Series and Parallel Natural Frequencies

Hi,

I encountered a lecture note in the web discussing about an oscillator circuit using LCCC. It looks like in parallel configuration with one capacitor in parallel with the L,C,C in series. But the text says the L,C,C,C are all in series and continues quickly to give its natural frequency formula without derivation.

So I did the derivation for the series case, and indeed it matched the formula. Then I derived for the parallel case and found it gave the same result. I am perplex. Does this means we can compute for the parallel and the series in either way?

Generally resonance is the condition where the reactance of the coil is the same as the reactance by the capacitor, which is the same for parallel and series circuits.

If you want to try something interesting, calculate the voltage across the coil or capacitor in a series resonant circuit if they are being excited by their resonant frequency. Hint: don't forget the series resistance, since it likely be the current limiting factor.

kaosad,

Series resonant circuits have only one frequency for resonance. Parallel resonant circuits have three frequencies. If the Q of the paralled resonant circuit is high, then the three frequencies are close together. Also parallel resonant circuits are unique with respect to series resonant circuits in that they can be made to be resonant at all frequencies. Ratch

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/parres.html

It's only true to say that series resonant circuits have only one resonance frequency if just a single definition of resonance is considered--in this case, the frequency at which the current and voltage are in phase. But, at least one other definition of resonance is reasonable for series circuits.

The American Standards Association offers 3 definitions of resonance.

1 The frequency of zero phase angle.

2. The frequency of maximum oscillation of charge; maximum capacitor voltage.

3. The frequency of natural oscillation of current in the circuit.

To determine the frequency of natural oscillation of the series circuit, apply a step of voltage to the circuit. If R^2 is less than 4*L/C (but greater than zero), then a damped sine wave of current will be produced. The frequency of that damped sine of current will be:

SQRT(1/(L*C) - R^2/(4*L^2))/(2*PI)

The resistance has a definite effect on this frequency. See the applet at:

http://www.coilgun.info/mark2/rlcsim.htm

Move the R slider up and you can see the frequency of the damped sine wave change.

This site also shows the natural frequency for three cases:

http://home.san.rr.com/nessengr/techdata/rlc/rlc.html]

So we have a definition of resonance for a series circuit where the resistance has an effect on the resonance frequency. This will also be a fourth resonance frequency for the circuit at hyperphysics.

As for making the particular parallel circuit shown at:

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/parres.html

resonant at all frequencies, not all parallel circuits have that property. For example, the circuit with an inductor in parallel with a capacitor, in parallel with a resistance. There is no resistance in series with either the capacitor or the inductor; it's just in parallel with both. This circuit can't be made to look like a pure resistance at all frequencies.