Engineering Acoustics/Electro-Mechanical Analogies

Why analogs to circuits?

Since acoustic devices contain both electrical and mechanical components, one needs to be able to combine them in a graphical way that aids the user's intuition. The method that is still used in the transducer industry is the Impedance and Mobility analogies that compare mechanical systems to electric circuits.

Two possible analogies

i) Impedance analog

ii) Mobility analog

                     Mechanical           Electrical equivalent

i)impedance analog

   Potential         Force     F(t)          Voltage    V(t)
   Flux              Velocity  u(t)          Current    i(t)

ii)Mobility analog

   Potential         Velocity  u(t)          Voltage    V(t)
   Flux              Force     F(t)          Current    i(t)

Impedance analog is often easier to use in most accoustical systems while mobility analog can be found more intuitively for mechanical systems. These are generalities, however, so it is best to use the analogy that allows for the most understanding. A circuit of one analog can be switched to the equivalent circuit of the other analog by using the dual of the circuit. (more on this in the next section).

The equivalent spring

Mechanical spring

$F=K\int udt$

Impedance analogy of the mechanical spring

$V=1/C\int idt$

Mobility analogy of the mechanical spring

$i=1/L\int Vdt$

The equivalent Mass

Mechanical mass

$F=M{x}^{\prime \prime }=M\frac{du}{dt}$

Impedance analogy of the mechanical mass

$V=L\frac{di}{dt}$

Mobility analogy of the mechanical mass

$i=C\frac{dV}{dt}$

The equivalent resistance

Mechanical resistance

$F=Ru$

Impedance analogy of the mechanical resistance

$V=Ri$

Mobility analogy of the mechanical resistance

$i=\frac{1}{R}V$

Review of Circuit Solving Methods

Kirchkoff's Voltage law

"The sum of the potential drops around a loop must equal zero."

This implies that the total potential drop around a series of elements is equal to the sum of the individual voltage drops in the series.

$e_{t}otal=dro{p}_1+dro{p}_2+dro{p}_3$

Kirchkoff's Current Law

"The Sum of the currents at a node (junction of more than two elements) must be zero"

Using the pipe flow analogy of circuits, this can be thought of as the continuity equation.

For example if there was a node with three elements connected to it (numbered 1,2 and 3) $i_1+i_2+i_3=0$ From the current law, their sum would equal zero.

Hints for solving circuits:

-Remember that certain elements can be combined to simplify the circuit (the combination of like elements in series and parallel)

-If solving a ciruit that involves steady-state sources, uses impedances! (This reduces the ciruit down to a bunch of complex domain resistor elements that can be combined to simplify the circuit.)

Examples of Electro-Mechanical Analogies

Additional Resources for solving linear circuits:

Thomas & Rosa, "The Analysis and Design of Linear Circuits", Wiley, 2001

Hayt, Kemmerly & Durbin, "Engineering Circuit Analysis", 6th ed., McGraw Hill, 2002

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