Resonances, waves and fields: 3.16a Electrical model of the acoustical circuit in Fig. 35 of posting 3.15

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3.16a Electrical model of the acoustical circuit in Fig. 35 of posting 3.15

Keywords: acoustic resonator, coupling iris, modeling, Q₀, coupling Q

Topics covered in this posting

This posting is the first step to mathematically analyze the first of the two resonant scattering animations in an earlier posting.
In this posting we develop an electrical circuit analog to the acoustical circuit in the original animation.
We also derive equations for the unloaded Q, i.e. Q₀ , and coupling Q, i.e. Q_C .

Contents of this posting

Modeling the coupling hole
- Equivalent coupling inductance
Modeling the acoustic circuit
- Eqn (3) - model of applied voltage
- Fig 3 - simplified circuit diagram
Q's of the above circuit
- Eqn (6) - coupling Q_C

1. Modeling the coupling hole
Fig. 1. Details of the resonator and coupling hole from Fig. 35 of a previous posting. The high velocity region in and near the coupling hole is similar to that of a Helmholtz resonator and has the Helmholtz or breathing mode. On the other hand, because the properties of the Helmholtz mode (resonant frequency, Q, etc.) are so dependent on the details of the hole for the resonance itself (as opposed to just the coupling), we focus on the next mode, the λ/2 mode (which is the first closed pipe mode) that can exist with or without the coupling hole and the hole for this mode mostly affects only the coupling. At the site of the resonator in Fig. 35 of the previous posting, incident waves travel upwards, hit a wall, and reflect back downwards. A small coupling hole in that wall allows the waves to excite the resonator and after the resonator is excited, for the resonator to radiate waves back out the coupling hole into the waveguide. The incident wave reflection process doubles the acoustic pressure at the wall and thereby doubles the excitation of the resonator. Waves traveling through the coupling hole see their acoustic velocity concentrated and increased over what they would have had in the waveguide. While the normal parameters to use with a plane wave in an open space are acoustic pressure p and acoustic velocity v, for an acoustic waveguide one uses acoustic pressure and acoustic volumetric velocity u = Av in units of m³/sec and where A is the cross sectional area of the acoustic waveguide. As discussed on Wikipedia the appropriate impedance to use with a waveguide is the characteristic impedance Z = p/Av = p/u . A coupling hole which is short compared with a wavelength acts primarily as a mass element in this acoustical circuit. At first glance, this may seem odd, since there is very little mass of air in the coupling hole compared with that in the wider waveguide. However the coupling hole acts similar to a pinch point (a constriction of lanes) in a traffic jam: it is the sluggishness of the cars in the pinch point, as few as they may be, that impedes the whole traffic flow for miles behind it and determines the throughput of the all these miles of traffic. We can derive the effective mass (i.e. inductance in the circuit analog) with some simple algebra. Starting with Newton's second law: , we use the fact that pressure is force divided by area: . (1) In the above equation len is the length of the coupling hole and ρ is the mass density (kg/m³) of air in the waveguide. The factor ρ len / A in the above equation plays the same role as an inductance L in circuit analysis (of electrical circuits) in the relation V = L(dI / dt) where the voltage V and current I are the potential and kinetic parameters, respectively. Thus we model the acoustic circuit in Fig. 35 by the electrical circuit in Fig. 2 below where a coupling inductor L_C plays the role that the coupling hole does in Fig. 35. Elmore and Heald (Amazon, Google books) on page 149 explains that the length len of the coupling hole should be extended by a factor 1.64a (where a is the hole's radius) to account for the high velocity region extending slightly past the length of the hole itself. Thus, the inductance to use is: . (2) where K = voltage/(acoustic pressure) is the conversion constant used to convert acoustical pressure into voltage. This conversion constant, although it is arbitrarily chosen, once it is chosen, the one value needs to be consistently used throughout a calculation or modeling.

