₐRₐ
230 = 240 − 10
ω
=
Eₐ
kΦ
100
=
230
2.3
T
=
kΦ
×
Iₐ
46 = 2.3 × 20
P
=
Eₐ
×
Iₐ
4,600 = 230 × 20
Formulas are worked in W. The step-by-step shows the conversions.
Assumptions
•
Steady state with a constant field; brush drop and armature inductance are left out.
•
kΦ is the machine constant in V·s/rad, the same as N·m/A.
•
Eₐ = kΦω is the back-EMF; EₐIₐ = Tω is the power converted, before friction and windage.
Step-by-step
WE KNOW
Vₜ = 240 V, Iₐ = 20 A, Rₐ = 0.5 Ω, kΦ = 2.3 V·s/rad
FIND
armature drop (IₐRₐ), back-EMF (Eₐ), speed (ω), torque (T), converted power (P)
Step 1 · Find armature drop (IₐRₐ)
Use IₐRₐ = Iₐ × Rₐ
Ohm’s law for the armature winding.
IₐRₐ = 20 × 0.5
IₐRₐ = 10 V
Step 2 · Find back-EMF (Eₐ)
Use Eₐ = Vₜ − IₐRₐ
Round the armature loop, the supply meets the resistance drop and the back-EMF.
Eₐ = 240 − 10
Eₐ = 230 V
Step 3 · Find speed (ω)
Use ω = Eₐ ÷ kΦ
The back-EMF grows with speed, Eₐ = kΦω, so divide it by kΦ.
ω
=
230
2.3
ω
=
100
rad/s
Step 4 · Find torque (T)
Use T = kΦ × Iₐ
The torque is the same constant times the armature current.
T
=
2.3
×
20
T
=
46
N·m
Step 5 · Find converted power (P)
Use P = Eₐ × Iₐ
The current flowing against the back-EMF is the power turned into motion.
P
=
230
×
20
P
=
4,600
W
Step 6 · Convert the answers
Change each answer back to the unit you chose.
P = 4,600 W = 4.6 kW (1 kW = 1000 W)
Check
Put the numbers back in. Both sides should match.
10 = 20 × 0.5 ✓
230 = 240 − 10 ✓
100
=
230
2.3
✓
46 = 2.3 × 20 ✓
4,600 = 230 × 20 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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