Proc R Soc A

electromagnetism

J. Priede

# Realization of Bullard’s disc dynamo

Ra l Alejandro Avalos-Z iga1
Jānis Priede2,3 1Research Centre in Applied Science and Advanced technology,
Instituto Polit cnico Nacional (CICATA - Quer taro), Cerro Blanco 141, Colinas
del Cimatario, Quer taro, Mexico
2Fluid and Complex Systems Research Centre, Coventry University,
Coventry, CV1 5FB, United Kingdom
3Department of Physics, University of Latvia, Riga, LV-1004, Latvia
<EMAIL_ADDRESS>
###### Abstract

We report experimental results from three successful runs of a Bullard-type
homopolar disc dynamo. The set-up consisted of a copper disc with a radius of
$30\,\mathrm{cm}$ and thickness of $3\,\mathrm{cm}$ which was placed co-
axially beneath a flat, multi-arm spiral coil of the same size and connected
to it electrically at the centre and along the circumference by sliding
liquid-metal contacts. The magnetic field was measured using Hall probes which
were fixed on the top face of the coil. We measured also the radial voltage
drop across the coil. When the disc rotation rate reached $\Omega\approx
7\,\mathrm{Hz,}$ the magnetic field increased steeply approaching
$B_{0}\approx 40\,\mathrm{mT}$ in the central part of the coil. This field was
more than two orders of magnitude stronger than the background magnetic field.
In the first two runs, the electromagnetic torque braking the disc in the
dynamo regime exceeded the breakdown torque of the electric motor driving the
disc. As a result, the motor stalled and the dynamo was interrupted. Stalling
did not occur in the third run when the driving frequency was set higher and
increased faster. We also propose an extended disc dynamo model which
qualitatively reproduces the experimental results.

###### keywords:

disc dynamo, homopolar generator, geomagnetism

††journal: rspa

##  1 Introduction

Bullard’s disc dynamo [1] is arguably the simplest model of the
magnetohydrodynamic dynamo. It is often used to illustrate the self-excitation
of the magnetic field by moving conductors [2]. This is how the magnetic
fields of the Earth, the Sun and many other cosmic bodies are thought to come
about [3]. In its basic form, the Bullard dynamo consists of a solid metal
disc and a wire: the former spinning about its central axis and the latter
twisted around and connected through sliding contacts to the rim and axis of
the disc. If the disc spins sufficiently fast and in the right direction, such
a set-up can amplify the electric current circulating in the system and, thus,
the associated magnetic field. This happens when the rotation rate of the disc
exceeds a certain critical threshold above which the potential difference
induced across the disc exceeds the voltage drop caused by the ohmic
resistance of the system. Then the current starts to grow exponentially in
time resulting in the self-excitation of the magnetic field. The growth stops
when the braking electromagnetic torque becomes so strong that it slows down
the disc. This is how the dynamo would operate in the ideal case with no
background magnetic field.

In the presence of a background magnetic field, the dynamo may manifest itself
somewhat differently. In this case, a current is induced in the set-up as soon
as the disc starts to rotate. If the disc rotates in the right direction, the
induced current amplifies the background magnetic field which, in turn,
amplifies the current. Thus, the rate of amplification increases with the
speed of rotation and becomes formally infinite at the dynamo threshold.

Despite its simplicity, the implementation of the disc dynamo is faced with
severe technical challenges. The main problem is the sliding electrical
contacts which are required to convey the current between the rim and the axis
of the rotating disc. The electrical resistance of sliding contacts, which are
usually made of solid graphite brushes, is typically several orders of
magnitude higher than that of the rest of the set-up. This results in
unrealistically high rotation rates which are required for the dynamo to
operate. Therefore, in contrast to the fluid dynamos, which have been realised
in several laboratory experiments using liquid metal [4, 5, 6] ([7] in
preparation), the disc dynamo was thought to be technically unfeasible [8, 9,
10].

We overcome this problem by using sliding liquid-metal electrical contacts
which are similar to those employed previously in the homopolar motors and
generators [11, 12, 13] as well as in the laboratory model of Herzenberg
dynamo [14] built by Lowes and Wilkinson [15, 16]. The set-up consists of a
coil made of a stationary copper disc which is divided into spiral-shaped
sections by thin slits [17]. The coil is placed co-axially above the solid
copper disc and connected to the latter by sliding liquid-metal contacts. The
slits make the conductivity of the coil anisotropic which allows this
essentially axially symmetric dynamo to generate an axially symmetric magnetic
field [18, 19].

In this paper, which is organised as follows, we report results from three
successful runs of such a dynamo. Set-up is described in Sec. 2. Experimental
results are presented in Sec. 3. In Sec. 4, we introduce an extended disc
dynamo model which is used in Sec. 5 for the interpretation and analysis of
experimental results. Summary and conclusions are presented in Sec. 6.

