Impedance Matching

[Roger Jolly]

Hi Bob,
            A great post, a pleasant change from the normal smoke and
mirrors that we hear on IMPEDANCE, a balanced review of the whole picture
Regards Roger


At 04:33 PM 11/08/99 -0400, you wrote:
>
>I'd like to try my hand at clarifying some of the issues related
>to impedance matching as applied to strings, bridge, and soundboard.
>
>In electronics, the impedance describes the relationship between
>voltage and current in a circuit, particularly when both are
>periodic sine waves.  The voltage and current sine waves can have 
>different amplitudes and different phases.  If an impedance is
>"purely resistive" that means the voltage and current have the
>same phase.
>
>To carry the notion of impedance over to the world of acoustics,
>we can replace voltage by force and current by velocity.  So the
>acoustic impedance describes the relationship between the force
>and the velocity, both in amplitude and in phase.
>
>To illustrate the phase relationship, suppose that you are exciting
>a soundboard by pushing down and up at the rate of once per second.
>At this very slow excitation rate (1 Hz) the response of the soundboard
>is essentially like a spring.  That is, the force is proportional to the 
>deflection.  And the point where the force is at its highest corresponds
>to the point where the velocity is zero.  The board is at its
>maximum deflection and has just stopped moving in one direction and
>is about to start moving in the other direction.  So clearly the
>force and the velocity are not in phase with each other.  They are,
>in fact, 90 degrees out of phase.  The velocity is at its maximum
>when the board passes through its resting position and the 
>instantaneous force is zero.
>
>To illustrate the opposite extreme, suppose that we raise the
>frequency of excitation to the point where board inertia is the
>dominate factor.  And to make sure that air resistance doesn't
>play a role, suppose we move the experiment into a vacuum chamber.
>Under these conditions, the force is also at its highest when the
>board is stopped, but now, instead of acting to push the board
>further away from its resting position, the force is acting to
>push the board back towards the center.  So at this higher frequency,
>the force and the velocity are again 90 degrees out of phase, but
>in the opposite direction.
>
>Now let's bring back the air.  No, let's do more than that.  Replace
>all the air around the soundboard with gear oil.  Very viscous 
>gear oil.  Under these conditions, and with any reasonable excitation
>frequency, the force required to wiggle the board will be dominated
>by the resistance of the oil.  The springiness of the board will not
>have much effect.  The inertia of the board will not have much effect.
>In this case, the velocity with be directly proportional to the force.
>That is, they will be in phase.
>
>Now let's get back to reality.  Suppose an ordinary soundboard has
>a resonance at 100 Hz.  If you try to excite it by pushing and
>pulling on it at 100 Hz, it will readily comply.  At a resonant
>frequency, the spring effect and the inertia effect of the board are
>exactly in balance.  Thus the only thing the force needs to oppose
>is the air resistance.  (If it weren't for air resistance, or other
>resistive effects, the resonant excitation would cause the deflection
>to grow without bound until the board destroyed itself, just like
>an opera singer breaks a wine glass by singing at its resonant
>frequency.)  At resonance, the force and velocity are in phase,
>so the impedance has a zero phase angle.  As you gradually change
>the frequency of excitation away from resonance the phase angle
>becomes non-zero.  As you go higher than resonance, the inertia
>dominates and the force leads the velocity.  As you go lower than
>resonance, the springiness dominates and the velocity leads the
>force.
>
>But not all resonances are created equal.  A 100 Hz resonance in
>a soundboard is not like a 100 Hz resonance in a carillon bell.
>The 100 Hz resonance in the bell has a long sustain.  The 100 Hz
>resonance in a soundboard, on the other hand, has such a short
>sustain that if you tap the board you will have a hard time hearing
>even an indication of a tone.  And that's a good thing.  If the
>100 Hz resonance of a soundboard was such a strong resonance then it
>would severely distort the sounds from strings that produce tones
>very near to 100 Hz.  In terms of impedance phase angle, this
>means that as you move around a resonance in a soundboard, the
>phase angle deviates only slightly from zero degrees.  In fact,
>one of the qualities of a good soundboard would be that the
>impedance phase angle never deviates very far from zero.
>
>Now the string also has an impedance.  A very strong one.  We don't
>generally think in terms of continuous excitation for strings, but
>if we did, we would find that the force and velocity would be
>nearly 90 degrees out of phase on once side of resonance and nearly
>90 degrees out of phase in the opposite direction on the other side
>of resonance.  The impedance of the string depends very much on
>where you choose attach to it.  Since we are interested in the
>interaction between string and soundboard, the relevant point of
>attachment is the bridge.  One thing you do not want is an
>impedance match between the string and soundboard.  If there were
>a perfect match, then the energy imparted by the striking hammer
>would travel down the string and be entirely absorbed by the
>soundboard - all in one cycle!  The travelling wave would not be
>reflected back down the string.  You would never even hear a tone -
>just a thud.  No, what we want is a for the impedance to be
>mismatched - terribly mismatched.  We want to have such a poor
>mismatch that most of the energy travelling down the string
>from the hammer gets reflect back from the bridge.  Then when it
>hits the agraffe, it encounters another terrible impedance mismatch
>and gets reflected again.  At each cycle, some of the energy
>leaks into the bridge and soundboard.  There is a direct tradeoff
>between acoustic power and sustain.  You can lengthen the sustain
>by stiffening the soundboard at the point of connection with the 
>bridge, but the result would be a weaker sound.  Of course, this is
>not the only factor in determining sustain/power. It is possible
>to lose energy through mechanical friction at a faulty soundboard
>support or in a loose bridge pin.  Or bad hammer voicing could
>prevent the optimum amount of energy from getting into the string
>in the first place.  But if these extraneous energy drains are under
>control, the only thing left is the sustain/power tradeoff based on
>the impedance mismatch at the bridge.
>
>
>Bob Scott
>Ann Arbor, Michigan
> 
Roger Jolly
Baldwin Yamaha Piano Centre
Saskatoon and Regina
Saskatchewan, Canada.
306-665-0213
Fax 652-0505