Yeah, I know it is a strange title! Actually there is a good reason of this title. Traditionally a parallel LC circuit is called a tank circuit because it is similar to water sloshing in a tank. The twisted sister part has to do with the duality principle I have talked about since we started talking about electrical theory. The idea is to learn only one half of the theory very well and remember the other half is similar but different.

Tonight we are going to talk about the parallel LC circuit shown above and make an analogy to the series LC circuit we recently discussed in “Two four six eight we are going to resonate“. As before I chose the values of L and C to both produce a reactance of 1000 Ω at 100 Hz. Before going any further, I do want to tell you I “cheated” in this analysis. The formula for parallel impedances is Z = (Z1 * Z2) /(Z1+ Z2). The probably came about at 100 HZ where Z1= ZC = (0-1000j) and Z2 =ZL= (0+1000j) and the sum of those is 0. Computers do not like 0 in the denominator and the program stopped and informed me that I am an idiot. (Actually it said “Divide by Zero Error”.) My work around was to add a resistance of 0.000001 Ω in series with the inductor. In the calculations this has no effect, except to make the computer happy. (A happy computer is a very good thing!)

When we graphed the total current in the series RLC circuit, the Xl and Xc cancelled and we reached the maximum current flow when the frequency in was at the resonance frequency of 100 HZ. So in keeping with our twisted sister theme, what should we expect the current to do in the parallel LC circuit?

As expected the twisted sister has to do things differently. The parallel LC circuit blocks the current at resonance, but conducts at well at frequencies above and below resonance. It is relatively easy to understand why it conducts off frequency because above resonance the capacitor becomes a good conductor and below resonance the inductor conducts well. We will get to an explanation of what happens at resonance soon. However, before that lets look at some other ways these are “similar but different.”

In the plot to the right I show the voltage across the inductor and capacitor in the series LC circuit. The next plot will be current through the parallel LC circuit.

The plot looks very similar to the above plot… but different. The obvious difference is we are looking at current and not voltage. In both cases the Inductor value is in black, and the capacitor value is in red and “the one on the left is on the right”. (Old folk-rock song title.) So as you can see duality is kicking in high gear on this analogy.

Ok, by now I have wore out the duality thing, so lets talk about why the circuit does not conduct at the resonance frequency.

This plot will help to explain what is going on. The first thing to notice is the current in the two devices is always 180 degrees out of phase. This means that while current is flowing out of one component it is flowing into the other component. Now also remember as we increase in frequency above resonance the capacitor will decrease in reactance and the inductor will increase. The stored energy balance between the two will not be equal so the excess energy can flow into the rest of the circuit.

The next picture shows the phase of the voltage across the inductor and capacitor. At resonance the phase is zero or exactly in phase with the voltage input. As expected above and below resonance either the capacitor or inductor will become dominate and shift the voltage accordingly. Meanwhile as show in the above graph the 90 degree phase shift will be maintained between current and voltage in each component.

To summarize it all: At resonance all of the current is being used back and forth between the two tank circuit components and it develops a voltage across the components that is exactly equal and in phase with the voltage of the source. So no current will flow.

In real life tank circuits do have internal resistance so some energy must be supplied. Tank circuits are the workhorses of radio communications and often act as the primary circuit of a air core transformer to drive the transmitting antenna. The natural resonance is a big part of what sets the frequency of the transmitter. Similar to pushing a child in a swing once the tank circuit is oscillating all it requires to keep it going through the cycle is a push at either the top or the bottom of the cycle. Most transmitter circuits only are “on” and conducting during this brief period. Another analogy is the tank circuit acts like a flywheel of an engine. The circuit keeps itself operating through out most of the cycle and the piston only “fires” during part of the cycle. In the case of the Tubes (valves for our British friends) or Transistors this means the total power dissipation in the electronics is much lower since it operates at full on or full off and very little partly on. Partly on wastes a lot of energy within the device and creates damaging heat within the device.

As always I hope you enjoyed my twisted way of presenting what could very well be some very dry material. We are getting very close to the end of pure theory and will very soon be getting back into application of the theory.

Gary

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