LC - RLC - RESONANCE
PUTTING IT ALL TOGETHER
SERIES LC AND RLC
In a series circuit, the current through all components will be the same (Kirchoff's current law, sometimes called the first). Both components will impede the current and both are reactive so reactances can simply be added.
XT = XL + XC ... OR ... XT = ωL - 1 / ωC
At some frequency the magnitude of both will be the same, ( ω L = 1 / ω C OR -ω L = -1 / ω C ) and the total reactance will be zero. Using simple algebra, the formula for resonant frequency can be given as:-
ω = 1 / √LC ... OR in Hz ... f = 1 / 2Π√LC
Because the reactance of an inductor increases with frequency, above resonance, most of the reactance will be in the inductor and the resulting reactance will therefore be INDUCTIVE.
The reverse is true below resonance and the circuit will appear CAPACITIVE.
NOTE:- This is the same with an antenna. Above resonance it will appear too long and be inductive. Below resonance it will appear too short and be capacitive. With the addition of radiating resistance, an antenna behaves exactly the same way as a series RLC circuit.
If the applied voltage is removed, energy will continue to bounce back and forth between the capacitor and inductor. Remember the reactance (or impedance) of a series resonant circuit is 0. There is therefore a virtual short circuit between them whether they are connected or not.
ADDING A RESISTOR - IMPEDANCE AT RESONANCE
The easiest way to calculate the impedance is to first consider the reactive components. At resonance, the inductance and capacitance represent zero reactance (and thus impedance) so the impedance at resonance will simply be the value of the resistor.
Above or below resonance, the total reactance will be non-zero and be either a capacitive or inductive. This can be worked out using the formula above.
|Z| = √ ( (XL + XC )2 + R2 )
At a non-resonant frequency or phase angle Ø ≠ Π/2 and Ø ≠ Π/2 and Ø ≠ 0. Above resonance 0 < Ø < Π/2 and below resonance -Π/2 < Ø < 0
At resonance (XL + XC ) = 0 so Z = √ R2 or simply R.
The phase angle will be arctan (reactance/resistance). That is, in this case ARCTAN ( ( XL + XC ) / R ). At resonance the phase angle will be ARCTAN ( 0 / R ) = 0
PARALLEL LC - RLC
The circuit shown above is commonly referred to as a tank circuit. At some frequency, such a circuit consumes almost no power because energy stored in one half a cycle in one component, is transferred to the other component in the other half cycle. This frequency is where the impedance of the capacitor and inductor are equal in magnitude and exactly opposite in sign. Any resistance in the wires and coil will be where any power is consumed. If the components are superconducting (no resistance) and somehow the laws of physics could be bypassed and the circuit doesn't produce EM, the circuit will resonate forever with no applied voltage other than to get it going.
The resonant frequency will be calculated in the same way as for the series circuit.
|ω = 1/ √ LC ... or in Hz ... f = 1 / 2Π√LC|
PARALLEL IMPEDANCE AT RESONANCE
The current through both will be determined by the applied voltage. IL = V / ZL and IC = V / ZC. These two currents will be Π out of phase but using the convention of adding -1 used here, they can be added thus IT = IL + IC (because IC will be a negative quantity.)
The total impedance of the circuit is now ZT = V / IT or ZT = V / ( V / ZL + V / ZC ). All those "V"s are a little annoying so simplifying we get simply ZT = 1 / ( 1 / ZL + 1 / ZC )
Substituting ZC and ZL we get:-
ZT = 1 / ( 1 / ωL + 1 / ( -1 / ωC ) ) ... OR ... ZT = 1 / ( 1 / ωL - ωC )
By inverting fractions, multiplying and complication we get:-
ZT = ωL / ( 1 - ω2LC ) ... OR ... ZT = -ωL / ( ω2LC - 1 )
|ADDING A RESISTOR|
The impedance of this circuit can be worked out from the reactive and resistive components for any frequency but resonance is the only frequency of interest. At resonance, the combined reactance of the capacitor and inductor is infinite. The only component left to conduct current is the resistor.
Above resonance the capacitor will conduct most of the current. The circuit will therefore appear capacitive. This is the opposite to the series circuit. Below resonance, the inductor will conduct most of the current and the circuit will appear inductive, again the opposite to a series circuit. At resonance the impedance will be purely resistive because the combined reactances don't conduct any current.
The circuit above is not typical though. That shown at right is more typical. Generally the resistance is in series with the inductor and is a result of the resistance of the wire the inductor is made of. This presents other problems.
There are now two resonant frequencies to consider. If a voltage is applied, the circuit will have a resonance. This will be the frequency where the impedance is purley resistive, that is, at a maximum. This is often referred to as the "driven resonant frequency".
There is also a "natural resonant frequency". The circuit will contain energy stored in the reactive components and so will continue to oscillate for a few moments after the applied voltage is removed until that energy is dissipated in the resistor. This will be the frequency where the capacitive reactance is equal to the inductive reactance. See previous page R - X - C.
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