Concept explainers
(a) Design a two-stage op amp filter circuit with a bandwidth of 1000 rad/s, a low-frequency cutoff of 100 rad/s, and a voltage gain of 20 dB. (b) Verify your design with an appropriate LTspice simulation.
(a)
Design a two-stage op amp filter circuit with a bandwidth of
Explanation of Solution
Given data:
The value of the bandwidth
The value of the lower cutoff frequency
The value of the voltage gain
Formula used:
Write the expression to calculate the impedance of the passive elements resistor and capacitor.
Here,
Calculation:
The two-stage op amp filter circuit can be obtained in a band-pass filter by cascading the low pass filter and high pass filter.
In order to design the band-pass filter of gain
In two-stage op amp filter, the low pass op amp filter is the first stage and the output of first stage filter is given as an input of the second stage high-pass op amp filter.
Low-pass filter design:
For low-pass filter, the voltage gain
The active low-pass op amp filter is drawn as Figure 1.
The s-domain circuit of the Figure 1 is drawn as Figure 2 using the equations (1) and (2).
Write the general expression to calculate the transfer function of the circuit in Figure 2.
Here,
Use nodal analysis on node
Rearrange the above equation to find
Use nodal analysis on node
Rearrange the above equation to find
It is known that for an ideal operational amplifier,
Substitute
Rearrange the above equation to find
Substitute
Therefore, the equation (4) is the transfer function of the active low-pass filter that is the product of the transfer function of general low-pass filter and the gain of the non-inverting amplifier.
From equation (4), the gain of non-inverting amplifier is,
Write the general expression to calculate the voltage gain in dB.
Substitute
Rearrange the above equation to find
Assume the resistor,
Substitute
n (5).
Rearrange the above equation to find
Write the expression to calculate the corner frequency of the filter circuit shown in Figure 1.
Assume the capacitor,
Write the expression to calculate the bandwidth of the filter.
It is known that the corner frequency
Substitute
Rearrange the above equation to find
Substitute
Rearrange the above equation to find
For first stage low-pass filter,
High-pass filter design:
For high-pass filter, the voltage gain
The active high-pass op amp filter is drawn as Figure 3.
The s-domain circuit of the Figure 3 is drawn as Figure 4 using the equations (1) and (2).
Write the general expression to calculate the transfer function of the circuit in Figure 4.
Here,
Use nodal analysis on node
Rearrange the above equation to find
Use nodal analysis on node
Rearrange the above equation to find
It is known that for an ideal operational amplifier,
Substitute
Rearrange the above equation to find
Substitute
Therefore, the equation (8) is the transfer function of the active high-pass filter that is the product of the transfer function of general high-pass filter and the gain of the non-inverting amplifier.
From equation (8), the gain of non-inverting amplifier is,
Write the general expression to calculate the voltage gain in dB.
Substitute
Rearrange the above equation to find
Assume the resistor,
Substitute
Rearrange the above equation to find
Write the expression to calculate the corner frequency of the filter circuit shown in Figure 4.
Assume the capacitor,
It is known that the corner frequency
Given that,
Substitute
Rearrange the above equation to find
Simplify the above equation to find
For second stage high-pass filter,
The two-stage op amp band-pass filter circuit is drawn as Figure 5.
Conclusion:
Thus, the two-stage op amp filter circuit with a bandwidth of
(b)
Verify the two-stage op amp filter circuit design with an appropriate LTspice simulation.
Explanation of Solution
Calculation:
Create the new schematic in LTspice and draw the circuit in Figure 5 as shown in Figure 6 by entering the corresponding values of resistors and capacitors.
Using SPICE Directive in Edit menu, mention the command .ac dec 1000 0.1 100k and .lib opamp.sub as shown in Figure 7 and Figure 8.
Use Label Net option and enter the Vout. After adding all the commands mentioned above, the circuit is shown as in Figure 9.
Now run the simulation and place the probe at resistor
Refer to Figure 10 the thick red line indicates magnitude response, the dotted red line indicates phase response, and the black dotted line represents the corner frequency of the magnitude response.
Conclusion:
Thus, the design of the two-stage op amp filter is verified by using the LTspice simulation.
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