##  2 Experiment set-up

The dynamo set-up shown in Fig. 1 consists of a rotating copper disc and a
coil. The latter is made of a flat copper cylinder of radius
$r_{o}=30\,\mathrm{cm}$ and thickness $d=3\,\mathrm{cm}$ which is sectioned
starting from the radius $\tilde{r}_{i}=7.5\,\mathrm{cm}$ by $40$ logarithmic
spiral slits with a constant pitch angle $\alpha\approx 58^{\circ}.$ The disc
has an annular channel along the rim and a cylindrical cavity at the centre.
The coil, which has a cylindrical solid electrode protruding $4\,\mathrm{cm}$
out from the centre of its bottom face, is placed $3\,\mathrm{mm}$ above the
disc. At the inner and outer radii, $r_{i}=4.5\,\mathrm{cm}$ and
$r_{o}=30\,\mathrm{cm,}$ there are two annular gaps of width
$\delta=0.25\,\mathrm{mm}$ and height $d=3\,\mathrm{cm}$ which separate the
coil from the disc. These gaps were filled with the eutectic alloy of
$\mathit{GaInSn,}$ which is liquid at room temperature. The coil and disc are
held by electrically insulated iron supports. Note that, in the initial design
with $3\,\mathrm{mm}$ annular gaps, electrical contacts failed before the
dynamo threshold was attained [20].

The disc was driven by a $3\,\mathrm{HP}$ $(2.2\,\mathrm{kW)}$ 6-pole AC motor
which had a synchronous rotation rate of $1200\,\mathrm{RPM}$ at
$60\,\mathrm{Hz}$ input frequency. The sense of rotation was opposite to the
orientation of the spiral arms of the coil which corresponds to a clock-wise
rotation in the set-up shown in Fig. 1(b). The rotation rate was changed using
a variable frequency drive (VFD) Delta VFD022EL23A with a rated output current
of $11\,\mathrm{A}$. The VFD output frequency was controlled from a PC using
the LabVIEW software. With default settings, the output voltage was reduced
directly with the driving frequency. In this standard VFD regime, the driving
torque was maintained constant while the motor power dropped off directly with
the rotation rate. It means that the maximal power the motor can produce, when
operated sufficiently close to the expected dynamo threshold of
$600\,\mathrm{RPM}$ $(10\,\mathrm{Hz),}$ is only half of its rated power,
i.e., $\approx 1\,\mathrm{kW.}$ The actual output power may be significantly
lower when the rotation rate starts to drop because of too high a load.

The rotation rate was determined using a self-made opto-mechanical tachometer
which consisted of a disc with periodic slits with a light source on one side
and a photodiode on the other. The magnetic field was measured using the
THM1176 3-axis Hall Magnetometer of Metrolab with the low- and medium-field
probes THM1176-LF and THM1176-MF. These probes have an active volume and the
upper magnetic field strength of
$6\text{$\times$}3.4\text{$\times$}3\,\mathrm{mm^{3},}$$8\,\mathrm{mT}$ and
$16.5\text{$\times$}5\text{$\times$}2.3\,\mathrm{mm}^{3},$ $100\,\mathrm{mT,}$
respectively. The probes were fixed with adhesive tape on the upper face of
the coil next to one of the crossed iron arms holding the coil. Three
components of field were acquired for a period of several minutes at the rate
of $30$ samples per second. The voltage between the inner and outer radii of
the spiral arms was measured using a digital multimeter Keithley 2100.

(a)(b)

Figure 1: (a) Set-up consisting of coil (1), disc (2), iron frame (3), gearing
system (4), and AC motor (5); (b) a cross-sectional view showing coil (1),
disc (2), and sliding liquid-metal contacts (3).

##  3 Experimental results

(a)(b)(c)

(d)(e)(f)

(g)(h)(i)

Figure 2: The magnetic field $(B_{x},B_{y},B_{z})$ at the inner radius of the
coil (a,d,g), the voltage between the centre and the rim of the coil (b,e,h),
and the electric current supplied to the motor (c,f,i) along with the driving
$(\Omega_{d})$ and rotation $(\Omega)$ frequencies recorded during the first
(a,b,c), second (d,e,f) and third (g,h,i) runs.

In this section, we present experimental results for three runs in which the
disc rotation rate was measured together with the induced voltage and the
magnetic field at the upper face of the coil. The runs differ mainly by the
way in which the frequency driving the motor was varied. In the first run, for
which the results are shown in the top row of Fig. 2, the driving frequency
was ramped up from zero to $\Omega_{d}=7.33\,\mathrm{Hz}$ nearly linearly in
$\approx 79\,\mathrm{s}.$ Although the rotation rate $\Omega$ closely followed
$\Omega_{d},$ there was a difference between both frequencies. This difference
increased as the magnetic field became stronger. In this run, the magnetic
field was measured using only the low-field probe which was placed in the
vicinity of the inner radius of the spiral slits. Before the disc started to
rotate, a background field $\vec{B}_{0}\approx-(0.18,0.77,0.57)\,\mathrm{mT}$
with $y$ component directed downwards and $x$ and $z$ components in the plane
of the coil was detected. This field, which was more than an order of
magnitude stronger than Earth’s magnetic field, was obviously due to the iron
frame holding the coil.

(a)(b)

Figure 3: The magnetic field $(B_{x},B_{y},B_{z})$ at the outer radius of the
coil along with the driving $(\Omega_{d})$ and rotation $(\Omega)$ frequencies
in the second (a) and third (b) runs.

