The previous sections explained filters by their properties, mathematical functions, time and frequency domain responses, and different types. Using the theoretical knowledge from the previous sections, we can implement it in the real world. However, this filter circuit will not perform as per your expectations.
The figure below shows how the ideal and real filter works. The red dotted line represents the ideal low-pass filter and the blue line represents the actual filter. According to the ideal filter curve, a frequency above 1 kHz should not pass. But in real life, this happens because the filter does not transition to cut the frequency sharply, as shown by the red dotted line. It happens as shown by the blue line.
Figure 1: Cutoff frequency transition
Multiple sections of orders two and one are used to design a higher order filter (mostly greater than two). In this case, the frequency and quality factor of each section must be the same. Otherwise, the overall response of the circuit will deviate from the desired response. This will not be accurate.
Therefore, in the typical engineering case, a trade-off must be made that is exactly what the designers need and can vary from project to project. Designers need to ask themselves whether their design should have the filter response or frequency exactly as desired, whether the response can be accepted despite some ripple in the passband, and whether the cutoff frequency deviates from the main frequency.
Some tradeoffs are intrinsic to filter design, mainly due to the manufacturing of semiconductor components.
Passive Components
The main problem is caused by the resistor, capacitors and inductors. As these components are available in standard values, the calculated value of capacitors, inductors and resistors will not be commercially available. However, they can be customized. Commercial passive components may not provide the exact calculated value of passive components, but they can be used in parallel or in series. This will increase the cost and size of the project and, in parallel, increase the manufacturing cost. Again, these components will have a tolerance of +-1%. Its dependence on temperature and current will also affect its performance.
A practical way to save time and costs is to use existing CAD programs to analyze circuits using standard values of passive components. The deviation of components can be analyzed in relation to temperature, and a combination of series and parallel components can be analyzed to obtain the desired response within limits.
The calculated passive component values determine the cutoff frequency and quality factor of a filter. These passive components will bias the cutoff frequency and quality factor of the filter. This situation occurs in higher order filters because they can drastically decrease the gain after the cutoff frequency compared to lower order filters. The next article “Practical Filter Implementation” will give a practical example of a high-order filter.
The higher the quality factor, the more critical and accurate the value of the components. A ratio of two or more components defines the quality factor, typically capacitors. In addition to component tolerance, the effect of temperature/time deviation can also be added. Especially capacitors, as they not only drift, but are also a function of temperature.
During filter construction, there are infinite options for selecting the value of the passive components, but there is a limit on the practical size of the components. Capacitor values between 20pF and 10uF are practical, but not outside this range. For filter designs, electrolytic capacitors are not recommended due to their leakage property.
The same type of problem occurs in certain resistors. Resistance values between 100 Ohm and 1 Mohm are recommended for designing the filter. Resistance below 100 Ohm requires more current to drive the inverter and therefore dissipates more power; In a filter design, these two situations must be avoided. Large resistance values are more prone to parasitics and can be easily coupled to small capacitances.
In electronic design, circuit performance is affected by layout design because it causes stray capacitances (unwanted capacitance within the electronic component part) in the design. This can be formed between two traces of a PCB on the same layer or on another side of the layer. Parasitic capacitances can be formed between terminals of adjacent components. As these capacitances have very small values, they have a large effect on high impedance nodes. This can be controlled by reducing the impedance of the circuit PCB. Remember that the effect of parasitic capacitances is frequency dependent and the effect increases with higher frequencies where the impedance drops.
Parasites are not only associated with external sources, but also with components themselves.
A capacitor is not just a capacitor. It has resistance and inductance in its conductor wire, as shown in the figure below. In the capacitor specification, a resistance indicates leakage and low power factor. In filter design or any other circuit design, it is recommended to have a capacitor with low leakage and good power factor. T The best is a polystyrene or Teflon film capacitor and a metal resistor.
The figure below shows the metal film resistor and the polystyrene capacitor.
Figure 2: Polystyrene Capacitor and Metal Film Resistor
It is best practice to use surface mount components to reduce component parasitics. Because the terminals of any component create inductance, SMD components are ideal for positioning. The main disadvantage is that not all capacitor types are available in SMD format. Ceramic capacitors are best for filter designs, and the NPO family has the best features. Ceramic capacitors are prone to feedback and can act as motion sensors that can transform vibration into electrical sensors, which will be noise.
Fig. 3: Capacitor Equivalent Circuit
Due to the terminals, the resistance also has parasitic inductance and capacitance.
Limitations of Active Components in Filters
There are some properties of operational amplifiers that can be better used in filter design.
1. Infinite gain
two. Infinite input impedance
3. Zero output impedance
In filter design, an operational amplifier is used in the design of active filters due to the below explained properties of operational amplifiers:
Infinite gain (ideal case): Op amps have gain, so this component is best to use in a filter if you have a significantly smaller amplitude signal. Gain from operational amplifiers can be added to the signal to amplify it.
Infinite input impedance (ideal case): Infinite input impedance (ideal case) or high input impedance (practical case) means that the input of the op amp does not require any current to power the signal.
Zero output impedance (ideal case): Op-amps have significantly less output impedance, which means they can supply enough current to drive the connected circuit.
The one These properties are true only for ideal operational amplifiers. Practical operational amplifiers always have high input impedance, near-zero output impedance, adjustable gain, and do not vary with frequency.
Amplifiers are built with the physical limitations of the devices and therefore become band limited. And the main limitation of the operational amplifier is the variation of gain with frequency. The gain must be first order for the stability of the operational amplifier.
The op amp's sullen switch configuration (a second-order filter design topology) is the least dependent on the op amp's frequency response. Since the amplifier is used in gain block mode, it is necessary for the response to be flat just after the frequency where the minimum attenuation begins.
Quality factor improvement and peak
When designing the filter, it is necessary not to exceed the dynamic range of the amplifier. The peak in filter response may be caused by increasing the quality factor above 0.707. High quality factors can cause input and output overload.
Fig. 4: Effect of the frequency response of operational amplifiers on the quality factor.
The quality factor multiplied by the gain must also remain below the loop gain plus some margin. The effective quality factor decreases as soon as the amplifier becomes overloaded and the transfer function changes even though the output appears undistorted. Therefore, it is clear that increasing the input level will change the transfer function.
These principles apply to all types of filters, such as low-pass, high-pass, band-pass, and band-reject filters. The improvement in quality factor will not affect high-pass filters since the resonance frequency will be low relative to the cutoff frequency.
