rests within the piano was mounted on a piece of oak laid
atop the large crossbars that connect to the piano frame.
The electromagnets themselves were originally intended
for holding applications.
Bar Magnet
DAC E-- - E^ //^Eitrn qnpe
D... I...................---- -- j~ T...I-aFigure 1. System block diagram for one channel
N, /
Air gap z Current i
Figure 2. Relationships between fz, i, and z
3.2. Analysis
We can simplify the analysis by using the superposition
principle to describe the flux density B(t) at time t at a
particular point on the string. That is, similarly to the linearized analysis of analog circuits containing transistors,
the signal variable B(t) can be split into a large signal
component BL (t) and a small signal component Bs (t).
B(t) = BL(t) + Bs(t)
(1)
3. ELECTROMAGNETIC TRANSDUCERS
3.1. Summary
The permanent magnets in the transducer magnetize the
portion of the steel piano string near of the transducer, so
that small currents flowing through the electromagnet can
push and pull on the string. Similar electromagnetic transducers are often modeled as variable reluctance transducers by finding the equivalent electrical circuit and deriving
its behavior [6]. The goal of our analysis is to determine
the relationships between the force fz on the element in
motion, the current i(t) flowing through the coil, and the
air gap z, which is the vertical distance between the string
and the transducer. In general, fz and i(t) will be nonlinearly related, as depicted in Fig. 2 (right). fz will be
approximately proportional to i2(t), except for fields so
high that the string saturates magnetically, in which case
fz becomes nearly linearly related to i (circle in Fig. 2).
By placing the permanent bar magnets such that the field
is focused on the string, we can make the transducer operate in the approximately linear region, which is key in
allowing the injection of arbitrary signals into the piano
string.
The force is commonly also a hyperbolic function of
z (Fig. 2, left). This is undesirable since this makes the
system time-varying. As measurements will show below,
when the transducer is placed close to the agraffe, the displacement of the string is small enough so that the system
is roughly time-invariant. Unfortunately our analysis is
not so simple as in [6] because the path length that the
magnetic flux flows along through the string is not constant. However, we will show that the analysis holds for
an arbitrarily small string element. In addition, we will
consider the case of only one string beneath the transducer
for simplicity's sake. The model could easily be extended
to multiple strings using coupled string methods [1].
BL(t) takes on the role of "biasing" the transducerstring system such that it operates nearly linearly. BL (t)
turns out to correspond to the permanent magnets. Bs (t),
on the other hand, causes the string to start vibrating according to the audio data from the sound card output, and
as such, Bs(t) is proportional to i(t).
3.2.1. Magnetization of the String
BL(t) magnetizes the piano string so that it can be more
easily acted upon by the solenoid. However, because the
magnetic field due to the permanent bar magnets is much
stronger than that due to the coil, we will neglect the coil's
contribution here. In addition, the piano string's excursion
near the agraffe is small compared with the distance between the string and the permanent magnets, so Bpm can
be approximated to be roughly constant:
BL(t) = Bpm + Bcoil(t) BL B Bpm
(2)
The qualitative plot in Fig. 3 shows the shape of the
magnetic field lines corresponding to BL as approximated
by a two-dimensional electromagnetic field simulator. The
field lines tend to flow along the string rather than near it
because the magnetic permeability of the string p is much
higher than that of free space Po.
Figure 3. Expected magnetic field lines (side view)