As the rotation rate increased, the magnetic field raised steeply, especially
its vertical $(y)$ component, which is seen Fig. 2(a) to exceed the upper
limit of the low-field Hall sensor by reaching $B_{y}\approx-9.3\,\mathrm{mT}$
at $\Omega\approx 6.5\,\mathrm{Hz}$ $(t\approx 76\,\mathrm{s})$. A few seconds
later, the increase in the driving frequency was halted at
$\Omega_{d}=7.33\,\mathrm{Hz.}$ At this point, the rotation rate, which had
reached $\Omega\approx 6.7\,\mathrm{Hz},$ started to fall while the $x$ and
$z$ components of the magnetic field kept growing for a few more seconds. The
same held also for the coil voltage which can be seen in Fig. 2(b) to rise
from $V\approx 7.7\,\mathrm{mV}$ at the velocity maximum to $V_{\text{max
}}\approx 10\,\mathrm{mV}$ at $t\approx 82\,\mathrm{s}.$ However, this time
delay between the rotation rate and the other two measured quantities was too
short to be determined reliably. Namely, it was comparable to the uncertainty
in the time synchronization between the measurements of these quantities which
were recorded using different devices.

The slowdown means that the electromagnetic braking torque acting on the disc
had shot past the equilibrium point corresponding to the driving torque
produced by the motor. As discussed before, this effect can be due to
transient eddy currents which, according to the previous estimates, decay over
the characteristic time $\sim$$1\,\mathrm{s}.$ They reduce the magnetic flux
through the disc and so the associated electromagnetic braking torque when the
disc accelerates while the opposite is the case when the disc decelerates.
Although the decrease in the rotation rate at the fixed driving frequency and
voltage results in the rise of the current running through the motor (seen in
Fig. 2c), it does not prevent the slowdown because the maximal torque the
motor can develop is lower than the electromagnetic braking torque. Actually,
the deceleration is escalated further by the relatively large difference
between the driving and rotation frequencies. When the frequency difference
becomes too large, the motor breaks down because the output torque, which is
described by Eq. (14), starts dropping with the rotation rate.

Although, as seen in Fig. (2c), the coil voltage drops together with the
rotation frequency, the rate of decrease is lower than that at which the
voltage increased when the disc accelerated. Both the voltage and the magnetic
field are noticeably higher than the respective values at the same rotation
rate in the acceleration stage. This difference is likely due to the remanent
magnetization of the iron frame. However, the transient eddy currents, which
delay the variation of the magnetic flux through the disc as discussed above,
can have a similar effect.

(a)(b)

(c)(d)

Figure 4: The magnetic field at the inner radius of spiral slits for three
runs (a,b,c) and at the outer radius for the second run (d) versus the coil
voltage.

Attempting to reduce the slip, at $t\approx 126\,\mathrm{s},$ we started to
ramp down $\Omega_{d}$ to $7\,\mathrm{Hz}$ which was reached in $\approx
7\,\mathrm{s}.$ This, however, just increased the slowdown rate without
noticeably reducing the slip. The subsequent reduction of $\Omega_{d}$ to
$6.7\,\mathrm{Hz}$ had a similar effect. The slowdown temporarily reversed at
$t\approx 156\,\mathrm{s}$ when $\Omega_{d}$ was ramped up to the original
level of $7.33\,\mathrm{Hz.}$ Acceleration lasted only for $\approx
15\,\mathrm{s}$ after which $\Omega$ resumed falling. When, at $t\approx
192\,\mathrm{s},$ $\Omega_{d}$ was ramped down in $\approx 6\,\mathrm{s}$ to
$6.73\,\mathrm{Hz},$ the falling rate of rotation frequency as well as that of
the magnetic field initially increased. After $\approx 6\,\mathrm{s},$ when
the rotation rate and the $y$ component of the magnetic field had dropped down
to $\Omega\approx 1.6\,\mathrm{Hz}$ and $B_{y}\approx 5.6\,\mathrm{mT},$ the
slowdown stopped and the disc started to re-accelerate. Unfortunately, a few
seconds later, the liquid metal contacts started to fail due to the excessive
oxidation caused by the exposure to the air. The failure started with sporadic
current interruptions which appear as irregular oscillations of the magnetic
field and coil voltage in Figs. 2(a,b). The electric contact vanished
completely in $\approx 8\,\mathrm{s}$ and so did the associated
electromagnetic braking torque. As a result, the disc quickly accelerated to
the prescribed VFD frequency, and the motor current dropped from $\approx
21\,\mathrm{A}$ to $5.7\,\mathrm{A}$ (see Fig. 2c). The motor was turned off
at $t\approx 244\,\mathrm{s}$ and slowed down by the VFD linearly to a
complete stop in $\approx 60\,\mathrm{s}.$ The magnetic field measured at this
point, $\vec{B}_{0}\approx-(0.23,1.0,0.72)\,\mathrm{mT,}$ was $26-36\%$ higher
than at the beginning of the run. This difference is obviously due to the
remanent magnetization of the iron frame.

