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Parallel Resonance Circuit and decibels.

Parallel LC "tank" circuit.

Parallel LC “tank” circuit.

In the post “Series RLC Resonance – The effect of Changing R“, I talked about the Quality factor, Q.  Q was defined as:  Q= 2*π * energy stored per cycle / energy dissipated per cycle.  Just like I did in that post we will take a look at what happens as we change the value of R. However, I have learned a thing or two and we only have to look at two graphs this time.  Since this post would be relatively short, I am also going to describe the unit the decibel.

First I probably need to introduce a new term.  In the graphs I have been producing for these talks about resonance I have been displaying frequency on the X axis. The term for this type of plot is:  “We are in the frequency domain.”  While in the frequency domain we are looking at sinusoidal waves being introduced into the circuit.  To do this I am plotting the output for a known input at various frequencies.  The frequencies are being increased by a constant multiplier from the previous frequency and that is why the frequency is displayed on a logarithmic scale.  This provides us with some very useful information on how the circuit operates at various frequencies.

However,  sometimes it is very useful to know what happens when we run a short pulse into the circuit.   That type of analysis is called transient analysis and is very important in control systems.  I will have to write a little different program to show what happens in one of these circuits when that occurs.   However, I have several important concepts to introduce while we are still in the frequency domain.   In other words we are at another junction. Eventually the two paths will cross again, but it makes sense right now hold the same path we are on.  However, good things are coming!

The current through the Parallel LC circuit for several resistance loads.

The current through the Parallel LC circuit for several resistance loads.

In the time since the series RLC circuit I have got a little smarter with the software and this time I am plotting all of the outputs on one graph.  I did have to change the voltage input into the circuit to maintain the same current at the while the circuit is not at resonance.  For example, when I increased R from 1K to 10K, I had to increase in voltage in from 1 V to 10 V to maintain the same reference current of 1mA.  This made a more meaningful plot.  As you can see in the graph as R increased the cutoff became sharper. The Q of the circuit was increased because more energy was stored in the reactive components at the resonance frequency,

The Current Phase shift Plot for various size resistance loads.

The Current Phase shift Plot for various size resistance loads.

In this plot I show how the current phase shifts at resonance. I did not plot enough points to show the full effect at the higher resistances because the phase change happens over a much smaller range of frequencies for the higher resistances.  However, there are enough plotted to give you the idea.
Often we are interested in how much the power output in a circuit is increased or decreased from the previous value due to a change.  As we already know, Output / Input = Gain.   The Output and Input are often voltages, so the gain would be voltage gain in that case.   However, we could be comparing power in and that would give us power gain.

As people became interested in the losses in telephone cables, they determined the ear seemed to perceive an increase in audio power by 10 to be twice as loud.  In other words ears seem to be sensitive to a the logarithm of the audio power.  The unit of 1 Bel was defined to be equal to a power increase 10.  Bel = log base 10 of (Actual power / A reference power).   A decibel or dB = one tenth of a Bel  so dB = 10*log base 10 of (actual power / reference power).  In spreadsheets and computer programs this is usually written as log( ) while the natural log is written ln().   Usually in the case of amplifiers and filters the reference power is the power input.   If the calculation returns a positive number then we have a dB gain.   If the calculation returns a negative number we have a dB loss or attenuation.    If the input and output see the same resistance the calculation is 20*log base 10(Vout/Vin).   (For a little more about the logarithm please refer to a previous post “Exponentials and Logarithms“.)

In filter circuits the -3 dB point is considered an important point because at -3 dB power has dropped by 1/2.   In audio devices the frequency response is often specified in dB. For example the frequency response of some speakers may be specified something like 44Hz to 16 KHz +/- 3 dB.   Note the use of +/- in that case:  They are applying a little “specmanship” to make the speaker look better than if they just used the – 3 dB drop off point.

Some other dB values to commit to memory are:  +3 db means a doubling of power;   +10 dB means power has increased by 10 times; and a -10 dB means power has decreased by a factor of 1/10.  Adding db values are the same as multiplying the actual power increase/decrease.  For example,  my father is a ham radio operator, he chose to spend his money on a directional antenna, a beam, instead of purchasing a power amplifier for his transmitter.  His reasoning is he can get more effective power output that way.   His transmitter puts out 120 watts of power.  If he puts up a directional antenna that has a 12 dB gain over the standard antenna, a dipole, he has:

12 db = 3 + 3 + 3 + 3 db gain and that is equivalent to 2 X 2 X 2 X 2 or 16 times increase in effective power.  16 X 120 = 1920 watts of effective power and the maximum amplifier a ham can use is 1000 watts.  Plus, he also attenuates signals received that are not from the one direction.

The dB plot of the attenuation of the parallel LC circuit.

The dB plot of the attenuation of the parallel LC circuit.

The final purpose of using dB is the fact that since it is based on logarithms the numbers are more reasonable to use in a graphical plot.  The graph to the right shows the db attenuation of our parallel RLC circuit over the frequency range.

 

An expanded plot of the Parallel LC showing the attenuation in the pass band.

An expanded plot of the Parallel LC showing the attenuation in the pass band.

 

In the final plot I zoomed in on upper part of the previous plot to show how the 100 Ohm resistor case even has a 3 dB loss at 10 Hz and 1000 Hz.

We are now at the point where I can explain some of the additional data on the op-amp spreadsheet and will be doing that very soon in a upcoming posts.   We have come a long long way.

As always in sincerely hope I have made this worth your time.

Gary


 

 

 

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