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3.22  Comparison of series and parallel LRC resonant circuits
Keywords: LRC resonator, series, parallel, Q, dual circuits
Topics covered in this posting
 The simple LRC circuit is widely used as the elemental model for resonances of all kinds. In this posting we compare the two types (series and parallel) of these resonators.
 We show that the graph of admittance versus frequency can be considered as "the resonance curve" for series LRC circuits. The graph of impedance versus frequency plays this role for parallel LRC circuits.
 Providing the parameters are properly adjusted, the two curves are identical.
 We summarize concepts of Q for these resonators.
 We show some of the possible variations in real, more complicated LRC resonators.
Contents of this posting
The most widely used simple model for a resonator is that of an LRC circuit. In this posting we compare various properties of the two simple forms of this circuit: the series circuit and the parallel circuit. In section 4 below we briefly look at other forms of the LRC circuit.
1. Elemental relationships for the two types of resonant circuits 

Fig. 1. Series and parallel LRC circuits. With the same component values, both circuits resonate at the same resonant frequency (if the effect of the resistor is ignored) ω_{0} = 1/√LC. As we shall discuss below, the resistance values required by the two circuits to achieve similar damping are very different. The two circuits can be considered electrical duals of each other meaning that they are governed by the same equations provided we exchange the roles of current and voltage (as well as L and C, and R and G). At left in Fig. 1, we see series and parallel LRC resonant circuits. Without the resistors, they are the same circuit. However with the resistors, they are different.

2. Resonant circuits in the weak coupling limit  

Series LRC circuit  Parallel LRC circuit 
For the series circuit shown in Fig. 2a, very weak coupling means decreasing R_{0} to a very small value, i.e. R_{0} << R.  In the parallel circuit on the right side of Fig. 2, very weak coupling means increasing R_{0} to a very large value, i.e. R_{0} >>R
Conductance is defined as the inverse of resistance: G = 1/R. 
To make the analysis easier, we convert the circuit in Fig. 2a into the equivalent circuit with an ideal AC voltage source as shown in Fig. 2. This equivalent circuit has the same value for R_{0} (still very small) and an amplitude of the ideal voltage source given by V = I R_{0} where I is the amplitude of the ideal current source of Fig. 2a. Given a fixed I, we see that the amplitude V decreases as R_{0} is reduced, meaning that weaker coupling via reducing R_{0} results in lessening the excitation of the resonator, as we would expect. In the limit of very small R_{0}, the series circuit has V applied across the LRC combination. In the limit of very small R_{0}, we can eliminate R_{0} from the circuit if we wish as is shown below in Fig. 3.  In the parallel circuit in the limit of very large R_{0} , the VR_{0} combination acts like an ideal AC current source with I = V/R_{0}. This ideal AC current source supplies current to the parallel LRC combination. We can redraw the circuit with only an ideal AC current source if we wish as is shown below in Fig. 4.

