A conceptual circuit for manually setting the speed of a dc electric motor is shown in Fig. 3.31a. The resistors R1 and R2 are inside a component called a potentiometer, or pot, which is nothing more than an adjustable resistor, for example, a volume control. Turning the knob changes the ratio α =

Question

A conceptual circuit for manually setting the speed of a dc electric motor is shown in Fig. 3.31a. The resistors [latex]R_{1}[/latex] and [latex]R_{2}[/latex] are inside a component called a potentiometer, or pot, which is nothing more than an adjustable resistor, for example, a volume control. Turning the knob changes the ratio α = [latex]R_{2}/(R_{1} + R_{2}),[/latex] but the total resistance, [latex]R_{pot} = R_{1} + R_{2}[/latex], is unchanged. In this way the pot forms a voltage divider that sets the voltage [latex]V_{speed}.[/latex] The power amplifier output, [latex]V_{M}[/latex] is four times [latex]V_{speed}.[/latex] Power amplifiers can output the high currents needed to drive the motor. Finally, the dc motor speed is proportional to [latex]V_{M}[/latex]; that is, the speed in rpm is some constant k times V. Without knowing the details of the power amplifier, can we analyze this system? In particular, can we develop a relationship between rpm and α?

Accepted Answer

Since the power amplifier output voltage is proportional to its input, we can model the amplifier as a simple dependent source. The resulting circuit diagram is shown in Fig. 3.31b. Now we can easily develop a relationship between motor speed and the pot position, α. The equations that govern the operation of the motor, power amplifier, and the voltage divider are

speed (rpm) = [latex]K_{M} V_{M}[/latex]
[latex]V_{M} = 4 V_{speed}[/latex]

[latex]V_{speed} = 5 \frac{R_{2}}{R_{1} + R_{2}} = 5 [\frac{R_{2}}{R_{pot}}][/latex] = 5 α
[latex]R_{2} = α R_{pot} \quad R_{1} = (1 - α)R_{pot}[/latex]

Combining these relationships to eliminate [latex]V_{speed}[/latex] yields a relationship between motor speed and α, that is, rpm = 20α. If, for example, the motor constant [latex]K_{M}[/latex] is 50 rpm/V, then

rpm = 1000α

This relationship specifies that the motor speed is proportional to the pot knob position. Since the maximum value of α is 1, the motor speed ranges from 0 to 1000 rpm. Note that in our model, the power amplifier, modeled by the dependent source, can deliver any current the motor requires. Of course, this is not possible, but it does demonstrate some of the tradeoffs we experience in modeling. By choosing a simple model, we were able to develop the required relationship quickly. However, other characteristics of an actual power amplifier have been omitted in this model.

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