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Approximation of RC or RL circuits.

RC circuit for Bode Analysis.

RC circuit for Bode Analysis.

We have learned some “high power” analysis of circuits containing capacitors and/or inductors using complex numbers and computers. However, sometimes you really don’t want to go to the computer immediately. It would be nice to have a way to approximate what is going to happen.  Back in the prehistoric days when people used slide rules, they had some methods to do this.  So… pull off your shoes while we walk five miles to school… in the snow and it is uphill … both ways.   Seriously, this is very easy stuff.

Computer Plot of the Gain of the circuit.

Computer Plot of the Gain of the circuit.

Imagine the circuit I have shown is connected to a amplifier input.  Way back there in “Capacitors and Alternating Current” we learned capacitors have a high reactance to low frequencies and have a very small current flow at low frequencies.  However they pass higher frequencies.  In the circuit we are using tonight this means capacitor effectively shorts out high frequencies so only the low frequencies show up in the next circuit.  A computer generated plot of the gain of this circuit is shown in the 2nd picture.   Basically it is a simple low-pass filter.

Before proceeding any further I need to define a couple of words the way we will use them.  The first word is Octave.  An Octave means double the frequency.  This turns out to be exactly the same definition used in music.  The “Oct” prefix is a word usually associated with eight as in Octagon, an eight sided regular polygon.   My only guess as to how it came to be associated with double the frequency is it is the 8th note on in a major scale.

The second word is Decade. In this context a decade is a multiple of 10.  This is different than the normal usage of a decade is 10 years.   So if our base frequency is 1, the first decade is 10, and the 2nd decade is 100, etc.

Application of some simple rules to the RC circuit.

Application of some simple rules to the RC circuit.

Now look at the marked up copy of the gain plot.  The vertical blue line marks the knee of the plot.  This is the point where Xc = R.  In this circuit I set that point at 1000 Hz.  Because we determined that the gain will fall off at higher frequencies our ideal line will be no loss at frequencies below the knee point or a gain of 0 dB.  At frequencies above the knee the gain will fall-off at a rate of 20 dB per decade.  This is shown by the diagonal line.

Now we will deal with the part where the quickly drawn curve doesn’t fit the computer generated curve.  The one point we know exactly is right at the knee frequency the curve will be 3 dB down.  The rest of the curve part we will have to free-hand, but a good fit is to make the free hand curve fit join with the two lines we have drawn at about 1/2 the distance on the graph of to the decade above and below the knee frequency.

Zoomed in graph at +/- 1 decade of the knee frequency.

Zoomed in graph at +/- 1 decade of the knee frequency.

The graph to the right shows a little closer detail how the curve acts in this region.

What about the phase shift information?   I am glad you asked!  The phase plot is just about as easy to draw.

The approximation of the phase plot.

The approximation of the phase plot.

 

 

To draw the phase plot, again the vertical line is drawn at the frequency where Xc = R.  At this frequency the phase shift will be 45 degrees as shown by the horizontal line on the graph.  Now it is time to get on the thinking cap.  Since the capacitor has a large reactance at low frequencies very little current will flow through the capacitor and it will have zero voltage shift across it.  This means we will draw a horizontal line at 0 degrees for the low frequencies below the knee point.   At high frequencies a large current will flow through the capacitor and it will create a phase change.  Since the total current is primarily affected by the resistor at these frequencies it will be in phase with the voltage input.   This means the voltage across the capacitor will lag by 90 degrees at high frequencies.  (Remember ELI the ICE man?)  We draw the -90 degree line at the frequencies above the knee frequency.

The diagonal line assumes that the complete phase change occurs from one decade below the knee frequency to one decade above the knee frequency.   This idealized curve does not fit the actual data as well as the gain curve, but it is probably good enough for a starting point.

So far this is not all that impressive, but it is going to grow very soon and as you will find it will be able to do some impressive stuff.  The next stop will be a circuit with two capacitors and two resistors.  Later we will talk about how this analysis can be helpful with feedback circuits.  (Op-amps — remember those?) You may want to try hand sketching a high-pass filter or a filter with an inductor instead of a capacitor.  The same principles work for all of those.  We have not yet talked about how to deal with both an inductor and a capacitor in the same circuit.

Who knows, maybe your Grandfather had things figured out while he was walking back and forth to school and had to think of something other than his frozen feet and that 5 mile walk up hill.

Gary

 

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