Quiz 4 Example Solution

The suction produced by a vacuum cleaner is the result of a partial vacuum created by fan blades driven by an electric motor. When the hose of a vacuum cleaner is covered, the pressure across the fan blades equalizes and suction is lost. When the motor is producing no suction, the load on the motor is negligible and its speed increases. This variation in the speed of the motor can be explained by an analogy to a Thevenin Equivalent Circuit.

A Thevenin Equivalent Circuit (TEC) is used to model a nonideal source; in class we considered an electrical power source---a battery. Electrical power is the product of voltage times current: P = V * I. A TEC can be used to predict the sort of voltage decrease that a battery exhibits under a heavy load.

A TEC is a model composed of two components: (1) an ideal voltage source, Vo, (2) a resistor, Ro. These 2 components connected in series represent the power source. Let Vt represent the voltage at the terminals of the power source as shown below. The equation for the circuit is: Vt = Vo - RoIL. When the power source is not connected to a load, the voltage across the terminals is equal to the ideal value, Vo, a.k.a. the open source voltage.

The motor of a vacuum is a power supply---a mechanical power supply. Mechanical power is the product of angular velocity times torque: P = w * T. We will use an electrical analogy involving a TEC to explain the increase in motor speed (i.e., angular velocity) that occurs when the load on a vacuum motor is removed (i.e., you cover the vacuum hose with your hand).

An ideal vacuum motor is expected to provide torque at a constant speed just as an ideal battery is expected to provide current at a constant voltage. (Remember from our class discussion of electric motors that torque is proportional to current and angular velocity is proportional to voltage.) We will express the analogy between voltage and speed in our TEC representation of the vacuum motor as shown pictorially below. In this representation, angular velocity is analogous to voltage and torque is analogous to current. The internal resistance, Ro, reduces the ideal angular velocity provided by the motor by an amount proportional to the required torque. When the motor is not loaded (your hand is placed over the hose), the angular velocity of the motor increases to the ideal angular velocity which is analogous to the open source voltage, Vo, in the TEC model.

The performance of a mechanical power source can be explained by analogy to an electrical power sources using a TEC. This is not surprising when the mechanical source under consideration is an electric motor.