1. Modeling the coupling hole

Fig. 1. Details of the resonator and coupling hole from Fig. 35 of a previous posting. The high velocity region in and near the coupling hole is similar to that of a Helmholtz resonator and has the Helmholtz or breathing mode. On the other hand, because the properties of the Helmholtz mode (resonant frequency, Q, etc.) are so dependent on the details of the hole for the resonance itself (as opposed to just the coupling), we focus on the next mode, the λ/2 mode (which is the first closed pipe mode) that can exist with or without the coupling hole and the hole for this mode mostly affects only the coupling.

At the site of the resonator in Fig. 35 of the previous posting, incident waves travel upwards, hit a wall, and reflect back downwards. A small coupling hole in that wall allows the waves to excite the resonator and after the resonator is excited, for the resonator to radiate waves back out the coupling hole into the waveguide. The incident wave reflection process doubles the acoustic pressure at the wall and thereby doubles the excitation of the resonator.

Waves traveling through the coupling hole see their acoustic velocity concentrated and increased over what they would have had in the waveguide. While the normal parameters to use with a plane wave in an open space are acoustic pressure p and acoustic velocity v, for an acoustic waveguide one uses acoustic pressure and acoustic volumetric velocity u = Av in units of m³/sec and where A is the cross sectional area of the acoustic waveguide. As discussed on Wikipedia the appropriate impedance to use with a waveguide is the characteristic impedance Z = p/Av = p/u .

A coupling hole which is short compared with a wavelength acts primarily as a mass element in this acoustical circuit. At first glance, this may seem odd, since there is very little mass of air in the coupling hole compared with that in the wider waveguide. However the coupling hole acts similar to a pinch point (a constriction of lanes) in a traffic jam: it is the sluggishness of the cars in the pinch point, as few as they may be, that impedes the whole traffic flow for miles behind it and determines the throughput of the all these miles of traffic.

We can derive the effective mass (i.e. inductance in the circuit analog) with some simple algebra. Starting with Newton's second law: ,
we use the fact that pressure is force divided by area:
. (1)
In the above equation len is the length of the coupling hole and ρ is the mass density (kg/m³) of air in the waveguide. The factor ρ len / A in the above equation plays the same role as an inductance L in circuit analysis (of electrical circuits) in the relation V = L(dI / dt) where the voltage V and current I are the potential and kinetic parameters, respectively.

Thus we model the acoustic circuit in Fig. 35 by the electrical circuit in Fig. 2 below where a coupling inductor L_C plays the role that the coupling hole does in Fig. 35. Elmore and Heald (Amazon, Google books) on page 149 explains that the length len of the coupling hole should be extended by a factor 1.64a (where a is the hole's radius) to account for the high velocity region extending slightly past the length of the hole itself. Thus, the inductance to use is:

. (2)

where K = voltage/(acoustic pressure) is the conversion constant used to convert acoustical pressure into voltage. This conversion constant, although it is arbitrarily chosen, once it is chosen, the one value needs to be consistently used throughout a calculation or modeling.

2. Modeling the acoustic circuit in Fig. 35 and the Q's of this circuit

Fig. 2. An electrical circuit which is equivalent to the acoustical circuit in Fig. 35.

At the right we see the electrical equivalent of the acoustical circuit of Fig. 35. We see three "legs" protruding out from a circulator in the center. Each leg is a transmission line of characteristic impedance Z₀ = R₀. The top left leg is connected to the wave source. The waves from this source propagate down the transmission line to port 1 of the circulator which sends them out port 2. At port 2 they travel to the right in the right most transmission line to the resonator. There they are partially absorbed and partially reflected. The reflected waves travel to the left, to the circulator's port 2 and are sent out port 3 to the third leg. In the third leg they travel down and to the left to the terminating absorber having resistance R₀, i.e. equal to the characteristic impedance of the transmission lines.

When viewed from the segment of the transmission line between points A and B, Fig. 2 above is similar to Fig. 3 below. The circulator mostly has the function of making sure there is a clean signal from the only source propagating to the right in the section between points A and B. The circuit to the right of point B labelled "load" (consisting of L_C and the resonator) has a a certain impedance, which presents a load to the waveguide to the left of point B. For the moment we will label this impedance Z_L.

