Hello guys………… now having learnt about the basics of dc motors , we will now go on to derive some of the very standard equations in the field of dc motors

**1. ***THE EMF EQUATION OF A MOTOR………….*

Let Ø=flux/pole in webers…….

Z=total number of conductors= no. of slots x no. of conductors/slot

P= no. of generator poles….

A= no. of parallel armature paths

N= armature rotations in revolutions per minute (r.p.m)

E= Back Emf of motor

So, Eb=emf generated in any 1 of the parallel paths i.e. E

Average emf per conductor= dØ/dt volts (N=1)

Now, flux cut/conductor in 1 revolution= dØ=Øp Wb

No. of revolutions /second= N/60

Hence, time for 1 revolution, dt=60/N second

So, according to Faraday’s Laws of EMI,

EMF for 1 conductor= dØ/dt= Øpn/60

Therefore, emf for Z conductors for A parallel paths= ØNZP/60A

For a simplex wave winding, A= 2

For a simplex lap winding, A=P

*2.*THE TORQUE EQUATION OF A DC MOTOR………

Consider the pulley of radius r metre as shown in figure acted upon by a circumferential force of F newton which causes it to rotate at N r.p.m

Then, torque, T= Fxr Newton- metre

Work done by this force in one revolution= Force x distance=F x 2Ԉr Joules

Power developed= F x 2Ԉr xN Watts

Now, 2ԈN=Angular velocity ώ in rad/s and Fxr= Torque, T

Hence,Power developed = Txώ watts or P=Tώ watts

Moreover if N is in r.p.m, then, ώ=2ԈN/60 rad/s

and P= 2ԈNT/60 OR P=(2Ԉ/60)xNT= NT/9.55

3.TORQUES IN A MOTOR………..

The torque on a current-carrying coil, as in a DC motor, can be related to the characteristics of the coil by the “magnetic moment” or “magnetic dipole moment”. The torque exerted by the magnetic force (including both sides of the coil) is given by

The coil characteristics can be grouped as
called the magnetic moment of the loop, and the torque written as The direction of the magnetic moment is perpendicular to the current loop in the right-hand-rule direction, the direction of the normal to the loop in the illustration. Considering torque as a vector quantity, this can be written as the vector product |

Since this torque acts perpendicular to the magnetic moment, then it can cause the magnetic moment to precess around the magnetic field at a characteristic frequency called the Larmor frequency.

If you exerted the necessary torque to overcome the magnetic torque and rotate the loop from angle zero to 180 degrees, you would do an amount of rotational work given by the integral

The position where the magnetic moment is opposite to the magnetic field is said to have a higher magnetic potential energy.

### Torque vs RPM

For permanet magnet DC motors, there is a linear relationship between torque and rpm for a given voltage.

The maximum torque occurs at 0 rpm, and is called *stall torque*. The minimum torque (zero) occurs at maximum rpm, reached when the motor is not under a load, and is thus called *free rpm*. The formula for torque at any given rpm is:

**T = T _{s} – (N T_{s} ÷ N_{f})**

where **T** is the torque at the given rpm **N**, **T _{s}** is the stall torque, and

**N**is the free rpm.

_{f}Power, being the product of torque and speed, peaks exactly half way between zero and peak speed, and zero and peak torque. For the above graph, peak power occurs at 1500 rpm and 5 ft-lbs of torque; 1.4 hp. However, you do not generally want to run a motor at this speed, as it will draw much too much current and overheat. The above motor might be rated for only 0.5 hp (1 ft-lbs of torque at 2700 rpm).

Knowing the stall torque and the free rpm, we can derive two important constants for the motor in question. First is the *induced voltage constant*, which relates the back-voltage induced in the armature to the speed of the armature.

**K _{e} = V ÷ N_{f}**

where **K _{e}** is the induced voltage constant,

**N**is the free rpm, and

_{f}**V**is the voltage.

The second important constant is the *torque constant* which relates the torque to the armature current.

**K _{t} = T_{s} ÷ V**

where **K _{t}** is the torque constant,

**T**is the stall torque, and

_{s}**V**is the voltage.

Using these two constants, we can write the motor equation (these are all the same equation, solved for different variables):

**T = K _{t} × (V – (K_{e} × N)**

**V = (T ÷ K _{t}) + (K_{e} × N)**

**N = (V – (T ÷ K _{t})) ÷ K_{e}**

where **T** is torque, **V** is voltage, **N** is rpm, **K _{t}** is the torque constant, and

**K**is the induced voltage constant. The units don’t matter, as long as they’re the same units you used to calculate the constants.

_{e}

Stall Torque | ft-lbs | |

Free RPM | rpm | |

Reference Voltage | volts | |

Torque constant | ||

Induced Voltage constant | ||

RPM | rpm | |

Torque | ft-lbs | |

Voltage | volts |

Note that these formulas *only* apply to shunt motors and permanent magnet motors. Series motors behave differently.

### Torque and Current

Torque is proportional to the product of armature current and the resultant flux density per pole.

**T = K × f × I _{a}**

where **T** is torque, **K** is some constant, **f** is the flux density, and **I _{a}** is the armature current.

In series wound motors, flux density approximates the square root of current, so torque becomes approximately proportional to the 1.5 power of torque.

**T = K × I _{a}^{1.5±}**

where **T** is torque, **K** is some constant, and **I _{a}** is the armature current.

### Speed, Voltage, and Induced Voltage

Resistance of the armature widings has only a minor effect on armature current. Current is mostly determined by the voltage induced in the windings by their movement through the field. This induced voltage, also called “back-emf” is opposite in polarity to the applied voltage, and serves to decrease the effective value of that voltage, and thereby decreases the current in the armature.

An increase in voltage will result in an increase in armature current, producing an increase in torque, and acceleration. As speed increases, induced voltage will increase, causing current and torque to decrease, until torque again equals the load or induced voltage equals the applied voltage.

A decrease in voltage will result in a decrease of armature current, and a decrease in torque, causing the motor to slow down. Induced voltage may momentarily be higher than the applied voltage, causing the motor to act as a generator. This is the essense of regenerative breaking.

Induced voltage is proportional to speed and field strength.

**E _{b} = K × N × f**

where **E _{b}** is induced voltage,

**K**is some constant particular to that motor,

**N**is the speed of the motor, and

**f**is the field strength.

This can be solved for speed to get the “Speed Equation” for a motor:

**N = K × E _{b} ÷ f**

where **N** is rpm, **K** is some constant (the inverse of the K above), **E _{b}** is the induced voltage of the motor, and

**f**is the flux density.

Note that speed is inversely proportional to field strength. That is to say, as field strength *decreases*, speed *increases*.

### Runaway

In a shunt-wound motor, decreasing the strength of the field decreases the induced voltage, increasing the effective voltage applied to the armature windings. This increases armature current, resulting in greater torque and acceleration. Shunt-wound motors run away when the field fails because the spinning armature field induces enough current in the field coils to keep the field “live”.

In a series-wound motor, the field current is always equal to the armature current. Under no load, the torque produced by the motor results in acceleration. As speed increases, induced voltage would normally increase until at some speed it equalled the applied voltage, resulting in no effective voltage, no armature current, and no further acceleration; in this case, however, increasing speed decreases field current and strength, stabilizing induced voltage. Torque never drops to zero, so the motor continues to accelerate until it self-destructs.

Runaway does not occur in permanent magnet motors. Starter motors, electric car motors, and some golf cart motors are series wound, however, and can run away

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