082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

2.4 Review of some common engineering functions and techniques 97
Section 3.4. The usual procedure is to apply a range of fixed-amplitude sinusoidal
signals with different frequencies in order to obtain information about the circuit or
system.InSection23.9wewillseethatifthecircuitislinearthen,afterithassettled
down, the output signal is also a sinusoidal signal of the same frequency but with a
different amplitude and phase (see Section 3.7 fordetails of these terms).
ABode plotconsists of two components:
(1) The ratio of the amplitudes of the output signal and the input signal is plotted
against frequency.
(2) Thephaseshiftbetweentheinputandoutputsignalsisplottedagainstfrequency.
A log scale is used for the frequency in order to compress its length; for example,
a typical frequency range is 0.1 to 10 6 Hz which corresponds to a range of −1 to 6
on a log scale. A log scale is also used for the ratio of the signal powers as this is
calculatedindecibels.Phaseshiftisplottedonalinearscale.Sothesignalamplitude
ratio versus frequency is a log--log graph and the phase shift versus frequency is a
linear-- log graph.
Anoperationalamplifierisanexampleofalinearcircuit.Inatechnicalcomputing
language such as MATLAB ® itisusually easytoproduce a Bode plot.
An example function which describes the behaviour ofsuch a device is:
A(f)= 100000
1 + j f 8
which gives the input-output voltage function. Do not worry about the meaning of
the ‘j’ in the equation for now. This will be covered in Chapter 9 when we discuss
complex numbers.
We could produce a Bode plot by typing:
f=1:1:10000;
semilogx(f, 20*log10(abs(100000./(1+j*f/8))));
The function semilogx plots a graph with a logarithmic scale on thex-axis. Chapter
21 examines Bode plotsinmore detail.
EXERCISES2.4.4
1 Evaluate
(a) log 2 8 (b) log 2 15
(c) log 16 50 (d) log 16 123
(e) log 8 23 (f) log 8 47
2 Simplify eachofthe following to asingle logterm:
(a) log7 +logx
(b) logx+logy+logz
(c) lny−ln3
(d) 2logy+logx
(e) ln(xy) +ln(y 2 )
(f) ln(x +y) −lny
(g) log(2x 2 ) +log(4x)
(h) 3logy+2logx
1
(i) 2 logx 4 −2logx
(j) 3lnx+2lny+4lnz
(k) logz−2logx+3logy
(l) logt 3 −log(2t) +2logt
3 Simplify each ofthe following to asingle log term:
(a) 3lnt−lnt
(b) 6logt 2 +4logt