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Approximation of RC & RL circuits — Adding Complexity

The General "Ladder" Circuit

The General “Ladder” Circuit

Tonight I am going to expand on the simple circuit I used in the last post “Approximation of RC or RL circuits” to show you the real power of this procedure.  First, I created a general four component “ladder” circuit as shown in the diagram.  Depending upon how I choose my components in this circuit I can create a bandpass filter, a band-reject filter, or a low-pass, or high-pass filter.

Our purpose in this post is to learn the simple procedure, but since I needed to write the program to show the exact solution this is a good opportunity to show how I calculated that.  In the diagram the dashed lines show the partial calculations used.  I first summed Z3 + Z4 to get a value I called ZA.   I then paralleled ZA and Z3 to get a value I called ZB.   ZB was then summed with Z4 to get Ztot (for total Z).  I then reversed the process to calculate the Voltage across and the current through each component.   All of these calculations were done using complex number math.  Obviously, I was very glad I was using a computer to do all that!

The Actual Circuit to Analyze.

The Actual Circuit to Analyze.

The actual circuit I created is a bandpass filter. C1 and R1 together form a high-pass filter with a cut off frequency of 100HZ. C2 and R2 together form a low-pass filter with a cut off frequency of 1000 Hz. This same analysis would work with inductors instead of capacitors, but usually in electronics capacitors are used because they are much smaller and cheaper components.

The computer output of the gain calculations.

The computer output of the gain calculations.

The output of the computer calculations is shown. As a side note, I am in the process of creating some videos showing how I create these graphs. All of the software used is Open-Source software and it is all available for Linux, Windows, and Mac.

The Approximation method of determining the gain.

The Approximation method of determining the gain.

On the next picture I show using the approximation method discussed in the previous post. My two cut off frequencies are shown in the blue vertical lines. On the left side of the graph the red diagonal line shows the 20 dB per decade drop off for the C1, R1 combination for the low-pass filter part. On, the right side of the graph, the green line shows the 20 dB per decade drop off for the C2, R2 combination fo the high-pass filter part. The biggest misfit is at the top of the graph where the rounding of the curve from the low-pass filter does not get near the 0 dB point before the high-pass filter starts to work. However, I think careful drawing of those curves following the procedure outline in the previous post would also show that correctly.

Another bandpass filter but the C2 -R2 combination has a cut off of 10 KHz

Another bandpass filter but the C2 -R2 combination has a cut off of 10 KHz

Just due to curiosity and the fact that I wrote the program in a way that it could easily be modified, I plotted the same circuit with the high frequency cut off at 10 KHz so the two rounded off corners would not affect each other.

 

 

Phase Plot of the 100 HZ and 1 KHZ bandpass filter.

Phase Plot of the 100 HZ and 1 KHZ bandpass filter.

 

I also plotted and approximated the phase shift of the 100 Hz to 1 KHz band-pass filter. Again the same rules were used as taught in the last post. The red lines are the C1 & R1 combination and the green lines are the C2 & R2 combination. Obviously there is a large overlap between the two curves.  I am not sure that I would have guessed the phase change between the two frequencies would have been a straight line as I added the two curves.
We could continue on and add more and more of these elements to the circuit if we wanted higher cut offs.   If you think back to the op-amp circuit, we can also add elements to the feedback loop circuit and that is the area where we will go next.

Since I have the luxury of using a program that produces the logarithmic plot of the frequency, I forgot to mention it is possible to buy graph paper that has a log scale on the horizontal axis.  It is possible to do this with pencil and paper.  If I were doing this, probably all I would do is the gain plot on paper.  Then, once I obtained the gain plot I thought I wanted, I would then look at what is happening to the phase changes.  Before I started soldering components I would simulate this with a computer program.

My motto…. It is a heck of a lot easier to change on paper than it is in real life.

Gary

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