# Conductance and resistance of an acoustic transducer

I have the conductance and susceptance vs. frequency data on an acoustic transducer. Also mentioned is the resonant resistance and resonant capacitance which is just the inverse of the conductance and susceptance value at the resonance frequency.

The given data is:

• Resonant Frequency(fc) = 44000 Hz
• Conductance = 6963.4 uS
• Susceptance = 4307.73 uS
• Resonant Resistance = 143.61 ohm (= 1/Conductance)
• Resonant Capacitance = Susceptance/(2 * pi * fc) = 15.581 nF

From the conductance and susceptance are given, if I calculate Y=G+jB and then Z= 1/Y or R = G/(G^2+B^2) and X = -j * (B/(G^2 + B^2)), I am getting different results than the resistance value mentioned in the data provided along with the transducer.

Am I missing something or calculating incorrectly?

If I am not wrong, R=1/G is only applicable when there is no reactance or susceptance.

simulate this circuit – Schematic created using CircuitLab

When you see "resonating capacitance = 15nf", there is implied an associated resonating inductor. The capacitor and inductor combine to resonate at 44000 Hz. It is very unlikely that C1 and L1 and R1 are actual physical components.

I'm assuming that "Resonant resistance" is in a series circuit, since a parallel resistance with that capacitor describes a network that is not particularly resonant:

simulate this circuit

So which is it...

• a parallel "resonant" circuit?
• a series resonant circuit?
• or something more complex?

You gain some more insight by measuring with DC, as an ohmmeter does.

As for something more complex, consider the simplified model (below) for a loudspeaker transducer. It does display a resonance at low frequency, yet an ohmmeter will measure about 6 ohms. Note that this model is incomplete - at higher audio frequency, this model deviates from reality. And the elements that convert electrical energy to acoustic energy are missing.

simulate this circuit

Another "more complex" model of a resonant transducer is sometimes used to describe a piezo-electric ceramic element:

simulate this circuit

In this case, resonance usually describes the series arm of R1, C2, L1. A measurement of transducer reactance at a much lower frequency than resonance gives a capacitive susceptance dominated by C1. C1 > C2....for the piezo properties of a crystal, the factor might be 100 - 250.

TLDR: The simple series circuit (top of answer) is the more likely model - a slightly resonant series circuit. Be aware that this simple model needs many more elements to properly describe how your acoustic transducer outputs acoustic energy from electrical energy.

When dealing with acoustic transducers, admittance is the customary units since the working bandwidth is readily seen in the conductance versus frequency plot. It is implied that admittance data for acoustic transducers are parallel components. Your transducer model in parallel and series formats at 44kHz looks like this:

simulate this circuit – Schematic created using CircuitLab

As you stated, $$\ Z = {1 \over Y} = 103.9 -j64.3 \$$ which is how you go from the parallel topology to series topology. If you're trying to tune out the reactive part of the transducer because you are using the transducer as a projector, the series topology is easier to work with since you will choose an inductor with the same inductive reactance magnitude as the capacitor, or $$\ L = {X_L \over \omega } = 233 \mu H \$$. If you add a series tuning inductor, you will get a voltage gain at resonance of $$\ \sqrt{1+ {\left({X_c \over R}\right)^2}} \$$.

The series topology is also convenient for figuring out the thermal noise of the transducer since that is the noise caused by the real part of the reactance, or about $$\ 0.13 \,\sqrt{103.9} \approx 1.3nV \$$.

If I am not wrong, R=1/G is only applicable when there is no reactance or susceptance.

You're correct in saying that ( R = \frac{1}{G} ) is valid only when there's no reactance or susceptance, i.e., when the transducer is purely resistive. In your case, however, both the real (conductance) and imaginary (susceptance) parts are non-zero. Therefore, the calculation for impedance needs to consider both.

Given: ( G ) = 6963.4 µS = 0.0069634 S ( B ) = 4307.73 µS = 0.00430773 S

1. First, you can calculate the admittance ( Y ) as: [ Y = G + jB ]

Using the given values, [ Y = 0.0069634 + j0.00430773 ]

1. The impedance ( Z ) is the inverse of admittance ( Y ): [ Z = \frac{1}{Y} ] However, when ( Y ) is a complex number, you need to use the formula for the inverse of a complex number: [ \frac{1}{a + jb} = \frac{a - jb}{a^2 + b^2} ]

Applying this formula, we get: [ Z = \frac{1}{0.0069634 + j0.00430773} = \frac{0.0069634 - j0.00430773}{0.0069634^2 + 0.00430773^2} ]

1. Now, compute the impedance: [ R = \frac{G}{G^2 + B^2} ] [ X = -j \times \frac{B}{G^2 + B^2} ]

Plugging in the values, you'll get ( R ) and ( X ).

Given ( R = \frac{1}{G} ) in the data seems to be a simplification which is incorrect when there's a non-zero susceptance. The approach you've taken for calculating ( R ) and ( X ) based on ( G ) and ( B ) is correct. The discrepancy you're observing is likely due to this simplification.

• Note, display math is written with $$....$$, inline math with $...$. Commented Oct 13, 2023 at 23:39