In an upcoming posting (3.21) we will derive an equation for the voltage of a complex load at the end of a transmission line. Using Equation 6 of 3.21, the voltage at point B is given by:

, (3)

where we have replaced Z₀ with R₀ because we are assuming the characteristic impedance of the transmission lines is real and equal to R₀. We can now draw a simpler equivalent circuit for the waves at point B to replace the more complicated circuit shown in Fig. 2. This simpler circuit is shown in Fig. 3 below. Following (3), we have a voltage source given by V = 2V_source with a resistor of value R₀ in series with V and in series with the "load" from Fig. 2. We do this because the term in parentheses in (3) is just a voltage divider equation, giving the voltage across Z_L when it is in series with a resistor R₀.

Note that Fig. 3 makes sense in several limits.

One hypothetical limit is if Z_L = R₀ so the load Z_L perfectly terminates the transmission line and the incident wave is entirely absorbed. Since in this case there is no reflected wave, the total voltage at point B is just the incident signal V_source, consistent with (3). Note then that in Fig. 3, half the voltage drop from the source will be across R₀ and half will be across Z_L.
In the limit that Z_L >> R₀, then the entire 2V_source will be across the load, i.e. across Z_L. This is consistent with the fact that all the wave will be reflected in this case with zero phase shift and as such the reflected wave will add to the incident wave doubling the amplitude on the load.
In the limit that Z_L → 0, then all the voltage drop in Fig. 3 will be across R₀ and none will be across Z_L. This is consistent with the fact that all the wave will be reflected but in this case it will have 180 degrees of phase shift and as such the reflected wave will exactly cancel out the incident wave making the total voltage across the load zero.
These limits are portrayed in the animation in an earlier posting.

Fig. 3. The electrical circuit which is equivalent to the circuit of Fig. 2 above, from the point of view of the resonator.

At the left we see the equivalent electrical circuit of the acoustic circuit, from the point of view of the resonator. We model the waveguide and source with an ideal AC voltage source and resistor. The amplitude of the voltage source is twice that of the sound source in Fig. 40 (with whatever conversion factor we wish to use between sound pressure and electronic voltage). That is V_equivalent = 2V_waveguide. The factor of two allows an open circuit at the end of the waveguide to experience twice the waveguide source voltage (source voltage in an actual waveguide and not this simplified equivalent circuit), while allowing a perfectly matched load resistor at the waveguide end to experience a voltage equal to the actual source voltage.

The waveguide "resistance" R₀ is set equal to the equivalent acoustic impedance of the waveguide in Fig. 40.

