Characteristics of dc motors

Torque and Speed Equations

For a d.c. motor >> Torque is directly proportional to the product of Armature current and main flux.

\displaystyle T_a = \phi I_a \biggl ( \frac {PZ}{2\pi A} \biggr )

\displaystyle T \propto \phi I_a 

Now \displaystyle \phi  is the flux produced by the field winding and is proportional to the current passing through the field winding.

\displaystyle \phi \propto I_f 

Similarly the back emf produced in the armature is given by

\displaystyle E_b = \phi N \biggl ( \frac {ZP}{60A} \biggr ) 

\displaystyle E_b \propto \phi N 

\displaystyle N \propto \frac {E_b}{\phi} 

\displaystyle V = E_b + I_a R_a 

\displaystyle \Rightarrow E_b = V - I_a R_a 

Speed equation becomes :

\displaystyle N \propto \biggl ( \frac{V - I_a R_a}{\phi} \biggr )

These relations play an important role in understanding the various characteristics of different types of motors.

DC Motor Characteristics

The performance of a d.c. motor under various condition can be judged by the following characteristics :

(i). Torque – Armature current characteristics

(ii). Speed – Armature current characteristics

(iii). Speed-Torque characteristics

Characteristics of DC Shunt Motor

Torque – Armature current characteristics

\displaystyle T \propto \phi I_a 

For a constant values of \displaystyle R_{sh}  and supply voltage \displaystyle V >> I_{sh}   is also constant and hence flux is also constant.

\displaystyle \phi \propto I_f 

\displaystyle T_a \propto I_a 

The equation represents a straight line, passing through the origin as shown in the figure. Torque increases linearily with armature current. It is seen that armature current is decided by the load, So as load increases >> Armature current increases >> Increasing the torque developed linearly.


To generate high starting torque, this type of motor requires a large value of armature current at the start. This may damage the motor hence d.c. shunt motors can develop moderate starting torque and hence suitable for such applications where starting torque requirement is moderate.

Speed – Armature current Characteristics

\displaystyle N \propto \biggl ( \frac{V- I_a R_a}{\phi} \biggr ) 

\displaystyle \phi \rightarrow    Constant

\displaystyle N \propto V - I_a R_a 

So as load increases >> The armature current increases and hence drop \displaystyle I_a R_a  also increases.

Hence for constant supply voltage, \displaystyle V- I_a R_a  decreases and hence speed reduces. But as \displaystyle R_a  is very small, for change I_a  in from no load to full load, drop I_a R_a  is very small and hence drop in speed is also not significant from no load to full load.


In d.c. shunt motor >> Speed regulation is order of 5 to 10 % from no load to full load >> That’s why d.c. shunt motor is also called as a constant drive motor.

Speed – Torque Characteristics

\displaystyle T \propto \phi I_a 

\displaystyle T_a \propto I_a 

\displaystyle N \propto \biggl ( \frac{V - I_a R_a}{\phi} \biggr ) 

\displaystyle \phi \rightarrow   Constant

\displaystyle N \propto V - I_a R_a 

So from these 2 equations, we can conclude that Speed and Torque both has a linear relationship. This characteristics is similar to speed – Armature current characteristics.


This curve shows that the speed almost remains constant through torque changes from no load to full load conditions.

Characteristics of DC Series Motor

Torque – Armature current Characteristics

In case of a series motor, the series field winding is carrying the entire armature current. So flux produced is proportional to the armature current.

\displaystyle \phi \propto I_a 

\displaystyle T_a \propto \phi I_a \propto I^2_a 

The torque in case of a series motor is proportional to the square of the armature current. This relation is parabolic in nature as shown in the figure.


As load increases, armature current increases and torque produced increases proportional to the square of the armature current up to a certain limit.

As the entire Armature current passes through the series field, there is a property of an electromagnet called saturation, may occur. Saturation means though the current through the winding increases, the flux produced remains constant. Hence after the saturation, the characteristics take the shape of the straight line as flux becomes constant.

These types of motors can produce high torque for a small amount of armature current hence series motor is suitable for the applications which demand high starting torque.

Speed – Armature current Characteristics

From the speed equation, we get

\displaystyle N \propto \frac{E_b}{\phi} 

\displaystyle N \propto \biggl ( \frac {V - I_a R_a - I_a R_{se}}{I_a} \biggr )

Now the values of R_a  and R_{se}  are so small that the effect of change in I_a   on speed overrides the effect of change in V - I_a R_a - I_a R_{se}  on the speed.

Hence in the speed equation, Eb ≈ V and can be assumed constant. So speed equation reduces :

\displaystyle N \propto \frac {1}{I_a} 

So speed – armature current characteristics is rectangular hyperbola as shown in the figure.


Speed – Torque Characteristics

In case of series motors,

\displaystyle T \propto I^2_a 

\displaystyle N \propto \frac{1}{I_a} 

Hence we can write,

\displaystyle N \propto \frac{1}{\sqrt{T}} 

Thus as torque increases when load increases, the speed decreases. On no load, torque is very less and hence speed increases to dangerously high value.


Thus the nature of the speed-torque characteristics is similar to the nature of the speed – armature characteristics.

Why Series motor is never started on No load?

It is seen earlier that motor armature current is decided by the load. On light load or no load, the armature drawn by the motor is very small.

In case of a dc series motor,

\displaystyle \phi \propto I_a 

and on no load armature current is small hence flux produced is also very small.

According to the speed equation,

\displaystyle N \propto \frac{1}{\phi} 

\displaystyle E_b \rightarrow   Almost Constant

So on very light load or no load as flux is very small, the motor tries to run at a dangerously high speed which may damage the motor mechanically. This can be seen from the speed – armature current and the speed-torque characteristics that on low armature current and low torque condition motor shows a tendency to rotate with dangerously high speed.

For this reason, it is not selected for belt drives as breaking or slipping of belt causes to throw the entire load off on the motor and made to run a motor with no load which is dangerous.

Practically pure d.c. series motors are not used, instead of them, Compound motor is used.

Characteristics of DC Compound Motor

Compound motor characteristics basically depend on the fact whether the motor is cummulatively compound or differential compound. All the characteristics of the compound motor are the combination of the shunt and series characteristics.

A cummulative compound motor is capable of developing a large amount of torque at low speed just like a series motor. However, it is not having a disadvantage of the series motor even at the light or no load.

So cummulative compound motor can run at a reasonable speed and will not run with dangerously high speed like series motor, on light or no load condition.


In a differential compound motor, as two fluxes oppose each other, the resultant flux decreases as load increases, thus the machine runs at a higher speed with an increase in load. This property is dangerous as on full load, the motor may try to run with dangerously high speed. So the differential compound motor is generally not used in practice.

The exact shape of these characteristics depends on the relative contribution of series and shunt field windings. If the shunt field winding is more dominant then the characteristics take the shape of the shunt motor characteristics. While if the series field winding is more dominant then the characteristics take the shape of the series characteristics.


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