Analyzing and Designing Filters with the Ti-84 plus (CE)
This page offers a glimpse into the potential of designing and redesigning filters, but it does not provide a comprehensive overview of all filtering concepts.
First order low-pass filter (example 1)
A method is demonstrated to analyze and design electrical filters, providing a valuable tool for Electrical Engineers. As an example, a low pass filter, consisting of a resistor of 1000 Ohm and a capacitor of 4,7 nF =0.00471uF is taken . See the schematic
The complex transfer function of this filter equals : Vout/Vin= 1/(1+i2πXR1C1) where X is the frequency (Hz) of the input Voltage. The value of R is stored in memory R, and the value of C is stored in memory C. Using Y1 as the logarithmic absolute value of the transfer function for the Bode plot and Y2=-3 (dB) the Ti calculates the bandwidth (33782,44 Hz) by using the command intersect. For low fequencies the gain is 1 = 0dB
Redesign of the the low-pass filter
Two examples illustrate the application of the numerical solver in the math menu to redesign the filter. In the first case, a new bandwidth of 25,000 Hz and a capacitor value of 4.7nF lead to a recalculated resistor value of 1,351.29 ohms. In the second scenario, a bandwidth of 10,000 Hz and a fixed resistor value of 1,000 ohms result in a recalculated capacitor value of 15.87nF."
A bandwidth of 1000Hz and a resistor of 1000 Ohm results in a capacitor of 15.87nF
First order high-pass filter (example 2)
By employing the same method, a high-pass filter can be designed by swapping the positions of the capacitor and resistor. Refer to the attached electrical diagram for a visual representation. The complex transfer function of this filter can be expressed as: Vout/Vin = (i2πXR1C1) / (1 + i2πXR1C1), where X represents the frequency (Hz) of the input voltage. The value of R is stored in memory R, and the value of C is stored in memory C
The Input and the Window properties are shown alongside the plotted results. As expected for high frequencies, the gain is close to unity (-0.11dB = 0.987 @ 200kHz) and diminishes at low frequencies (-10.96dB= 0.28 @ 10kHz). The -3dB gain occurs at 33.9kHz.
Analyze and design higher order filters
Often, the transfer functions of higher order filters are quite complex. Utilizing the Y editor as a formula editor, the transfer function can be entered step by step, making it easier to handle the intricate mathematical expressions. For a double low-pass filter, the transfer function of Vout/Vin = 1/((RCi2πX)^2 +3*(RCi2πX) +1) is shown below. Deriving the correct transfer function took me half an hour, even in this case where R1=R2 and C1= C2. Utilizing the step by step method, I was able to enter the transfer function in just a few minutes
The first step in the process is calculating the complex impedance of capacitor C1 using Y1 of the Y-editor.
1: Y1=1/(i2πX*0.0047E-6), with X= the frequency (Hz), i= the complex unity vector.
(For Ti84, the symbol i=used for j). 0.0047E-6 Farad is the capacitor value of C1.
2:The impedance of resistance R2 in series with C2= is stored in Y1
3: Y3=parallel impedance of C1 and R2 in series with C2 : Y3=Y1*Y2/(Y2+Y3). You find Y1, Y1 with the key VARS- Y-Vars-Function-Y1 etc.
4:The voltage V1= Y3/(Y3+R)* Vin. Assume Vin = 1 V than V1=Y4= Y3/(Y3+R)
5: Vout= V1*Y1/Y2= Y5= Y4*Y1/Y2
6 : Disable Y1, Y2, Y3, Y4 Y5 (complex functions can not be plotted)
7 Plot Y6=20log(absY5)). Abs(Y5)= a real number
8: The -3dB line, represented by Y7 = -3, is employed for the calculation of the filter's bandwidth.
9:Y8, the analytically derived transfer function, is employed here to verify Y6."
The results demonstrate that the Y-editor is an invaluable tool for analyzing higher-order filters.
Finally, the filter is adapted to a wider Bandwidth utilizing the built-in numeric solver within the math menu by modifying the capacitor value. The "new" bandwidth remains consistent with example 1 of the first-order filter. Comparing the first and second order within a single plot demonstrates that the second order filter effectively attenuates higher frequencies compared to the first-order filter.
Analysis and design of a second-order low-pass Butterworth filter
The Butterworth filter is a type of filter intentionally crafted to exhibit a frequency response as flat as possible within the pass band. The most basic electrical schematic for a second-order filter is presented below. Numerous resources on the internet provide valuable insights into filters. For a second-order Butterworth filter, the transfer function is readily available for a 1Hz bandwidth. For R1=1kOhm and a bandwidth of f=33,872 Hz (bandwidth of the first-order low-pass filter example 1), the formulas for L1 and C1 are presented, leading to the values of L1 and C1 illustrated in the schematic. For more comprehensive information on Butterworth filtering, please refer to the following resource:
To illustrate the superior performance of the Butterworth filter over the first-order low-pass filter, both amplitude responses across the frequency spectrum are presented. The Butterworth filter consistently outperforms the simpler first-order filter throughout the entire frequency range.
Analysis and design of a second-order high-pass Butterworth filter
By employing the same method, a high-pass filter can be designed by swapping the positions of the capacitor and inductance in the electric circuit and by exchanging s by 1/s in the transfer. Refer to the attached electrical diagram for a visual representation. The complex transfer function of this filter can be expressed as: Vout/Vin = (i2πXL1C1)^2 / ((i2πXL1C1)^2 + i2πXL1/R1+1), where X represents the frequency (Hz) of the input voltage. The value of R1 is stored in memory R, the value of C1 is stored in memory C and L1 in memory L. For R1=1kOhm and a -3dB gain @ f=3,333 kHz, the formulas for L1 and C1 are presented, leading to the values of L1 and C1 illustrated in the schematic.
Again, the Butterworth filter consistently outperforms the simpler first-order filter throughout the entire frequency range.
Analysis and design of a second-order band-pass Butterworth filter
Finally, the band-pass, consisting of a low-pass filter in series with a high-pass filter, is illustrated. In an ideal situation, the filters in series should not influence each other. This can be realized by using an amplifier with very high input impedance and very low output impedance between the 2 filters. The electric diagram is shown. In this example, we use the second order Butterworth filters low-pass with bandwidth of 33782 Hz (LDF) and the high-pass filters with -3dB frequency of 3333Hz (HDF), already explained on this page. The transfer function of the band-pass filter is the product of the transfer function of both filters. Therefore, a band-pass frequency of 33.7kHz-3.3kHz=30.4kHz is expected. The component values can be calculated by the formulas below. The diagrams generated by the Ti84 are shown.
Logarithmic horizontal scale
Up to now, a linear scale is used in the examples. In practice, not only the vertical scale is logarithmic but, at most times, the horizontal scale is also logarithmic. If a logarithmic horizontal scale is used on the Ti-84, then the belonging frequency can not be read out immediately. For this problem, a solution is found as will be shown in the next example. To obtain a horizontal logarithmic scale, we substitute 10^X in the formulas instead of X which is shown in the next figures. The figures of a band-pass filter look much more common now.
Finally, an extra function Y8 =10^X is defined. This function gives the belonging frequency to the point which has been selected in the frequency diagram. Most times, Y8 is not visible in the diagram because of the high value.
In the figures above, the maximum of the band-pass filters is found at X (logarithmic) =4.03. Choosing the blue curve Y8 gives the belonging frequency of 10722,672 Hz.
Resume
On this page, an impression is given of the possibilities of the Ti-84 calculator. It appears a helpful tool at analyzing filters.