# 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. 