Impedance Matching

[Robert Scott]

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