In this page, we learn more about the the frequency response of other second-order circuits, listed below.
A second-order band-pass filter circuit can be viewed as the combination of the first-order low-pass circuit and the first-order high-pass circuit. The first-order low-pass filter circuit and its asymptotic response are presented in Fig. 48.
The transfer function of the circuit in Fig. 48 is
The first-order high-pass filter circuit and its asymptotic response are presented in Fig. 49.
The transfer function of the circuit in Fig. 49 is
We can combine the two plots as shown in Fig. 50, when the pole frequency of the high pass circuit is lower than that of the low-pass circuit.
The band-pass filter circuit can be obtained by cascading the low-pass and the high-pass filter circuits, as shown in Fig. 51.
The transfer function of the circuit in Fig. 51 is
Since the pole frequency of the high pass circuit is lower than that of the low-pass circuit, it is necessary that R1 > R2. A band-pass circuit can have either the low-pass section as the first stage followed by a high-pass section or vice versa.
It is possible to build a band-pass circuit, using a second-order RLC circuit shown in Fig. 52.
We can get the following transfer functions for the circuit in Fig. 52.
The voltage across the resistor is the response of a band-pass circuit. A second-order RLC circuit can be over-damped, critically-damped or under-damped. An over-damped circuit behaves as a band-pass circuit, whereas an under-damped behaves like a resonant circuit, with a peak response at resonant frequency. The behaviour of a resonant circuit is described later in detail. There is no point in using a critically-damped circuit as a band-pass circuit, since both the cut-off frequencies shown in Fig. 50 coincide in the case of a critically-damped circuit.
A low-pass filter passes only low frequency signals to the output, whereas a high-pass filter passes only high frequency signals to the output. On the other hand, a band-pass blocks both the low frequency and high frequency signal, but passes the signals between the two-cut off frequencies. The range between the two cut-off frequencies is known as the pass band. At a low frequency, the high-pass section blocks the passage of signal to the output. As the frequency of source increases, it passes the cut-off frequency of the high-pass section. So long as the source frequency is higher than the cut-off frequency of the high-pass section, but lower than the cut-off frequency of the low-pass section, both the low-pass section and the high-pass section do not block the passage of signal to the output. It can also be seen why the cut-off frequency of the low-pass section should be higher than cut-off frequency of the high-pass section. If the cut-off frequency of the low-pass section is lower than cut-off frequency of the high-pass section, the high-pass section blocks the low frequency signals, the low-pass section blocks the high frequency signals and the output gets very little signal strength.
From equation (3.2), the magnitude and the phase angle of a second-order over damped system can be expressed as a function of frequency, as shown below.
The approximate log-magnitude of the band-pass system described by equation (4.7) can be obtained as follows.
From equations (4.9), (4.10) and (4.11), we get the equation presented below.
The approximate phase angle plot of the band-pass system described by equation (4.7) can be obtained as follows.
It can be seen that it is not that easy to draw the approximate phase angle plot. For the band-pass transfer function, it is easier to sketch the actual phase angle plot.
The plots obtained using a Matlab program for a band-pass system is presented below.
% Bode Plots for s*p2/[(b+ a*s+ b*s^2)]
% Band-pass Second-order system: Equation (4.7)
% Low-pass cut-off frequency 100 rad/s
% High-pass cut-off frequency 1 rad/s
clear
p2 = 100;
p1 = 1;
a=101; % a = p2 + p1;
b=100; % b = p2 * p1;
y = 1.025;
omega = 0.1;
for m=1:375;
theta(m) = omega;
if (omega< p1) asGain(m)= omega / p1;
elseif (omega< p2) asGain(m)= 1.0;
else asGain(m)= p2/omega;
end;
actVal= omega*p2/(b+j*a*omega- omega*omega);
mag(m)= actVal;
phase(m)=90 +180/pi*angle(mag(m));
omega = omega*y;
end;
subplot(2,1,1)
loglog(theta,abs(mag),theta,asGain)
ylabel('magnitude')
axis([0.1 1000 0.1 2])
grid on
subplot(2,1,2)
semilogx(theta,phase)
ylabel('phase angle')
xlabel('w, angular frequency')
axis([0.1 1000 -100 100])
grid on
gtext('slope + 1')
pause
gtext('slope - 1')
pause
gtext('p1=1')
pause
gtext('p2=100')
The plots obtained are shown below.
Next we find out how we can obtain the polar plot for equation (4.7) .