3. Q's of the above circuit
To calculate Q₀ (the intrinsic Q of the resonator alone) and Q_C (the coupling Q) of our circuit (Fig. 3) we rearrange equation (125) of an earlier posting: . (4) The resonator is modeled as a parallel LRC circuit. The intrinsic Q, i.e. Q₀ (the quality factor if the LRC resonator were in a circuit all to itself but magically powered or in free decay) is given by: , (5) where ω₀ = 1/√LC and V_res is the voltage amplitude across the parallel elements of the resonator. The factor of ½ in the numerator is part of the formula for energy in a capacitor, an energy that shuttles between the capacitor and inductor during resonance. The factor of ½ in the denominator is due to time averaging the power loss in the resistor. This power loss oscillates between a minimum value of 0 and maximum value of I_Ro²R₀. The coupling Q_C is calculated by assuming that the source voltage is turned off (replaced by a short) and the resonator is magically excited by an ideal voltage source (zero output resistance). The coupling Q equals ω₀ times the energy in the resonator divided by the power loss in the coupling resistor: . (6) Fig. 4. Coupling Q and loaded Q versus the coupling inductor showing the effect the coupling has on these Q's. Both Q's have been divided by Q₀ (which equaled 20 in our example) and the coupling inductor was multiplied by the factor ω₀/R₀ . As expected, the coupling inductor L_C affects the coupling Q_C. Larger L_C means larger Q_C's and weaker coupling. This corresponds to a smaller coupling hole in the acoustic circuit of Figs. 35 and 1, consistent with (2) remembering that the hole's area A in the denominator of (2) is proportional to the hole's radius squared. Equation (6) is plotted at the right. On the vertical axis, we plot a normalized coupling Q_C defined as Q_C/Q₀ , i.e. (6) divided by (5) or Q_C/Q₀ = 1/β = (R₀/R)×(1 + x²) where x = ω₀L_C/R₀ . On the horizontal axis we plot x, the normalized coupling inductance. For the graph we assumed R₀/R = 1. To obtain normalized coupling Q's for other R₀/R values, multiple the value from the graph by R₀/R . We also plotted the normalized loaded Q i.e. Q_L/Q₀ . Note that a larger coupling inductor mean weaker coupling, a higher coupling Q and higher loaded Q. In a future posting (3.16c), we calculate the steady state equations for this equivalent circuit above and graph the results. There, we shall see that the resonant frequency is given by ω₀ = 1/√LC for high loaded Q's but at lower Q's the resonant frequency is affected by the coupling inductor L_C . Before we get to the entire circuit, we first examine just the resonator part in the next posting.

3. Q's of the above circuit

To calculate Q₀ (the intrinsic Q of the resonator alone) and Q_C (the coupling Q) of our circuit (Fig. 3) we rearrange equation (125) of an earlier posting:

. (4)

The resonator is modeled as a parallel LRC circuit. The intrinsic Q, i.e. Q₀ (the quality factor if the LRC resonator were in a circuit all to itself but magically powered or in free decay) is given by:

, (5)

where ω₀ = 1/√LC and V_res is the voltage amplitude across the parallel elements of the resonator. The factor of ½ in the numerator is part of the formula for energy in a capacitor, an energy that shuttles between the capacitor and inductor during resonance. The factor of ½ in the denominator is due to time averaging the power loss in the resistor. This power loss oscillates between a minimum value of 0 and maximum value of I_Ro²R₀.

The coupling Q_C is calculated by assuming that the source voltage is turned off (replaced by a short) and the resonator is magically excited by an ideal voltage source (zero output resistance). The coupling Q equals ω₀ times the energy in the resonator divided by the power loss in the coupling resistor:

. (6)

Fig. 4. Coupling Q and loaded Q versus the coupling inductor showing the effect the coupling has on these Q's. Both Q's have been divided by Q₀ (which equaled 20 in our example) and the coupling inductor was multiplied by the factor ω₀/R₀ .

As expected, the coupling inductor L_C affects the coupling Q_C. Larger L_C means larger Q_C's and weaker coupling. This corresponds to a smaller coupling hole in the acoustic circuit of Figs. 35 and 1, consistent with (2) remembering that the hole's area A in the denominator of (2) is proportional to the hole's radius squared.

Equation (6) is plotted at the right. On the vertical axis, we plot a normalized coupling Q_C defined as Q_C/Q₀ , i.e. (6) divided by (5) or Q_C/Q₀ = 1/β = (R₀/R)×(1 + x²) where x = ω₀L_C/R₀ . On the horizontal axis we plot x, the normalized coupling inductance. For the graph we assumed R₀/R = 1. To obtain normalized coupling Q's for other R₀/R values, multiple the value from the graph by R₀/R . We also plotted the normalized loaded Q i.e. Q_L/Q₀ . Note that a larger coupling inductor mean weaker coupling, a higher coupling Q and higher loaded Q.

In a future posting (3.16c), we calculate the steady state equations for this equivalent circuit above and graph the results. There, we shall see that the resonant frequency is given by ω₀ = 1/√LC for high loaded Q's but at lower Q's the resonant frequency is affected by the coupling inductor L_C .

Before we get to the entire circuit, we first examine just the resonator part in the next posting.

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