1. The
power we use at home has a frequency of 60 Hz. The period of this sine wave can
be determined as follows:
Solution
T = 1/f = 1/60 = 0.0166 s
= 0.0166 x 103 ms = 16.6 ms
2. The
period of a signal is 100 ms. What is its frequency in kilohertz?
Solution
First we change 100 ms to seconds, and then
we calculate the frequency from the period (1 Hz = 10−3 kHz).
100 ms = 100 x 10-3 s = 10-1
s
f = 1/T = 1/10-1 Hz = 10 Hz
= 10 x 10-3 kHz = 10-2
kHz
Note:
Frequency is the rate of change with respect
to time.
Change in a short span of time means high
frequency.
Change over a long span of time means low
frequency.
3. A
sine wave is offset 1/6 cycle with respect to time 0. What is its phase in
degrees and radians?
Solution
We know that 1 complete cycle is 360°. Therefore,
1/6 cycle is
1/6 x 360 = 60o
= 60 x 2π/360 rad = π/3 rad = 1.046 rad
Note:
Phase describes the position of the waveform
relative to time 0.
Three sine waves with the same amplitude and
frequency, but different phases
4. If
a periodic signal is decomposed into five sine waves with frequencies of 100,
300, 500, 700, and 900 Hz, what is its bandwidth? Draw the spectrum, assuming
all components have a maximum amplitude of 10 V.
Solution
Let fh be the highest frequency, fl
the lowest frequency, and B the bandwidth. Then
B = fh - fl = 900 – 100
= 800 Hz
The spectrum has only five spikes, at 100,
300, 500, 700, and 900 Hz (see Figure).
Note:
The bandwidth of a composite signal is the
difference between the highest and the lowest frequencies contained in that
signal.
5. A
periodic signal has a bandwidth of 20 Hz. The highest frequency is 60 Hz. What
is the lowest frequency? Draw the spectrum if the signal contains all
frequencies of the same amplitude.
Solution
Let fh be the highest frequency, fl
the lowest frequency, and B the bandwidth. Then
B = fh - fl => 20 =
60 - fl => fl = 60 – 20 = 40 Hz
The spectrum contains all integer
frequencies. We show this by a series of spikes (see Figure).
0 Comments