The peak value of $B_{y},$ which was not measured because it exceeded the
upper detection limit of the low-field Hall probe, can be estimated by
extrapolating its variation with the coil voltage. In a quasi-stationary
regime, both the voltage and the associated magnetic field are expected to
vary linearly with the current. This is confirmed by their mutual dependence
plotted in Fig. 4(a). Extrapolating the rising part of $V$and $B_{y},$ which
corresponds to the lower branch, we have $B_{y}\approx 22\,\mathrm{mT}$ at the
maximal coil voltage $V_{\text{ }}\approx 10\,\mathrm{mV.}$ Extrapolating the
falling part, which corresponds to the upper branch, we respectively have
$B_{y}\approx 25\,\mathrm{mT.}$ The dependence of the magnetic field on its
direction of variation is most likely due to the remanent magnetization of the
iron frame.

It is interesting to note that the relation between the coil voltage and the
magnetic field is nearly linear except the $z$ component which appears to
saturate close to $9\,\mathrm{mT,}$i.e., the upper detection limit of the low-
field Hall probe. Also note that both the iron frame and the spiral slits made
the magnetic field strongly non-uniform at the inner and outer edges of the
coil where the probes were placed. As a result, the strength, as well as the
direction of the magnetic field, were found to vary noticeably with the
location of the Hall sensor which was not fixed and may vary between the runs.

In the second run, the low-field probe was moved to the outer radius and the
medium-field probe was placed near the inner radius of the coil. The driving
frequency was ramped up through intermediate steps rather than continuously as
in the first run. As seen in Fig. 2(d,e), this resulted in the rise of the
magnetic field and the coil voltage at an increasing rate. A particularly
steep increase was observed when $\Omega_{d}$ was ramped up from
$7\,\mathrm{Hz}$ at $t\approx 194\,\mathrm{s}$ to $7.67\,\mathrm{Hz}$ at
$t\approx 200\,\mathrm{s}.$ As the rotation rate increased from
$6.1\,\mathrm{Hz}$ to $6.4\,\mathrm{Hz},$ $B_{y}$at the inner radius rose
steeply from $9.5\,\mathrm{mT}$ up to $26.5\,\mathrm{mT.}$ The magnetic field
at the outer radius can be seen in Fig. 3 to increase in a similar way. The
coil voltage, which is plotted Fig. 2(e), increased respectively from
$2.2\,\mathrm{mV}$ to $6.1\,\mathrm{mV.}$ After that the rotation rate started
to fall and so did the magnetic field along with the coil voltage. It
indicates that the electromagnetic braking torque had again exceeded the
torque that the motor can produce at the given parameters. Thus, the motor was
not able to sustain the previously attained field strength and went into the
breakdown regime. The rotation rate kept falling until $\Omega\approx
5.6\,\mathrm{Hz}$ was reached at $t\approx 255\,\mathrm{s}.$ At this point,
the slowdown ended and the rotation rate started to increase slowly. This
increase lasted only for $t\approx 25\,\mathrm{s}$ after which the rotation
rate dropped again reaching $\Omega\approx 5.5\,\mathrm{Hz}$ at $t\approx
297\,\mathrm{s}.$ The subsequent increase in the rotation rate was enhanced at
$t\approx 304\,\mathrm{s}$ by ramping up the driving frequency to
$8.47\,\mathrm{Hz}$ which was reached in $\approx 6\,\mathrm{s}.$ As a result,
the rotation rate raised forming a plateau at $t\approx 315\,\mathrm{s}$ with
$\Omega\approx 6.3\,\mathrm{Hz}$ which lasted for $\approx 8\,\mathrm{s}.$ The
vertical magnetic field, which is shown in Fig. 2(d), reached $B_{y}\approx
30\,\mathrm{mT}$ at $t\approx 319\,\mathrm{s}$ and stayed close to this value
for $\approx 6\,\mathrm{s}.$ After the plateau, the rotation rate can be seen
to increase slightly reaching $\approx 6.5\,\mathrm{Hz}$ in $\approx
2\,\mathrm{s}.$ This caused a steep rise in the magnetic field which reached a
peak of $\approx 38\,\mathrm{mT}$ at $t\approx 333\,\mathrm{s}.$ As before,
the motor could no longer sustain the attained rotation rate which, thus,
started to fall closely followed by the magnetic field until reaching
$\Omega\approx 6.1\,\mathrm{Hz}$ and $B_{y}\approx 32.5\,\mathrm{mT}$ at
$t\approx 359\,\mathrm{s}.$ At this point, we started to ramp down the driving
frequency thus disrupting the subsequent growth phase. At $t\approx
386\,\mathrm{s},$ when the rotation rate had slowed down to $\Omega\approx
5.2\,\mathrm{Hz},$ oscillations in the magnetic field increased abruptly and
the field started to fall sharply. The same may be seen in Fig. 2(e) to happen
to the coil voltage. This introduces the breakdown of electric contact which
was completely lost at $t\approx 400\,\mathrm{s}.$ At this point, the magnetic
field dropped down close to its initial background value. As the
electromagnetic braking torque vanished, the disc rotation rate increased
approaching the VFD driving frequency. Although the motor closely followed the
driving frequency in this regime, the VFD caused some oscillations in the
rotation rate around $\Omega\approx 7.5\,\mathrm{Hz}$ which may be seen in
Figs. 2(d-f).

