There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.

Origins of Newton's laws of motion

History of mechanical clocks with animations
Understanding a mechanical clock with animations
includes pendulum, balance wheel, and quartz clocks

## Thursday, July 18, 2013

### 3.22 - Comparison of series and parallel LRC resonant circuits

 All postings by author previous: 3.21 Reflection and absorption coefficients for transmission lines with loads up: Contents

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.

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) ω0 = 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.

• In the case of the series circuit, the quantity that all three elements of each circuit share is the current I . In the case of the parallel circuit, the common quantity is the voltage V  .

• The energy stored in the series circuit is:  U = ½|I |2L     (1)
and that in the parallel circuit is:    U = ½|V |2C   ,     (2)
where I and V are the amplitudes of the oscillating current and voltage in the two circuits.

• The power loss in the series circuit is:  P = ½|I |2R     (3)
and in the parallel circuit:     P = ½|V |2/R   ,     (4)
both averaged over a cycle.

• Using the equations just above we get that in the series circuit:
Q0 = ω0U/P = ω0L/R    ,    (5)
while in the parallel circuit:      Q0 = ω0RC     .    (6)

• Solving ω0 = 1/√LC  for C  and substituting this into Q0 for the parallel circuit, we get and alternate expression for Q0 for the parallel circuit:
Q0 = R/ω0L     ,    (7)
Note that (7) is the inverse of equation (5) for Q0 for the series circuit.

• We see that in the series circuit Q0 is inversely proportional to R and high Q0's require small resistances, i.e. R << ω0L . In parallel circuits, the criteria is reversed, Q0 is directly proportional to R and high Q0's require large resistances, i.e. R >> ω0L .

• The decay time constant and bandwidth will, for both circuits, be given by the same equations if the equations are expressed in terms of Q's:

τ = QL/ω0   (8)      and   Δω½/ω0 = 1/QL     (9)

where      1/QL = 1/Q0 + 1/Qcoupling   .    (10)

Fig. 2. Series and parallel LRC circuits with AC sources.
• To provide steady state excitation in the series LRC resonant circuit one normally breaks the circuit and places a sinusoidal source into the break as shown in Fig. 2. Fig. 2 shows this source as being composed of usual ideal voltage source plus an output resistor. An alternative model for a real source would be an ideal current source in parallel with a leakage resistor.

In the parallel circuit one normally places a sinusoidal source in parallel with the other elements as also shown in Fig. 2. Again we show a ideal voltage source in series with an output resistor.

• To understand coupling strength in the parallel LRC circuit, it is most instructive to start with an ideal voltage source plus output resistance as is shown on the right hand side of Fig. 2. With this combination, weaker coupling means larger values of R0. This causes both the excitation delivered to the LRC resonator to be weaker AND the damping due to R0 to be less, both as we would expect from weaker coupling of source and resonator.

Fig. 2a. Series LRC circuit with AC current source plus leakage resistor.

To lessen the coupling with the series LRC resonator (shown on the left side of Fig. 2) we need to reduce the resistance of R0 . This will lessen the damping due to the resistor and raise the QL. So we see that a smaller R0 does cause reduced coupling in a technical sense. However a smaller R0 also increases the excitation of the resonator, something that is normally not associated with reduced coupling.

A more instructive approach for treating the series resonator is to start with a parallel ideal AC current source and leakage resistor as shown in Fig. 2a. With this approach, reducing R0 reduces both the damping and the excitation of the resonator and so is more obviously the route to reduced coupling between source and resonator.

• Next, we analyze response of the resonator by itself by considering the circuits in the limit of extremely weak coupling:

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 R0 to a very small value, i.e. R0 << R.

In the parallel circuit on the right side of Fig. 2, very weak coupling means increasing R0 to a very large value, i.e. R0 >>R
OR alternately stated, weak coupling means a very small conductance: G0 << G.

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 R0 (still very small) and an amplitude of the ideal voltage source given by V = I R0 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 R0 is reduced, meaning that weaker coupling via reducing R0 results in lessening the excitation of the resonator, as we would expect.

In the limit of very small R0, the series circuit has V  applied across the LRC combination. In the limit of very small R0, we can eliminate R0 from the circuit if we wish as is shown below in Fig. 3.
In the parallel circuit in the limit of very large R0 , the V-R0 combination acts like an ideal AC current source with I = V/R0. 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 V-R0 combination by the equivalent parallel I-R0 combination that can be done in all cases. The equivalent current amplitude is I = V/R0 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 R0 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:

.     (12a)

The impedance of a parallel LRC resonant circuit is:

.     (12b)

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 ω0 = 2.5rad/sec and same Q0 = 20. The component value are C = 0.08F, L = 2H, Rparallel = 100Ω, and Rseries = 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:

,    (13)

where A is the global adjustable constant and Rparallel can be adjusted independently from Rseries to insure the same Q0 in both circuits.

Setting the Q0's from Equations (5) and (7) equal to each other and rearranging, we get the relationship:

,    (14)

as the condition to insure equal Q0's in the two circuits. Note that we met this condition in Fig. 5 by choosing values of Rseries and Rparallel 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/Rparallel will equal Rseries. We re-arrange A/Rparallel = Rseries to get:

,    (15)

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 Yseries (equivalent to multiplying the denominator of Zparallel ).

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):

.    (16)

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 ω0 = 1/√LC for C = 1/(ω02L) and substitute this into A = L/C to have:

,    (17)

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 Q0's of the two circuits are equal. Note that one condition, Equation (14), stems from requiring that the Q0'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 QL is related to the intrinsic Q0 via the equation:

,    (18)

where QC is the coupling quality factor (determined by R0 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. Q0, 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 Q0.

• We can use the ratio between the stored energy and power dissipated inside the resonator (as we did in (5) above) to determine Q0 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 3.12-3.19.

• 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.

 Fig. 6. Some possible configurations of LRC resonators. High Q requires that component values be picked to minimize the power loss in the resistance elements and maximize the energy stored in the L and C parts of the circuits.

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 off-resonance part (the tails) of the resonance curves.

 All postings by author previous: 3.21 Reflection and absorption coefficients for transmission lines with loads up: Contents