DC Motor Characteristics PDF

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motor calculation, torque, rpm, etc...

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DC Motor Characteristics Anurag Purwar Mechanical Engineering, Stony Brook University [email protected] This document details the relationships between current, voltage, rotational speed, torque, power and efficiency for a given DC motor. A sample DC motor datasheet is displayed below. We will be using the 101-G DC toy motor as an example in the following discussions.

Figure 1: A Toy Motor sample specifications

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Relationship between Torque and Current

A typical brushed DC Motor construction includes many coils which help convert flow of current within a magnetic field into mechanical motion. Fig 2 displays a single winding within a motor. The force couple exerted on each coil within the motor is given by F = iLB

(1)

where i is the current flowing within each coil, L is the length of coil perpendicular to magnetic field and B is the magnetic field.

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z y B

F

x

i

S

i

L

N

F Figure 2: Motor winding diagram This force couple can be converted into Torque by the following relation τ = 2rF

(2)

where r is the radius of coils and F is the force couple exerted on each coil. Substituting the expression for Force gives τ = 2riLB

(3)

for each coil. If our motor has n coils, the formula for torque changes to τ = 2nriLB.

(4)

In this formula n,r,L,B are fixed for a given motor. We assume the product 2nrLB = K which is constant for a given motor. Thus τ = Ki

(5)

This relation tells us that the torque produced by a given motor is directly proportional to the current drawn. Thus, for a motor rated for a specific voltage, the current drawn by it is dependent on the load applied. In reality, even in zero-load condition, the motor will draw some current due to the resistance of the winding. Thus the Torque-Current relationship for motors is more aptly given by the formula is − i0 i= τ + i0 (6) τs where is =stall current, i0 =no-load current and τs =stall torque. The torque vs current graph of Toy DC motor 101-G has been plotted in Fig 3. The values of (torque, current) used to plot the graph are (0, 0.26) and (8, 0.62). These values are given in the Fig. 1. From this graph, we can clearly see that the current drawn at the stall torque is almost 1.5 A.

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Figure 3: Torque vs Current

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Relationship between Voltage and the Rotational Speed of the shaft

Resistance of all the coils within the wire is given by R = ρL/A. Voltage drop due to load is called Back EMF (Eb ) which is dependent on mechanical loading. Thus for the total circuit powered by a V volt battery, we have the relationship V = iR + Eb

(7)

Eb can be estimated theoretically as power used up due to voltage drop Eb is equal to the mechanical power outputted. By equating electrical and mechanical power, we get the relationship Eb .i = τ.ω

(8)

and our circuit expression becomes V = iR +

τω i

(9)

Using the relationship τ = K i, and substituting i, we get V =

Rτ + Kω K

(10)

rearranging them gives us R V − 2τ (11) K K which gives us the linear relation between τ and ω for a given Voltage. We note that as torque increases, ω decreases. Thus, if you increase the load on the ω=

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robot, such as making a robot go up an incline, the speed will decrease until at a point when it would simply not move or worse starts to slip. Using the table in Fig. 1, this linear relationship can be redefined as ωm τ + ωm (12) ω=− τs where ωm is the maximum rpm and Ts is the stall torque The torque vs RPM graph of Toy DC motor 101-G has been plotted in Fig 4. The values of (torque, rpm) used to plot the graph are (0,12000) and (28, 0). These values are for nominal voltage 3V.

Figure 4: Torque vs RPM Practically, if your robot does not move or moves very slowly due to excessive load, increasing the voltage supplied to motor will shift the curve up giving you a speed increase for the same load. However, this voltage should be below rated voltage or else you risk burning out your motor.

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Power Consumprion

Power consumption can be calculated using Torque and RPM as follows P = τ.ω = τ (−

ωm τ + ωm ) τs

(13)

ωm 2 τ + ωm τ. (14) τs To find the max power output, we differentiate P (τ ) w.r.t. τ and equate it to zero. dP/dτ = −2τ /τs = 0 (15) =−

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τ = τs /2

(16)

for this τ , ω is computed to be ω=−

ωm ωm τs + ωm = 2 τs 2

(17)

The torque vs power graph of Toy DC motor 101-G has been plotted in Fig 5. It can be easily evaluated as we know the equations for RPM in terms of torque which gives us power as a function of torque.

Figure 5: Torque vs Power Output

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Efficiency

Efficiency can be calculated by ratio of Power Output to Power Input. Thus, η=

τω . Vi

(18)

We know ω in terms of τ . We know i in terms of τ . Substituting these expressions gives τ (− ωτm τ + ωm ) s η= = f (τ ) (19) is −i0 V ( τ τ + i0 ) s which is the expression of efficiency in terms of torque. dη/dτ = 0 gives us the τ at which η is max. After doing the math, we get ηmax at aprox τs /4 The torque vs efficiency graph of Toy DC motor 101-G has been plotted in Fig 6. 5

Figure 6: Torque vs Power Output

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