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• Analysis of the above circuit indicates that,
XC = 1/2pfC @ 0 W
• Thus, Vo = Vi at high frequencies.
• At f = 0 Hz, XC = , Vo = 0V.
• Between the two extremes, the ratio, AV = Vo / Vi will vary.
As frequency increases, the capacitive reactance decreases and more of the input
voltage appears across the output terminals.
The output and input voltages are related by the voltage – divider rule:
Vo = RVi / ( R – jXC)
the magnitude of Vo = RVi / ÖR2 + XC2
• For the special case where XC = R,
Vo =RVi / RÖ2 = (1/Ö2) Vi
AV = Vo / Vi = (1/Ö2) = 0.707
• The frequency at which this occurs is determined from,
XC = 1/2pf1C = R
where, f1 = 1/ 2pRC
• Gain equation is written as,
AV = Vo / Vi
= R / (R – jXC) = 1/ ( 1 – j(1/wCR)
= 1 / [ 1 – j(f1 / f)]
• In the magnitude and phase form,
AV = Vo / Vi
= [1 /Ö 1 + (f1/f)2 ] Ð tan-1 (f1 / f)
• In the logarithmic form, the gain in dB is
AV = Vo / Vi = [1 /Ö 1 + (f1/f)2 ]
= 20 log 10 [1 /Ö 1 + (f1/f)2 ]
= - 20 log 10 Ö [ 1 + (f1/f)2]
= - 10 log10 [1 + (f1/f)2]
• For frequencies where f << f1 or (f1/ f)2 the equation can be approximated by
AV (dB) = - 10 log10 [ (f1 / f)2]
= - 20 log10 [ (f1 / f)] at f << f1
• At f = f1 ;
f1 / f = 1 and
– 20 log101 = 0 dB
• At f = ½ f1;
f1 / f = 2
– 20 log102 = - 6 dB
• At f = ¼ f1;
f1 / f = 4
– 20 log102 = - 12 dB
• At f = 1/10 f1;
f1 / f = 10
– 20 log1010 = - 20dB
• The above points can be plotted which forms the Bode – plot.
• Note that, these results in a straight line when plotted in a logarithmic scale.
Although the above calculation shows at f = f1, gain is 3dB, we know that f1 is
that frequency at which the gain falls by 3dB. Taking this point, the plot differs
from the straight line and gradually approaches to 0dB by f = 10f1.
Observations from the above calculations:
• When there is an octave change in frequency from f1 / 2 to f1, there exists
corresponding change in gain by 6dB.
• When there is an decade change in frequency from f1 / 10 to f1, there exists
corresponding change in gain by 20 dB.
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