In the third run, which is documented in Figs. 2(g-j), the driving frequency
$\Omega_{d}$ was set higher and ramped up faster than in the second run.
Firstly, the driving frequency was ramped up in $5.4\,\mathrm{s}$ to
$\Omega_{d}\approx 3\,\mathrm{Hz}.$ Secondly, after $\approx 12\,\mathrm{s},$
when the rotation rate had stabilized at $\Omega\approx 2.8\,\mathrm{Hz},$ the
driving frequency was ramped up to $\Omega_{d}\approx 8.9\,\mathrm{Hz,}$ which
was reached nearly linearly via a few short intermediate steps at $t\approx
60\,\mathrm{s}.$ The motor rotation rate also increased nearly linearly but at
a slightly lower pace than the driving frequency and only up to $\Omega\approx
6.7\,\mathrm{Hz}$ which was attained at $t\approx 45\,\mathrm{s}.$ At this
point, $\Omega$ ceased to increase further with $\Omega_{d}.$ The subsequent
results show that only the magnetic field was affected by $\Omega_{d}$ while
$\Omega$ stayed nearly constant. Namely, the magnetic field, which saturated
at $B_{y}\approx 45\,\mathrm{mT}$ when the driving frequency was set to
$\Omega_{d}\approx 8.9\,\mathrm{Hz,}$can be seen in Fig. 2(g) to reach
$B_{y}\approx 48.6\,\mathrm{mT}$ at $t\approx 74\,\mathrm{s}$ when the driving
frequency was ramped up $\Omega_{d}\approx 9.7\,\mathrm{Hz}.$ Although $B_{y}$
fell off subsequently, it remained well above the previous equilibrium level
until the motor was switched off at $t\approx 92\,\mathrm{s}.$

##  4 An extended disc dynamo model

In this section, we extend our original quasi-stationary disc dynamo model
[17] by including several effects which are essential for the interpretation
and analysis of the experimental results. In order to capture the temporal
evolution and saturation of the disc dynamo, first of all, we need to take
into account two transient inductive effects. The first is an additional
potential difference between the inner and outer rim of the coil which is
induced by a time-dependent electric current $I_{1}(t)$ flowing along the
spiral arms of the coil and then connecting radially through the disc. Owing
to the fixed distribution of $I_{1}(t)$ in the coil, this voltage can be
expressed analytically in terms of the rate of variation of the associated
magnetic flux (Eq. (8) in [17]). The second effect, which is neglected in the
original disc dynamo model [1], is the eddy current induced in the disc by a
changing magnetic field [21]. The linear density of this purely azimuthal
current is governed by Ohm’s law

$J_{2}(r,t)=\bar{\sigma}E_{\phi},$

where $\bar{\sigma}=\sigma d$ is the effective electric conductivity of the
disc and $E_{\phi}=-\partial_{t}A_{\phi}$ is the induced electric field which
follows from Faraday’s law with $A_{\phi}$ standing for the azimuthal
component of the vector potential. Using the Biot-Savart law, the latter can
be written as follows

$A_{\phi}(r,t)=\frac{\mu_{0}}{4\pi}\intop_{\tilde{r}_{i}}^{r_{o}}\intop_{0}^{2\pi}\frac{(J_{1}(r^{\prime},t)-\bar{\sigma}\partial_{t}A_{\phi})r^{\prime}\cos\phi}{\sqrt{r^{2}+r^{\prime
2}-2rr^{\prime}\cos\phi}}\thinspace\mathrm{d}\phi\thinspace\mathrm{d}r^{\prime},$
(1)

which is an integro-differential equation governing the evolution of
$A_{\phi}$ over the radius of the system; $J_{1}(r,t)=\beta I_{1}(t)/2\pi r$
stands for the azimuthal current distribution in the coil and
$\mu_{0}=4\pi\times 10^{-7}\,\mathrm{H/m}$ is the permeability of vacuum. Note
that the vertical gap between the disc and the coil is supposed to be much
smaller than the radial size. Thus, the magnetic flux passing through the disc
is approximately the same as that passing through the coil [17]. In the
following, instead of solving Eq. (1), which is numerically complicated, we
pursue a simplified approach by assuming that the aforementioned magnetic flux
can be written as

$\Phi=L_{1}I_{1}+L_{2}I_{2}+\Phi_{0},$ (2)

where the first two terms on the RHS represent the fluxes generated by the
coil and the disc, respectively, and $\Phi_{0}$ is a background flux due to
external magnetic which may be Earth’s magnetic field as well as to that
produced by the magnetized iron frame. This model differs from that of Moffatt
[21] first, by taking into account the external field and, second, by
neglecting the leakage of the magnetic flux through the gap between the coil
and the disc.

