Shunt DC Motor

Below we the schematic diagram of a shunt DC motor and its equivalent circuit diagram.

In shunt DC motors, the field circuit is connected in parallel with the armature circuit, and both circuits are fed from the same DC voltage source. The resistance $R_f$ in (b) is the combination of the field resistance and an adjustable resistance, which is used for speed control purposes.

The main relations for this type of motor are:

$$\begin{array}{rl}{I}_f& =\frac{V_L}{R_f}\\ \varphi & =\frac{N_f{I}_f}{\mathcal{R}}\end{array}$$

where $N_f$ is the number of turns of field winding and $\mathcal{R}$ is the magnetic reluctance.

Then:

$$\begin{array}{rl}{V}_L& =I_a{R}_a+E_c\\ I_L& =I_a+I_f\\ E_c& =\frac{ZP}{2\pi a}\varphi {\omega }_m=\frac{ZP}{60a}\varphi n\\ {\tau }_m& =\frac{ZP}{2\pi a}\varphi I_a\end{array}$$

Thus, if $\varphi$ is constant, ${\tau }_m\propto I_a$.

Speed-Torque Characteristics

Consider the equivalent circuit diagram of the shunt DC motor shown below.

We have:

$$\begin{array}{rl}{V}_L& =I_a{R}_a+E_c\\ & =I_a{R}_a+\frac{ZP}{60a}\varphi n\end{array}$$

Using

$${\tau }_m=\frac{ZP}{2\pi a}\varphi I_a$$

we have:

$$I_a=\frac{2\pi a}{ZP}\frac{{\tau }_m}{\varphi }$$

Then, we have

$$V_L=\frac{2\pi a}{ZP}\frac{{\tau }_m}{\varphi }{R}_a+\frac{ZP}{60a}\varphi n$$

The rotor speed can be found from above as:

$$\begin{array}{rl}n& =\frac{V_L}{\frac{ZP}{60a}\varphi }-\frac{\frac{2\pi a}{ZP}\frac{{\tau }_m}{\varphi }{R}_a}{\frac{ZP}{60a}\varphi }\\ & =\frac{V_L}{K^{\prime }\varphi }-\frac{R_a}{K{K}^{\prime }{\varphi }^2}{\tau }_m\end{array}$$

This means that $n$ plotted as a function of $\tau$ is a straight line, provided that $\varphi$ is constant:

Thus, the speed of a shunt DC motor drops under load. This drop, however, is limited to below 8% of the rated speed, from no load to full load. Shunt DC motors are well-known for their fairly constant speed under load.