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3.16c Steady state response of the circuit: transmission line, coupling inductor and resonator

**Keywords:** resonator, steady state, ac circuit analysis, coupling inductor

**Topics covered in this posting**

- The entire equivalent circuit including resonator, coupling inductor, transmission line impedance, and source is mathematically analyzed.
- The results of the analysis is graphed: the resonator response as a function of coupling inductor value.
- A trick is used to quickly analyze the resonator response at unity coupling.
- The size of the coupling hole in the acoustical circuit as required for unity coupling and efficient coupling of the waves into the resonator is determined.

**Contents of this posting**

- The circuit
- The equations
- Graphs of the equations
- Simple calculations of response at unity coupling
- Coupling hole size for acoustical circuits of Fig. 35 of posting 3.15 for unity coupling

16c. Steady state response of the circuit in Fig. 42 of the posting 3.16a | |||||||||
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In the previous posting, we have considered only the resonator part of our equivalent circuit, labeled as "resonator" in Fig. 1 at the right. In this posting, we consider the whole circuit in Fig. 1, including the coupling inductor, driving voltage source, and its output resistance. The "output resistance"
We start with the fact that the current flowing from the wave source equals that flowing through the total impedance of the rest of the circuit as presented to the wave source, V/_{S}Z where _{total}Z = _{total}R_{0} + jωL + _{C}Z and _{resonator}Z = _{resonator}Z from Eqn. (1) of the previous posting_{res}:
The source voltage R_{0}, L and the resonator as a whole._{C}The voltage generated across the resonator alone indicates the excitation level of the resonator. We call this voltage I times the impedance of the resonator (or divided by the admittance of the resonator):_{S}
Fig. 2, at the right, plots the voltage across the resonator ω of the driving sound source, for three different values of coupling inductors L (in Henries)._{C}We notice two effects that changing the coupling inductor causes: - shifts the resonance curve and
- also broadens it.
To be quantitative concerning the first effect, we see that L = 7.82Henries for unity coupling, i.e. maximum power into the resonator. The previous calculation also indicated that the resonant frequency at unity coupling would be 2.79radians/sec which agrees with the green curve in Fig. 2._{C}As to the second effect, the broadening of the resonance curve in Fig. 2 for smaller coupling inductor Q's as indicated by the graph in Fig. 4 of posting 3.16a._{L}Fig. 3 plots the resonator amplitude versus the coupling inductance assuming that the drive frequency is adjusted at each value of inductance for the maximum response of the resonator, i.e. for the peaks shown in the last graph (Fig. 2). The graph in Fig. 3 was made using an approximation to calculate the optimal frequency for each coupling inductance. We started each calculation assuming a frequency of ω_{0}L . We then set _{C}X equal to the negative of the imaginary part of Eqn. (2) of the last posting, assuming that at the optimal frequency, the reactance of the coupling inductor would null out the reactance of the resonator. We then used the quadratic formula to calculate _{C}δ and used the approximation in the first part of Eqn. (9) to calculate ω i.e. used ω ≅ ω_{0} + ½ω_{0}δ . We then used this ω in (1) and (2) to calculate V for the particular inductance value._{res}Fig. 4, at the right, shows a more precise calculation of the same quantity as the previous graph showed. For this graph, we used Octave to find the peak value of V for each inductance value and plotted these peak values versus the coupling inductance. We see that this "exact curve" roughly agrees with the approximate plot, although the agreement could be better. The previous two graphs were made with gnuplot.
_{res}
In general, it is messy to calculate the voltage across the resonator at the resonance peak. We would need a formula for the frequency which would involve taking the derivative of the magnitude of (2) (which involves (2) ) with respect to the frequency On the other hand, at resonance and critical coupling ( The incident power traveling from the source towards the resonator is: where 30V is the amplitude we used for V of the ideal voltage source in our modeling circuit shown above in Fig. 1. (V is the symbol for unit of voltage, i.e. for volt, and W is the symbol for watt, the unit of power.)_{S}At unity coupling, all the incident power is absorbed by the resonator. Since the only dissipative element in the resonator is the resistor Solving for which agrees with the peak value of the green curve in the Figs. 3 and 4 above.
To be more concrete, we now relate the above to actual physical dimensions of the acoustic circuit of posting 3.15. The details of the coupling hole are shown in Fig. 1 of posting 3.16a. We start with the equation for the equivalent inductance (eqn. 2 of 3.16a) of a small hole: Solving for where we have taken the limit for holes of very short lengths. Also ρ is the density of air (in kg/m^{3}) and K is the equivalence factor we choose between the acoustical circuit and electrical circuit.We next change this into a dimensionless quantity Z_{0}:, (7) where the equivalent characteristic impedance of an acoustical transmission line is Using the circuit parameters in the caption of Fig. 2 above, we calculate this same normalized coupling inductance for the electrical circuit of our graphs above at the optimal coupling: Equating (7) and (8) we get: To give a concrete example, in the last line we assumed We see that to produce the condition for optimal coupling (unity coupling) we need to restrict the opening to the resonator to a radius of about a third of that of the acoustical transmission line. This means that the cross sectional area of the coupling hole is about one tenth that of the transmission line. In spite of its relatively small size, This exercise is intended to demonstrate the power of a resonator. The higher the |

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