Fig. 3. Equivalent circuit to the first circuit of Fig. 2 in the limit of very weak coupling. 
(i)
(ii)
Fig. 4. Two circuits equivalent to the second circuit of Fig. 2. The first, (i), shows replacing the series VR_{0} combination by the equivalent parallel IR_{0} combination that can be done in all cases. The equivalent current amplitude is I = V/R_{0} where V is the voltage amplitude of the original circuit shown on the right side of Fig. 2. The second circuit (ii) is the equivalent circuit in the case of very weak coupling where R_{0} is very large and can be eliminated. 
Let's consider the effect of varying the frequency while keeping the source constant. In the series circuit shown in Fig. 3 the AC voltage amplitude across the resonator is fixed, set by the amplitude of the ideal voltage source, independent of whether the frequency of the AC voltage source is adjusted to the resonant frequency of the resonator or not.  The effect of varying the drive frequency of the parallel LRC resonator (shown in Fig. 4ii) is a little different. Here it is the AC current amplitude across the parallel LRC resonator that is fixed by the ideal current source. This current amplitude is independent of whether the frequency of the AC current source is adjusted to the resonant frequency of the resonator or not. 
The AC current amplitude of the series circuit is related to the fixed AC voltage V of the source by: I = V/Z = V Y , (11a) where Z is the impedance of the LRC series combination and Y is its admittance  The AC voltage amplitude of the parallel circuit is given by: V = I Z , (11b) where Z is the impedance of the LRC parallel combination. 
In the series circuit V is fixed, independent of frequency, but I varies via the standard resonance curve. I indicates the response of the resonator.  In the parallel circuit I is fixed, independent of frequency, but V varies via the standard resonance curve. V indicates the response of the resonator. 
In the series circuit, I is proportional to the admittance as per (11a) and varies in frequency as the admittance does.  In the parallel circuit, V is proportional to the impedance as per (11b) and varies in frequency as the impedance does. 
The admittance of a series LRC resonant circuit is:  The impedance of a parallel LRC resonant circuit is: 
Fig. 5. At the right, we graph both the admittance of the series LRC circuit and the impedance of the parallel LRC circuit. We have adjusted the scale so that both have almost the same peak value (slightly offset to show both curves), which involved multiplying the series admittance by 25. Both circuits have the same resonant frequency ω_{0} = 2.5rad/sec and same Q_{0} = 20. The component value are C = 0.08F, L = 2H, R_{parallel} = 100Ω, and R_{series} = 0.25Ω.  
How do the two above equations (12a) and (12b) produce identical graphs? The numerators of both are 1, so all we need to do is show that their denominators can be made equal using two "adjustable constants". The first adjustable constant stems from the fact that in order for the two circuits have the same Q, the resistances, i.e. the R's in the two equations will in general differ as explained in Eqns. (5)(7) above. Also admittance and impedance have different units and can never be said to be equal in a strict sense, so we can adjust the magnitude of one of the equations (multiplying by 25 as explained in the caption above).
In summary, we need to show that: where A is the global adjustable constant and R_{parallel} can be adjusted independently from R_{series} to insure the same Q_{0} in both circuits. Setting the Q_{0}'s from Equations (5) and (7) equal to each other and rearranging, we get the relationship: as the condition to insure equal Q_{0}'s in the two circuits. Note that we met this condition in Fig. 5 by choosing values of R_{series} and R_{parallel} which met the condition of (14). The real parts of the two sides of (13) will be equal if A is chosen such that A/R_{parallel} will equal R_{series}. We rearrange A/R_{parallel} = R_{series} to get: using (14) for the last substitution. Note that for the values we used in Fig. 5, A should be 25 which explains the 25 we used to multiply Y_{series} (equivalent to multiplying the denominator of Z_{parallel} ). Next we check the equivalence of the imaginary parts of (13). We start by factoring out C/L from the imaginary part of the left side of (13): Comparing (16) with (13), we see that the imaginary parts of the two sides of (13) are equal if AC/L equals 1, i.e. that A = L/C. We solve ω_{0} = 1/√LC for C = 1/(ω_{0}^{2}L) and substitute this into A = L/C to have: as the condition for the imaginary parts of the two sides of (13) to be equal. This is the same condition as (15) above for the real parts to be equal, which in turn is consistent with the condition the Q_{0}'s of the two circuits are equal. Note that one condition, Equation (14), stems from requiring that the Q_{0}'s of the two circuits be equal and allows A to be chosen so that (13) is true, i.e. that the graph of the admittance of the series circuit overlaps the graph of the impedance of the parallel circuit as shown in Fig. 5 above. 
3. Summary of concepts from an earlier posting concerning Q's:
Q's are most tightly connected to decay and bandwidth. If we measure either decay time or the bandwidth, we can accurately assess the loaded Q of the circuit at the time of measurement.
The loaded Q_{L} is related to the intrinsic Q_{0} via the equation:
where Q_{C} is the coupling quality factor (determined by R_{0} in the circuits in Fig. 2 above).
If we can reduce the coupling to a minimum and measure either the decay time or the bandwidth, we can determine the intrinsic Q, i.e. Q_{0}, provided we reduce the coupling sufficiently. Alternatively, if we could shock excite the resonator and totally remove the excitation source and its output losses, then we could measure the decay time in order to determine Q_{0}.
We can use the ratio between the stored energy and power dissipated inside the resonator (as we did in (5) above) to determine Q_{0} if we can determine both of these. Note that in the case where the excitation comes as a wave, the internal dissipated power is not always equal to the wave power that is incident on the resonator as discussed in postings 3.123.19.
While the coupling Q is easy to determine experimentally (measure the decay time or bandwidth with very weak coupling and then at the desired coupling) it may not be easy to determine the coupling Q based on math and the equivalent circuit. Our earlier derivation was done in the source free case, so to be safe (correct) one should use the free decay situation with the resonator excited but with the source turned off, while its output impedance is still there. This involves setting the voltage source to zero output volts, i.e. replacing the source with a wire.
4. Variations in real resonators
Many real resonators are not exactly equivalent to a simple series or parallel LRC resonator as discussed above. For example they may have their loss element (the R element) somewhere other than the locations discussed above. Often the loss may be due to resistance in the inductance itself, or in the capacitance, or due to leakage in the capacitance, or sluggishness of the inductor or capacitance, or due to electromagnetic radiation from the circuit. In most of these cases, we can model these losses with a resistor placed close as possible to the loss element.
Another possible variation from the simple circuits above is that the coupling may vary from that shown above. Instead of coupling across the whole resonator the coupling may be through part of the inductor or part of the capacitor, or magnetically through the inductor to name a few possibilities.
We show some possibilities in Fig. 6. Also there are resonators that do not even involve inductors, capacitors and resistors, but are still considered to be electronic resonators. These include cavity resonators, transmission line resonators, and physical effect resonators (such as quartz crystal resonators and yig resonators). Of course there are many, many types of mechanical resonators. For many of their properties we can model them as LRC resonators, keeping in mind some of their unique aspects.
The circuit equations will depend on the exact details of the resonator and coupling. In any case, approaching the resonator through the parameters of the Q's, through the power incident, power absorbed, stored energy, bandwidth, and decay time allows use of the more universal equations that we have derived here. In general, all resonators will display similar decay curves and resonance curves, although the resonance curves can differ somewhat from resonator to resonator particularly the offresonance part (the tails) of the resonance curves.
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