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3.16b Looking at the resonator part of the circuit
Keywords: resonator, impedance, matching, unity coupling, signal transformer, middle ear
Topics covered in this posting
 In this posting we restrict our analysis to the resonator part of the equivalent electric circuit.
 An equation for the impedance of the resonator is derived.
 This impedance is graphed in a few different ways.
 Using the graphs we discuss a possible strategy for coupling to the resonator, using a coupling inductor to efficiently match the waveguide's characteristic impedance to that of the resonator.
 The required inductance for matching is algebraically derived, as well as the new resonant frequency, shifted because of the coupling to the transmission line.
 The required inductance value and resonant frequency is graphed versus the characteristic impedance of the transmission line.
 We discuss the problem of matching a particular range of characteristic impedance values.
 The problems of coupling in this range of characteristic impedance values can be overcome by using a transformer instead of coupling inductor. This is related to the mechanical signal transformer inside the human ear.
Contents of this posting
1. The impedance of the resonator  


2. Getting to unity coupling, i.e. a perfect match  

a. Graphical solution A perfectly matched load for the transmission line means that all the waves launched towards the load will be absorbed and none reflected. In resonator language this means unity coupling. In Fig. 4 at the right, the vertical dotted line indicates the needed drive frequency (2.78 radians per second) to have the real part of the impedance equal 5Ω (the characteristic impedance of the transmission line that we will be using) and so be perfectly "matched" to the transmission line. Perfect matching also requires that we set the coupling inductor to an impedance ωL_{C} = 22Ω to cancel the negative reactance of the resonator at ω = 2.78 radians per second. A different coupling inductor will result in undercoupling or overcoupling. The frequency of 2.78 radians per second will be a new resonant frequency when we have perfect match. Changing the inductor's value from that required for perfect match will shift the resonant frequency as we shall see in the next section. b. Algebraic/computational solution To algebraically calculate the required impedance and resonant frequency of Fig. 4 we follow the same steps algebraically as the graphical ones above:
To begin, we separate out the real and imaginary parts of Eqn. (1):
, (2) where detuning factor, δ , is given by: and the reactance of the L or C inside the resonator, X_{res} , is given by: We can use the quadratic formula to solve (3) for drive angular frequency, ω , in terms of the detuning factor δ : To find the detuning needed for unity coupling, we set the real part of the resonator impedance Re{Z_{res}} = R_{0} = Z_{0} , i.e. the characteristic impedance, assumed real, of the transmission line to be used: and solved for δ to give the detuning factor required for unity coupling: where everything on the right side of the equation is known. The detuning faction, δ, from (6b) can be substituted into (5) to yield the resonant frequency of the circuit with unity coupling. To calculate the coupling inductance required for unity coupling we set the magnitude of the second term of (2), the imaginary part, equal to X_{Lc} = ωL_{C}: and solve for the L_{C} required for unity coupling: where everything on the right side of the equation is known. If we use (9) and then (7b) with the parameter value given in the caption of Fig. 1, we calculate the approximate values of ω = 2.77rad/secs and L_{C} = 7.86H. An exact solution is had by:
With this procedure, we get:
Graphs of the equations just above are shown below. 
4. Signal transformers and the ear  

How does one match a resonator or load in the case that the transmission line impedances are greater than the resistance of the resonator? The standard electrical engineering solution is to use a signal transformer, 2nd ref. This is a magnetic component, similar to an inductor, that employs two coils. It is similar to a power transformer used to change the AC voltage in the power grid. A signal transformer is usually much, much smaller than a power transformer and is used to change the impedance of an AC source. When used to match impedances it is placed at the site of the load and its turns ratio is set to change the impedance of an incident wave on a transmission line from the transmission line's impedance to the load's impedance. Signal transformers were widely used in the days of vacuum tubes to match the high output impedance of tubes to the low impedance of speakers in an old sound system. Today, because of their size and cost, most circuit designers shy away from them if possible. Still they are used at microwave frequencies for difficult matching problems. Interestingly, the human ear (and the ears of most animals), while not resonant, uses a mechanical version of a transformer to match the impedance of air borne sound waves to the very high impedance of the fluid filled inner ear to overcome this problem and allow efficient transfer of the acoustic energy into the inner ear. This mechanical transformer is located in the middle ear and consists of three tiny bones acting as a lever connecting one diaphragm (the ear drum) to another diaphragm of a different size covering the entrance of the inner ear. The diaphragms and bones act similarly in function to the lever shown in Fig. 26 of an earlier posting. 

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