Related RC circuits.
Cuthbert Nyack
Here the transfer functions of some related RC circuits are plotted
as a function of frequency. The Lead circuit is shown below, this circuit is used to produce a phase Lead.
The transfer function is
a(Ts + 1)/(aTs + 1) with a = R2/(R1 + R2) and T = R1C.
Setting Fn = 0 in the applet below shows the transfer function.
with parameters (3.0, na, 0.08, na, na, na, 0, 384, na, 1.0) a maximum phase lead of 58.4° occurs at 1.181rad/s. The gain changes from
a at low frequencies to 1 at high frequencies.
The Lag circuit below can be used to produce a phase lag or a reduced gain above a certain frequency. The transfer function is
(Ts + 1)/(bTs + 1) with
b = (R1 + R2)/R2 and T = R2C.
eg parameters(1.0, na, 10.0, na, na, na, 1, 312, na, 1.0) show
a maximum phase lag of 54.9° at 0.3136rad/s. The gain changes from
1 at low frequencies to 1/b at high frequencies.
The circuit below is used in the Wien bridge and Wien bridge
oscillator, it is characterised by having a real transfer function
= 1/(1 + c21 + 1/r21) at
w = 1/(TÖr21c21). T = R1C1, on applet
r21 = R2/R1 and c21 = C2/C1.
eg parameters(1.0, na, na, 1.0, 1.0, na, 2, 375, na, 5.0) show the
circuit has a real transfer function of 1/3 at
w = 1/T.
eg parameters(1.0, na, na, 0.5, 2.0, na, 2, 375, na, 5.0) show the
circuit has a real transfer function of 1/5 at
w = 1/T .
eg parameters(1.0, na, na, 2.0, 2.0, na, 2, 318, na, 5.0) show the
circuit has a real transfer function of 2/7 at
w = 1/2T.
eg parameters(1.0, na, na, 3.0, 3.0, na, 2, 285, na, 5.0) show the
circuit has a real transfer function of 3/13 at
w = 1/3T.
The 3CR circuit below can be used to produce a phase lead. It is
characterised by a real transfer function at
w = 1/(CR6½)
eg parameters(1.0, na, na, na, na, na, 3, 302, na, 20.0) show the
circuit has a real transfer function of ~ 1/29 and a
phase of ~180° at 0.4082rad/s (1/(CR6½) = 0.4082).
The bridged T circuit below can be used to
produce a minimum gain at a frequency
w =
1/(ÖC1R1C2R2) where the imaginary
part of the transfer function is zero.
eg parameters(1.0, na, na, 2.0, 2.0, na, 4, 375, na, 1.5)
have a real minimum of 0.333333 at 1rad/s.
eg parameters(1.0, na, na, 32.0, 32.0, na, 12, 375, na, 1.5) have a
real minimum of 0.03034 at 1 rad/s.
The Twin or parallel bridged T circuit below can be used to
produce a sharper minimum than can be obtained by the bridged T.
eg parameters(7.34, na, na, 0.5, 0.5, 722, 5, 447, na, 1.0)
have a minimum 1.75e-4 at ~60Hz.
eg parameters(8.0, na, na, 1.0, 0.25, 937, 5, 447, na, 1.0)
have a minimum 7.712e-5 at ~60Hz.
When applet is enabled its appearance is illustrated by the gif image below.
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COPYRIGHT © 2007 Cuthbert A. Nyack.