EMF equation of DC machines is a mathematical expression to calculate the EMF induced in a DC machine. As you know, the amount of EMF induced in a conductor depends on the number of turns of the coil and the rate of change of flux.
Derivation of EMF equation of DC machines
Consider a motor that has ‘P’ number of poles and a ‘Z’ number of conductors per pole spinning at a speed of ‘N’ radials per second.
If ‘φ’ is the flux produced in ‘webers’, according to Faraday’s law, the EMF induced per conductor is given by:
![\[ E_{av} \propto \frac{d \phi}{dt} \]](https://www.electricalclassroom.com/wp-content/ql-cache/quicklatex.com-83efabd6f24f930138ac4479a550c3c1_l3.svg "Rendered by QuickLaTeX.com")
The above equation states that the induced EMF is directly proportional to the change in flux. The flux generated by a DC is constant. The total flux generated by all the poles is given by:
![\[ d \phi = \phi \times P \]](https://www.electricalclassroom.com/wp-content/ql-cache/quicklatex.com-5f4de2515ac5c09d67e443d23d9fbebc_l3.svg "Rendered by QuickLaTeX.com")
As the machine is spinning at ‘N’ rpm (radials per minute), the time taken for one revolution is given by:
![\[ dt =\frac{60}{N} \]](https://www.electricalclassroom.com/wp-content/ql-cache/quicklatex.com-c5adadc46ae80ba839250df8c6bd1f6e_l3.svg "Rendered by QuickLaTeX.com")
Therefore the EMF induced per conductor is given by:
![\[ E_{av} = \frac{\phi \times P }{\frac{60}{N}} = \frac{PN \phi}{60} \]](https://www.electricalclassroom.com/wp-content/ql-cache/quicklatex.com-a27c3b45c0a1ea5903600d8993486fc6_l3.svg "Rendered by QuickLaTeX.com")
The total EMF induced in the machine, if there are ‘Z’ number of conductors and ‘A’ number of parallel paths is:
![\[ E = \frac{PN \phi}{60} \times \frac{Z}{A} = \frac{PN \phi Z}{60.A} \]](https://www.electricalclassroom.com/wp-content/ql-cache/quicklatex.com-524c356e9f29f8c84f47c22dbacf8ceb_l3.svg "Rendered by QuickLaTeX.com")
The above is the EMF equation of DC machines.