Generated Voltage in DC Generator

We saw how the mechanism of how DC generators generate voltage, and derived a formula for the voltage induced in a loop of wire was derived. Here, an expression for the terminal voltage of a general DC generator in terms of the structural parameters of the machine, field flux and rotor speed will be derived.

The voltage voltage induced in a conductor of length $l$ moving at the linear speed of $v$ under a pole face, where a magnetic flux density of $B$ exits, under the conditions that $v\perp B$ and $v\times B$, is given by:

$$e_{1\text{ conductor}}=vBl$$

The output voltage of a DC generator is determined by the number of conductors connected in series and the voltage induced in each conductor:

$$E_{\text{generated}}=\text{Number of Series Conductors}\times vBl$$

The number of series conductors can be found with:

$$\text{Number of series conductors}=\frac{\text{Total number of conductos}}{\text{Number of parallel paths}}=\frac{Z}{a}$$

where Z is the total number of conductors and a is the number of parallel paths in the armature between the machine terminals.

Thus, we have:

$$E_{\text{generated}}=\frac{Z}{a}vBl$$

If $r$ is the radius of the rotor, and ${\omega }_m$ is the angular speed of the rotor, we have:

$$v=r{\omega }_m$$

If $\varphi$ is the flux under a pole face and $A_p$ is the area per pole face,

$$B=\frac{\varphi }{A_p}$$

where

$$A_p=\frac{\text{Rotor Surface Area}}{\text{Number of Poles}}=\frac{2\pi rl}{P}$$

In the equation above, the air gap between stator and rotor and the gaps between successive pole faces have been neglected, making the total surface area of $P$ poles equal to the surface area of rotor.

Combining everything, our expression for generated voltage becomes

$$\begin{array}{rl}{E}_{\text{generated}}& =\frac{Z}{a}vBl\\ & =\frac{Z}{a}r{\omega }_m\frac{\varphi }{\frac{2\pi rl}{P}}l\\ & =\left(\frac{ZP}{2\pi a}\right)\varphi {\omega }_m\\ & =\boxed{K\varphi {\omega }_m}\end{array}$$

where $K=\frac{ZP}{2\pi a}$.

We can further write ${\omega }_m=\frac{2\pi n}{60}$, where $n$ is the rotor speed in RPM. Thus, we can write the above as:

$$\begin{array}{rl}{E}_{\text{generated}}& =\left(\frac{ZP}{2\pi a}\right)\varphi {\omega }_m\\ & =\left(\frac{ZP}{2\pi a}\right)\varphi \frac{2\pi n}{60}\\ & =\left(\frac{ZP}{60a}\right)\varphi n\\ & =\boxed{K^{\prime }\varphi n}\end{array}$$

where $K^{\prime }=\frac{ZP}{60a}$.

$K$ and $K^{\prime }$ are constants determined by the machine structure and cannot be changed during machine operation, whereas $\varphi$, ${\omega }_m$, and $n$ can be controlled during the operation of the machine, to vary/adjust the output voltage magnitude.

An interesting and important observation that can be made is that the magnitude of generated voltage in a DC generator is proportional to the product of speed and number of poles:

$$E_{\text{generated}}\propto {\omega }_{m}P,E_{\text{generated}}\propto nP$$

This means that besides the number of conductors in series, the products ${\omega }_{m}P$ or $nP$ can also affect the magnitude of the generated voltage.

• To produce a DC voltage of certain magnitude using a low-speed prime mover, a large number of poles are required.
• In the case of a high-speed prime mover, a small number of poles are needed. A similar observation can be made in other types of electric machines, as well.

A prime mover is a source of mechanical power, such as a diesel engine, gas turbine, steam turbine, or wind turbine that drives the shaft of the rotor.

Example 6-1

• Note that this example makes reference to lap and wave windings.