Before describing the formulae applicable to motors it is necessary to be aware of basic electric circuit formulae.
The first key electrical formula is that in a direct current (dc) circuit ohm’s law states that for a constant temperature V=IxR where V is the voltage, I is the current and R is the resistance; by transposing this formula any of the three variables can be deduced if the other two variables are known. (I=V/R : R=V/I)
The second key formula is W=VxI where W is the power in watts again by transposing this formula any of the three variables can be deduced if the other two variables are known. (I=W/V : V=W/I)
In an alternating current (ac) circuit there is an additional factor to be taken into account as a voltage applied to the circuit elements acts differently when compared to a direct current voltage being applied to the same circuit. That factor is the power factor which is derived from the ratio of the power divided by the apparent power.
Apparent power is defined as the result of multiplying the voltage and current as applied to the load and is measured in volt-amps (VxI=VA).
In an ac circuit the resistance in the Ohm’s law formula is replaced by the impedance whereby Ohm’s law for an ac circuit is V=IxZ where Z is the circuit impedance. The impedance is comprised of the resistance and reactance, which have the symbols R and X respectively.
Reactance is defined as the opposition of a circuit element to a change of current or voltage due to that element’s capacitance or inductance.
Impedance is calculated by the formula
Z=√(R2+X2)
| where | R=resistance |
| X=reactance |
Power factor is calculated by the formula
cos φ=W/VA
| where | cos φ=the power factor |
| VA=apparent power |
The power factor is the angle between the voltage and current waves at a given instant when those waves are represented by vectors. That angle is calculated by Cos-1 φ.
Utilising the above formulae the impedance, resistance and reactance can be calculated from the power, apparent power and voltage.
Z=V/I : R=ZxCos φ : X=√(Z2-R2) or X=ZxSin φ
For three phase systems it is necessary to take the effect of the three phases into account.
It is important to remember that impedance, resistance and reactance are single phase values.
Power in a three phase circuit is calculated by the formula
kW=(√3xVlxIlxCos φ)/1000 or kW=(3xVpxIlxCos φ)/1000
| where | kW=electrical power in kW |
| Vl=line voltage (phase to phase voltage) | |
| Vp=phase voltage (phase to neutral voltage) | |
| Il=line current (the supply current in one phase) | |
| Cos φ=power factor |
The apparent power in a three phase circuit is calculated by the formula
kVA=(√3xVlxIl)/1000 or kVA=(3xVpxIl)/1000
| where | kVA=apparent power in kVA |
The power factor in a three phase circuit is calculated by the formula
Cos φ=kW/kVA
The active current component of the current is calculated by the formula
Ia=(kWx1000)/(√3xVl) or Ia=(kWx1000)/(3xVp) or Ia=IxCos φ
| where | Ia=active current |
The re-active current component of the current is calculated by the formula
Ir=√[{(kVAx1000)/(√3xVl)}2-Ia2] or Ir=√[{(kVAx1000)/(3xVp)}2-Ia2] or Ir=√(I2-Ia2) or Ir=IxSin φ
| where | Ir=reactive current |
The impedance per phase is calculated by the formula
Z=Vl/√3xIl or Z=Vp/Il
| where | Z=impedance in ohms |
| Vp=phase voltage (phase to neutral) |
Again from the impedance and power factor the resistance and reactance can be determined by the following formula
Resistance R=ZxCos φ and Reactance X=ZxSin φ
As previously mentioned in this article motor output power is shown as kWo the (electrical) power input is shown as kW. The difference in input power to output power is the friction and windage losses of the motor plus the core losses and magnetising reactance this is the motor efficiency. Thus the standard three phase power formula can be used for the determination of motor output power by including a factor for efficiency.
Motor output can be calculated using the following formula
kWo=(√3xVlxIlxεxCos φ)/(100x1000)
| where | kWo=motor output power in kilowatts |
| ε=motor efficiency in percent |
By transposing the formula the motor current may be determined if it’s output is known
Il=(100x1000xkWo)/(√3xVlxεxCos φ)
A motor’s speed is determined by the number of pairs of poles and it is normal to quote a motors nominal speed as it’s synchronous speed which is calculated by
Nro=Fx60/P
| where | Nro=rotating field speed in revolutions per minute (rpm) |
| F=alternating current supply frequency in cycles per second (Hz) | |
| P=number of pairs of poles |
However a motor does not run at synchronous speed, the difference between the synchronous speed and rotor speed is known as the slip speed.
The slip speed can be calculated using the following formula
Nrs=Nro-Nrr
| where | Nrs=the slip speed in rpm |
| Nrr=the rotor speed in RPM |
The factional slip can be calculated using the following formula
s=Nrs/Nro or (Nro-Nrr)/Nro
| where | s= slip, for a motor at rest the slip s=1 |
The percentage slip can be calculated using the following formula
s%=(Nrs/Nro)x100 or [(Nro-Nrr)/Nro]x100
For a motor at rest the percentage slip s%=100
Torque is the rotational force created by a motor’s rotor when power is applied to it’s stator. Once the various losses are deducted from the input power to the stator then the remaining power is converted into the torque produced by the rotor. Therefore in summary the torque is produced by the rotor which is driven by the rotating magnetic field of the stator.
The torque produced by a motor can be calculated using the following formula
T=(kWox1000)/2xΠxNrr/60
| where | T=torque in Newton metres |
It must be emphasized that the input power to a motor is not constant at all speeds and it is power that is important, the power factor will also vary with speed.
The pdf version of this article contains an additional formulae which determines the maximum amount of current that may be drawn from the distribution system whilst limiting the volt drop to a set percentage at the point of common coupling to the system. It also contains the equivalent circuit of a motor and related information.