Impedance Matching

[Richard Brekne]

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Stephen Birkett wrote:

>
> The topic of this thread has rankled everytime I saw it come up. Finally
> someone has spotted the error.
>
> The art in this design issue is how to achieve a desirable impedance
> *mis-match* between soundboard and bridge.  This is far from impedance
> matching, since that situation would have zero sustain, all energy being
> rapidly passed from bridge to soundboard.
>
> Stephen
>

Hmmm.. lets see.. I had used the expression "impedance matching" because
this is what was used in the article by Klaus Wogram (included as one of the
5 Lectures on the Acoustics of the Piano). I didnt think that this refered
to matching the impedance of the strings to the impedance of the board in
such a way that these were equal. But I think I see how this idea might
easily get started and thereby your point is taken.

Re-reading the  article I tried to see if I could infer this idea from it,
and couldnt really. Correct me if I am wrong, but it seems to me that
Wogram  relates the soundboard impedance to the strings, in the sense that
the strings represent a Force applied to the board.

he states

        "The vibration energy of the string is transfered to the soundboard,
transformed from mechanical to acoustic energy, and radiated as airborne
sound. The rate at which this energy flow takes place is determined by the
soundboard (the consumer) in relation to the properties of the string (the
source). In engineering terms this relationship is referred to as "impedance
matching" The loading exerted by the soundboard on the string can be
expressed by means of its input impedance Z, which is defined as the
excitation force F, divided by the resulting velocity v, at the point of
excitation."

He goes on to say

    "A high value of the input impedance means that a large force must be
expended in order to achieve a certain volume"

This part of his discussion seems to be from the perspective of describing
some positive attributes of a good soundboard, and not of any "matching" to
the strings, above and beyond the general statement above. In the section he
entitles The Influence of String Tension , there is a description of what
effects raising and lowering of string tension has on the soundboards'
impedance curve and its resonant frequencies. Again this seems to me to be
in the perspective of analysing the characteristics of the soundboard
itself.

Where he does specifically define "matching" between the strings and the
board he states (in the section entitled Decay Of Piano Tone

    "Hence, a good match between the string and the soundboard is found when
the input impedance is high and the phase angle is positive"

He demonstrates in this section how the tension of the string can be
"matched" to the input impedance of the board to accomplish this. (tho no
method by which to design a scale in this manner is given) But again, this
section seems to be mostly concerned with the soundboard itself. From the
same section:

    "Consequently, one characteristic of a high quality soundboard is that
the impedance curve exhibits as few dips as possible, avoiding sharp
zigzaging. In addition the overall level of the impedance curve should be
high enough to ensure sufficient reflexion of the string energy at the
bridge, resulting in long decay time"

The point being that a soundboard that Does have an impedance curve with
large dips and sharp zigzaging will have strong resonances at these dips,
and suck the energy from any string that has partial frequencies that
correspond to these resonant frequencies., (regardless of its tension,
string impedance etc)

The final section on the Modal analysys of soundboards seems to go in the
same general direction, but taken a step further. By demonstrating that the
soundboard actually behaves like it was several interdependant boards,
vibrating in seperate sections (the number of which depends on the resonance
mode) in opposite phases which each other (with the exception of the first
mode). This creates nodelines between the sections for any given mode which
do not move much and hence represent high values of input impedance at those
points on the board which lie along that nodal line. If these points are
directly under the bridge then strings delivering their energy at these
points will meet high impedance, and this will result in long decay time.
(too long perhaps) Sections of the board away from these nodelines behave in
the opposite manner. The jist of all this again seems to point in the
direction soundboard itself. In this case more in the direction of how to
identify and adjust problem areas in the impedance characteristics of the
board.

This is part of my understanding of the Wogram article, and I would greatly
appreciate hearing where I am mistaken and where I am not. In anycase I do
not see any direct correlation to how one would go about "matching" string
impedance to soundboard impedance in this article. If I were to jump to
conclusions I would be tempted to think that a stringing scale should,
ideally, be designed after knowing the impedance properties of the
soundboard so as to be strong enough to drive the board, within tension
parameters allowed for the construction of the instrument as a whole, and
with a smooth as possible inharmonicity curve. How to go about this is
something I would also appreciate very much hearing about.

For those of you in the know, please understand I am just opening these
doors, and need direction and your input, even if that be only suggestions
for further readings.  For all such help I would be very gratefull.

Richard Brekne
I.C.P.T.G.  N.P.T.F.
Bergen, Norway

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