Applying Kirchhoff’s voltage law to the primary dynamo circuit, which carries
the current $I_{1},$ and to the disc, which constitutes the secondary circuit,
we obtain

$\displaystyle\dot{\Phi}+R_{1}I_{1}$ $\displaystyle=$
$\displaystyle\Omega\Phi,$ (3) $\displaystyle\dot{\Phi}+R_{2}I_{2}$
$\displaystyle=$ $\displaystyle 0,$ (4)

where $R_{1}$ and $R_{2}$ are the effective ohmic resistances of respective
circuits. The system is completed by the angular momentum equation:

$\mathcal{I}\dot{\Omega}=G-I_{1}\Phi-k\Omega,$ (5)

where $\mathcal{I}$ is the moment of inertia of the disc, $G$ is the driving
torque produced by the motor and $I_{1}\Phi$ is the electromagnetic braking
torque [21]. The last term with the coefficient $k$ represents a hyrodynamic-
type friction. It can also model the variation of the driving torque with the
rotation speed as in asynchronous electric motor controlled by the VFD with a
fixed voltage and frequency ratio. The system can be reduced to two equations
for $\Phi$ and $\Omega$ as follows. First, substituting
$I_{2}=-\dot{\Phi}/R_{2}$ from Eq. (4) into Eq. (2) we have

$I_{1}=(\Phi-\Phi_{0}+\dot{\Phi}L_{2}/R_{2})/L_{1},$

Second, substituting this expression for $I_{1}$ into Eqs. (3) and (5), after
a few rearrangements, we obtain:

$\displaystyle\tau_{0}\dot{\Phi}$ $\displaystyle=$
$\displaystyle\Phi(\tau_{1}\Omega-1)+\Phi_{0}$ (6)
$\displaystyle\mathcal{I}\dot{\Omega}$ $\displaystyle=$ $\displaystyle
G-\Phi(\Phi+\tau_{2}\dot{\Phi})/L_{1}-k\Omega,$ (7)

where $\tau_{i}={\textstyle L_{i}/R_{i}}$ are the characteristic
electromagnetic decay times (the time constants) for the primary $(i=1)$ and
secondary $(i=2)$ circuits and $\tau_{0}=\tau_{1}+\tau_{2}$. Note that the
kinematic dynamo problem, which corresponds to a prescribed rotation rate
$\Omega,$ is posed just by Eq. (6). According to this equation, a small
initial current perturbation grows exponentially with time if
$\Omega>\Omega_{c},$ where

$\Omega_{c}=\tau_{1}^{-1}$ (8)

is a critical rotation rate defining the dynamo threshold. Scaling rotation
rate and time with $\Omega_{c}$ and $\tau_{0},$ respectively, Eqs. (6) and (7)
can be written in the dimensionless form as follows

$\displaystyle\dot{\Phi}$ $\displaystyle=$
$\displaystyle\Phi(\Omega-1)+\Phi_{0},$ (9) $\displaystyle\dot{\Omega}$
$\displaystyle=$
$\displaystyle\Gamma(1-\kappa\Omega)-\Phi\left(\Phi-\Phi_{0}-\lambda\dot{\Phi}\right),$
(10)

were $\Gamma$ is a dimensionless driving torque, $\kappa$ is a friction
coefficient defined relative to $\Gamma$ and
$\lambda=\tau_{2}/(\tau_{1}+\tau_{2})$ is a parameter defining the relative
significance of eddy currents in the disc. In the following, we assume
mechanical friction to be negligible, i.e., $\kappa\ll\Gamma,$ and use
$\kappa=0$ unless stated otherwise. Finally, using Eq. (9) to eliminate
$\dot{\Phi}$ on the RHS of Eq. (10), we obtain a second-order dynamical
system.

If the disc is segmented by spiral slits like the coil, or by radial
insulating strips as suggested by Moffatt [21], so that no azimuthal eddy
currents can circulate in the disc, we have $\lambda=0.$ In this case, our
model, like that of Moffatt [21], reduces to the original Bullard dynamo model
[1]. If the disc is solid and well conducting, as in our experiment, so that
$\tau_{2}\gg\tau_{1},$ which will be shown in the following, we have
$\lambda=1.$

The driving torque $\Gamma$ can conveniently be defined using the dynamic
response time $\tau_{d},$ i.e., the time required by the motor to spin up the
disc to the critical rotation rate (8). In the case of constant driving torque
and dominating inertia, Eq. (10) yields $\Omega=\Gamma t.$ Taking into account
that $\Omega_{c}=1,$ we have $\Gamma=(\tau_{d}/\tau_{0})^{-1},$ which is a
dimensionless counterpart of $\tau_{d}^{-1}$. Similarly, in the case of a
constant power $\Pi=\Omega\Gamma$, Eq. (10) yields $\frac{1}{2}\Omega^{2}=\Pi
t$ and, thus, we have $\Gamma=\frac{1}{2}(\Omega\tau_{d}/\tau_{0})^{-1}.$

For a disc of radius $r_{o}$ and thickness $d\ll r_{o},$ which can be treated
as a thin sheet with the effective electric conductivity $\bar{\sigma}=\sigma
d,$ the characteristic time over which eddy currents decay can be estimated as
$\tau_{2}\sim\mu_{0}\sigma dr_{o}.$ For our set-up with
$r_{o}=10d=0.3\,\mathrm{m}$ and $\sigma=58.5\,\mathrm{S/m},$ the
electromagnetic time constant of the secondary circuit is $\tau_{2}\approx
0.7\,\mathrm{s}.$

The time constant of the primary circuit $\tau_{1}$ can be estimated using Eq.
(8) and the corresponding critical magnetic Reynolds number, which can be
written as follows

$\mathit{Rm}_{c}=\mu_{0}\sigma dr_{o}\Omega_{c}=\tau_{2}/\tau_{1}.$

Then taking into account that $\mathit{Rm}_{c}\approx 40$ [17], we have
$\tau_{2}=\tau_{1}\mathit{Rm}_{c}\gg\tau_{1}$ and, thus, $\lambda\approx 1.$

(a)(b)

Figure 5: Temporal evolution of the rotation rate $\Omega$ and the magnetic
flux $\Phi$ with a small initial value and no background magnetic field
$(\Phi_{0}=0)$ and no mechanical friction $(\kappa=0)$ when the disc is driven
either by a constant torque $\Gamma=0.1\,\text{(a)},0.01\,\text{(b)}$ or a
constant power $\Pi=0.1\,\text{(a)},0.01\,\text{(b);}$ $\lambda$ characterizes
the effect of eddy currents in the disc which is negligible when $\lambda=0$
and significant when $\lambda=1.$

The characteristic mechanic response time in which the disc attains the
expected critical rotation frequency $f_{c}\approx 10\,\mathrm{Hz}$ when the
motor is run at $P\approx 1\,\mathrm{kW,}$ i.e. half of its rated power, can
be estimated as

$\tau_{d}=\frac{1}{2}\mathcal{I}\Omega_{c}^{2}/P\approx\pi^{3}\rho
dr_{o}^{4}f_{c}^{2}/P\approx 7\,\mathrm{s},$

where $\rho=8.96\times 10^{3}\,\mathrm{kg/m^{3}}$ is the density of copper.
This time being by an order of magnitude longer than the electromagnetic
response time $\tau_{1}$ corresponds to $\Gamma\sim 0.1.$

There are two distinct ways how the dynamo can manifest itself. In the
classical scenario, the background field is assumed to be absent
$(\Phi_{0}=0).$ In this case, as soon as the rotation rate exceeds
$\Omega_{c}=1$, the magnetic field starts to grow exponentially in time at the
rate $\Omega-1$. This scenario, which corresponds to an instability of the
base state $\Phi=0$ at $\Omega>1,$ is captured by Eqs. (9,10) with the initial
conditions $\Omega(0)=0$ and $\Phi(0)=\Phi_{1},$ where $\Phi_{1}$ is a small
initial perturbation.

(a)(b)

Figure 6: Temporal evolution of the rotation rate $\Omega$ and the magnetic
flux $\Phi$ in the presence of a background field with
$\Phi_{0}=0.01\Gamma^{1/2},$ $\lambda=1,$ $\kappa=0,0.5,0.75,1$ when the disc
is driven by a constant torque $\Gamma=0.01\,\text{(a)},0.1\,\text{(b).}$

The second scenario, which more adequately describes the experiment, involves
a non-zero background flux $\Phi_{0}$. In this case, Eq. (9) for a fixed
rotation rate $\Omega$ has a stationary solution $\Phi=\Phi_{0}/(1-\Omega),$
which shows that the magnetic flux increases with the rotation rate as
$\sim(1-\Omega)^{-1}$ and becomes unbounded when $\Omega\rightarrow 1.$ This
is how the dynamo manifests itself in the second scenario.

In reality, the growth of the magnetic field is limited by the electromagnetic
braking torque which at a certain point outweighs the driving torque. The
stationary solution for the flux defined by Eq. (10) is

$\bar{\Phi}=\Phi_{0}/2+\sqrt{\Phi_{0}^{2}/4+\Gamma}.$ (11)

The respective equilibrium rotation rate following from Eq. (9) is
$\bar{\Omega}=1-\Phi_{0}/\bar{\Phi}.$ If the background field is negligible,
which is the case when $\Phi_{0}\ll\Gamma^{1/2},$ this equilibrium rotation
rate becomes equal to the threshold value $\Omega_{c}=1.$ It means that in
this case, the driving torque changes only the equilibrium magnetic flux
$\bar{\Phi}=\Gamma^{1/2}$ but not the disc rotation rate which stays equal to
$\Omega_{c}=1$ [1]. It is important to note, however, that in the Bullard
model, which corresponds to a constant driving torque $\Gamma$ with
$\lambda=\Phi_{0}=0,$ this equilibrium is never reached and the dynamo keeps
oscillating around the equilibrium state unless it coincides with the initial
state [1]. Periodic oscillations of the Bullard dynamo are due to an energy-
like conserved quantity:

$(\Omega-1)^{2}+\Phi^{2}-\Gamma\ln\Phi^{2}=\text{const},$

which makes Eqs. (9, 10) integrable. This is no longer the case when the
dynamo is driven by a constant power $\Pi=\Omega\Gamma,$ which also acts as a
mechanical damping. Namely, for small-amplitude oscillations around the
equilibrium rotation rate with $\Omega=1+\tilde{\Omega}$, we have

$\Gamma=\Pi/\Omega\approx\Pi-\Pi\tilde{\Omega},$

where the term $\Pi$ represents the constant part of the driving torque
whereas the term $-\Pi\tilde{\Omega}$ is equivalent to the damping with a
coefficient equal to $\Pi$. There are two additional electromagnetic damping
effects in the modified Bullard dynamo model. They become obvious when Eqs.
(9, 10) are combined into the following equation for small-amplitude magnetic
flux perturbation $\tilde{\Phi}=\Phi-\bar{\Phi}$ around the equilibrium flux
defined by Eq. (11):

$\partial_{t}^{2}\tilde{\Phi}+\tilde{\kappa}\partial_{t}\tilde{\Phi}+(2\bar{\Phi}^{2}-\Phi_{0}\bar{\Phi})\tilde{\Phi}=0.$

This equation features an effective damping coefficient

$\tilde{\kappa}=\lambda\bar{\Phi}^{2}+\Phi_{0}/\bar{\Phi}$ (12)

with the two terms on the RHS being due to the eddy currents induced in the
disc and the background magnetic flux, respectively.