% Program: Polar Plot of Equation 4_7
% Band-pass Second-order system: Equation (4.7)
% Low-pass cut-off frequency 100 rad/s
% High-pass cut-off frequency 1 rad/s
clear
p2 = 100;
p1 = 1;
a=101; % a = p2 + p1;
b=100; % b = p2 * p1;
y = 1.025;
omega = 0.1;
for m=1:375;
actVal= omega*p2/(b+j*a*omega- omega*omega);
ReH(m) = real(actVal);
ImH(m) = imag( actVal);
omega = omega*y;
end;
plot(ReH, ImH)
title('Polar Plot of Equation 4.06')
xlabel('Real Part')
ylabel('Imaginary Part')
grid on
The operation is just the reverse of the operation of the band-pass circuit. A band-pass circuit blocks the low-frequency and high-frequency signals and passes the intermediate frequency signals. On the other hand, a notch filter passes the low-frequency and high-frequency signals and blocks the intermediate frequency signals. A notch filter can be built by the combination of a low-pass filter and high-pass filter operating in parallel and then summing their outputs. In the case of a band-pass circuit, the cut-off frequency of the high-pass filter is much lower than the cut-off frequency of the low-pass filter. But in the case of a notch filter circuit, the cut-off frequency of the high-pass filter is much higher than the cut-off frequency of the low-pass filter.
A notch filter circuit is presented below.
For this circuit,
When the cut-off frequency of the high-pass section is much higher than the cut-off frequency of the low-pass section, the output of the notch filter at any frequency is either the output of the high-pass section or the output of the low-pass section.
A Matlab script can be written to obtain the response of notch filter and the script is presented below.
% Bode Plots for s/(s + p1) + (p2)/(s+p2)
% Band-pass Second-order system: Equation (4.18)
% Low-pass cut-off frequency, p2 = 1 rad/s
% High-pass cut-off frequency, p1= 100 rad/s
clear
p2 = 1;
p1 = 100;
y = 1.025;
omega = 0.1;
for m=1:375;
theta(m) = omega;
if (omega< p2) asGain(m)= 1.0;
elseif (omega< sqrt(p1*p2) ) asGain(m)= p2/omega;
elseif (omega< p1) asGain(m)= omega/p1;
else asGain(m)= 1.0;
end;
actVal= j*omega/(j*omega + p1) + p2/(j*omega + p2);
mag(m)= actVal;
phase(m)= 180/pi*angle(mag(m));
omega = omega*y;
end;
subplot(2,1,1)
loglog(theta,abs(mag),theta,asGain)
ylabel('magnitude')
axis([0.1 1000 0.01 2])
grid on
subplot(2,1,2)
semilogx(theta,phase)
ylabel('phase angle')
xlabel('w, angular frequency')
axis([0.1 1000 -90 90])
grid on
gtext('slope + 1')
pause
gtext('slope - 1')
pause
gtext('p2=1')
pause
gtext('p1=100')
The frequency response plots obtained are shown below.
It can be seen that it is possible to get the approximate asymptotic log-magnitude plot, but it is difficult to obtain the approximate phase angle plot. The phase angle changes fast near the geometric mean of the cut-off frequencies, p1 and p2.
The polar plot of notch filter defined by equation (4.18) can obtained with the use of a Matlab script, as shown below.
% Polar Plots for s/(s + p1) + (p2)/(s+p2)
% Band-pass Second-order system: Equation (4.18)
% Low-pass cut-off frequency, p2 = 1 rad/s
% High-pass cut-off frequency, p1= 100 rad/s
clear
p2 = 1;
p1 = 100;
y = 1.025;
omega = 0.1;
for m=1:375;
actVal= j*omega/(j*omega + p1) + p2/(j*omega + p2);
ReH(m) = real(actVal);
ImH(m) = imag( actVal);
omega = omega*y;
end;
plot(ReH, ImH)
title('Polar Plot of Equation 4.18')
xlabel('Real Part')
ylabel('Imaginary Part')
grid on
The plot obtained is presented below.
It is possible to build notch filter with a higher roll-off greater than 1. A second-order low-pass section and a second-order high-pass section can be used, and then the circuit will have a roll-off of 2.
A filter that passes all frequencies is an all-pass filter. The circuit in Fig. 53 can be used as an all-pass filter, if R1 = R2. Then the transfer function, shown by equation (4.18) has a value of unity. Since it is independent of frequency, the circuit acts like an all-pass filter.
This page has described how band-pass, notch and all-pass filter circuits can be built and analyzed. The next page presents some examples.