Download ELE 427 Homework #1 Fall `07

ELE 427
Homework #1
Fall ’07
1. Consider an ideal potentiometer with resistance Rp . The output of this potentiometer is
Vout = αVin . Suppose that a load resistance of RL is connected between the wiper terminal
and ground. Show that for this loaded potentiometer,
Vout
α
=
Vin
1 + α(1 − α)Rp /RL
2. Consider an unloaded potentiometer with total resistance Rp = 1KΩ and 5 volts supplied
across this resistance. The output of the unloaded potentiometer is 5α, where 0 ≤ α ≤ 1
is proportional to the angular position of the potentiometer shaft. If this potentiometer
is connected to a load resistance RL and α = 0.5, what is the minimum value of RL
needed to keep the difference between the unloaded (ideal) potentiometer output and the
loaded potentiometer output less than 2% of the ideal potentiometer output? [Hint: set
the difference between the ideal pot output and the loaded pot output to be equal to 2%
of the unloaded pot output. When you solve for RL , this value is the minimum acceptable
value. Larger values of RL will make the loaded pot output even closer to the unloaded pot
output.]
3. Consider unloaded and loaded potentiometers, both with Rp = 1KΩ. On the same graph,
plot the output of the unloaded potentiometer as a function of α (as a solid line), the output
of the loaded potentiometer with RL = 1KΩ (plot with asterisks for discrete values of α
spaced every 0.1), and the loaded potentiometer with RL = 10M Ω (plot with circles for
discrete values of α). Use Matlab, Excel, or some other graphing software to create the
plot.
4. Suppose a potentiometer with an input voltage of 5V and total resistance Rp = 1KΩ, is used
to measure angular position as a function of α. Suppose that the (unloaded) potentiometer
output is measured by an ideal voltmeter that displays to the nearest tenth of a volt. What
is the resolution of this potentiometer/voltmeter sensor? That is, what is the largest change
in α that can occur without the voltmeter reading changing?
5. Consider two systems for measuring angular position. The first is a 1,000-line optical encoder with a single decoding counter, and the second is a 1K potentiometer connected to
a voltmeter that displays to the nearest millivolt. For this problem, you may assume that
the potentiometer does not have a deadzone.
(a) What is the resolution, in units of degrees, of the encoder system?
(b) With 5 volts supplied across the potentiometer, what is the resolution (in degrees) of
the potentiometer/voltmeter system?
(c) With 10 volts supplied across the potentiometer, what is the resolution of the potentiometer/voltmeter system?
(d) If the measurement systems are all initialized to zero and you slowly begin to increase
the angular position, which of the above sensor systems will be the first to display a
nonzero value? Which will be the second and third?
6. In this problem, all voltages and currents are rms values of AC signals. Suppose that 10 KW
of power at 200V is delivered over a transmission line which has a resistance of 1Ω. (That
is, 10 KW of power is dissipated in the load resistance RL .) Find the power supplied by
the generator in arrangements (a) and (b) in the figure below.
Calculate the efficiency (i.e. power supplied to the load divided by power delivered by the
generator) for each arrangement. [Hints: power = V I. From the information in (a) you can
calculate the value of the load resistance, RL . For (b), you can find the currents flowing in
each of the three loops, as well as the voltages across the primaries and secondaries of the
transformers.]
7. Find I1 , V1 , and I2 in the following circuit: