3.9. Thévenin's theorem
It appears possible, from the foregoing discussions, to represent the operation of a transducer as a combination of electrical, mechanical, and acoustical elements. The connection between the electrical and mechanical circuit takes place through an electromechanical transducer. Similarly, the connection between the mechanical and acoustical circuit takes place through a mechanoacoustic transducer. A Thévenin's theorem may be written for the combined circuits, just as is written for electrical circuits only.
The requirements which must be satisfied in the proper statement and use of Thévenin's theorem are that all the elements be linear and there be no hysteresis effects.
In the next few paragraphs, we shall demonstrate the application of Thévenin's theorem to a loudspeaker problem. The mechanical radiation impedance presented by the air to the vibrating diaphragm of a loudspeaker or microphone will be represented simply as Z
MR
in the impedance-type analogy or Y
MR
=
l/Z
MR
in the admittance-type analogy. The exact physical nature of Z
MR
will be discussed in Chapters 4, 12, and 13.
Assume a simple electrodynamic (moving-coil) loudspeaker with a diaphragm that has only mass and a voice coil that has only electrical resistance (see Fig. 3.43(a)). Let this loudspeaker be driven by a constant voltage generator. By making use of Thévenin's theorem, we wish to find the equivalent mechanical generator
and the equivalent mechanical admittance Y
MS
of the loudspeaker, as seen in the interface between the diaphragm and the air. The circuit of Fig. 3.43(a) with the transformer removed is shown in Fig. 3.43(b). The Thévenin's equivalent circuit is shown in Fig. 3.43(c).
and the equivalent mechanical admittance Y
MS
of the loudspeaker, as seen in the interface between the diaphragm and the air. The circuit of Fig. 3.43(a) with the transformer removed is shown in Fig. 3.43(b). The Thévenin's equivalent circuit is shown in Fig. 3.43(c).We arrive at the values of
and Y
MS
in two steps.
and Y
MS
in two steps.Step 1. Determine the open-circuit velocity
by terminating the loudspeaker in an infinite admittance, Y
MA
=
∞ (that is, Z
MA
=
0), and then measuring the velocity of the diaphragm
. As we discussed in Part II, Z
MA
=
0 can be obtained by acoustically connecting the diaphragm to a tube whose length is equal to one-fourth wavelength. This is possible at low frequencies. Inspection of Fig. 3.43(b) shows that
by terminating the loudspeaker in an infinite admittance, Y
MA
=
∞ (that is, Z
MA
=
0), and then measuring the velocity of the diaphragm
. As we discussed in Part II, Z
MA
=
0 can be obtained by acoustically connecting the diaphragm to a tube whose length is equal to one-fourth wavelength. This is possible at low frequencies. Inspection of Fig. 3.43(b) shows that
(3.44)
Step 2. Short-circuit the generator e without changing the mesh impedance in that part of the electrical circuit. Then determine the admittance Y
MS
looking back into the output terminals of the loudspeaker. For example, Y
MS
for the circuit of Fig. 3.43(b) is equal to the parallel combination of 1/jωM
MD
and R
E
/(Bl)2, that is,
(3.45)
The Thévenin's equivalent circuit for the loudspeaker (looking into the diaphragm) is shown schematically in Fig. 3.43(c), where
and the admittance Y
MS
are given by Eqs. (3.44) and (3.45), respectively.
and the admittance Y
MS
are given by Eqs. (3.44) and (3.45), respectively.The application of Thévenin's theorem as discussed above is an example of how general theorems originally applying to linear passive electrical networks can be applied to great advantage to the analogs of mechanical and acoustic systems, including transducers.
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