Real Resistors, Capacitors and Inductors.
The equivalent circuits of real Resistors, Inductors and Capacitors
are shown below. The Resistor has a capacitor Cr across its leads. The produces 1 break frequency fr = 1/(2 p R Cr) above which the impedance of the real resistor decreases by 20 dB/dec.
The real Inductor has a capacitor Cl across its leads and a resistor
Rl of the coils of the wire. The Impedance has 2 break frequencies.
at the low frequency end the impedance does not go to zero as
jwL suggests but is limited by Rl below the break frequency
fl1 = Rl/(2 p L).
At high frequencies, the impedance does not increase indefinitely but
has a second break frequency
fl2 = 1/Ö(L Cl)
where the Inductor has a parallel(maximum impedance)
self resonating frequency with Cl. At this break frequency the impedance slope changes from +20dB/dec to -20dB/dec.
The capacitor has a lossy resistor resistor RL in
parallel with it which is important for charge storage and
low frequency applications but is not included in the analysis here.
Because of the resistor Rc, the impedance of the real capacitor does not
decrease indefinitely but is limited by Rc for frequencies higher
than the break frequency
fc1 = 1/(2 p Rc C).
At still higher frequencies, the impedance of the inductor Lc becomes larger than Rc and the capacitor has a second break frequency at
fc2 = Rc/(2 p Lc). If fc2 is less
than fc1 then the capacitor has a series(minimum impedance) self
resonating frequency at the average of fc1 and fc2. (Arithmetic average
of logs, geometric average of frequencies).
Applet below show the behaviour of the Impedance of the
Components with frequency. The numerical values of the impedances
are given in terms of R, C and L. In practice the break frequencies will
be measured and the components calculated as shown on the applet.
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COPYRIGHT © 2007 Cuthbert A. Nyack.