The temporal evolution of the rotation rate and the magnetic flux when the
disc is driven either by a constant torque $\Gamma$ or power $\Pi$ with no
background magnetic field $(\Phi_{0}=0)$ and negligible friction $(\kappa=0)$
is shown in Fig. 5 for the eddy current parameter $\lambda=0$ and $\lambda=1$.

The effect of a background magnetic field with $\Phi_{0}=0.01\Gamma^{1/2}$ on
the evolution of the dynamo driven by a constant torque with various friction
coefficients is illustrated in Fig. 6. The characteristic feature of the
temporal evolution of disc dynamo, which is present in all cases except the
marginal case of $\kappa=1,$ is the overshooting of the equilibrium state and
the subsequent oscillations of the rotation rate and the magnetic flux. These
oscillations, which decay unless $\kappa=\lambda=0$ and the dynamo is driven
by a constant torque, imply that deviation from the quasi-stationarity becomes
significant when the disc rotation rate approaches the dynamo threshold
$\Omega_{c}=1.$ This is because the effective magnetic relaxation time being
reciprocal of the growth rate diverges as $\sim(\Omega_{c}-\Omega)^{-1}$ when
$\Omega$ approaches $\Omega_{c}.$ As a result, the quasi-stationarity
inevitably breaks down in the vicinity of the dynamo threshold regardless of
how slowly the rotation rate is ramped up. Deviation from the quasi-stationary
solution of Eq. (9):

$\Phi(\Omega)=\frac{\Phi_{0}}{\Omega_{c}-\Omega},$ (13)

becomes obvious when the magnetic flux is plotted against the rotation rate as
in Fig. 7. This numerical data can be used to assess the possibility of
recovering $\Omega_{c}$ from the best fit to the quasi-stationary solution.
Limiting the fit to $\Omega\leq\Omega_{c}/2$, where the solution is relatively
quasi-stationary, we recover the values of $\Omega_{c}$ which slightly vary
with $\kappa$ and lie in the shaded strips Fig. 7. As seen, $\Omega_{c}$
produced by the best fit is just slightly higher than the true value
$(\Omega_{c}=1)$ for $\Gamma=0.01$ whereas the difference is about $20\%$ when
$\Gamma=0.1.$

(a)(b)

Figure 7: The magnetic flux $\Phi$ versus the rotation rate $\Omega$ from Fig.
6 along with the quasi-stationary solution (13) for $\Omega_{c}=1$ and
$\Gamma=0.01\,\text{(a)},0.1\,\text{(b)}$. The shaded strips show the range of
$\Omega_{c}$ resulting from the best fit of numerical results for various
$\kappa$ and $\Omega\leq\Omega_{c}/2$, where the solution is relatively quasi-
stationary.

When the dynamo is powered by an induction motor supplied with constant
frequency and voltage, the developed torque varies non-monotonously with the
rotation rate. The torque predicted by the basic asynchronous electric motor
model [22] can be written as

$\Gamma=2\Gamma_{\text{max}}\frac{(\Omega_{d}-\Omega)(\Omega_{d}-\Omega_{m})}{(\Omega_{d}-\Omega)^{2}+(\Omega_{d}-\Omega_{m})^{2}},$
(14)

where $\Omega_{d}$ and $\Omega$ are the synchronous and actual rotation
frequencies, $\Omega_{m}$ is the frequency at which the torque attains its
peak value $\Gamma_{\text{max}}.$ When $\Omega\approx\Omega_{d}$, the torque
varies approximately linearly with $\Omega_{d}-\Omega$:

$\Gamma\approx
2\Gamma_{\text{max}}\frac{\Omega_{d}-\Omega}{\Omega_{d}-\Omega_{m}}.$

When the frequency difference $\Omega_{d}-\Omega$ becomes sufficiently large,
the driving torque attains a maximum $\Gamma_{\text{max}}$ at the critical
frequency $\Omega_{m}$ and then drops off asymptotically as

$\Gamma\sim
2\Gamma_{\text{max}}\frac{\Omega_{d}-\Omega_{m}}{\Omega_{d}-\Omega}.$

If the equilibrium rotation frequency $\Omega_{c}=1$ happens in the latter
operating regime, for small perturbations of the rotation rate
$\tilde{\Omega}=\Omega-1$, we have

$\Gamma\approx
2\Gamma_{\text{max}}\frac{\Omega_{d}-\Omega_{m}}{\Omega_{d}-1}\left(1+\frac{\tilde{\Omega}}{\Omega_{d}-1}\right),$

where the second term is analogous to a friction force but with negative
coefficient

$-2\Gamma_{\text{max}}\frac{\Omega_{d}-\Omega_{m}}{(\Omega_{d}-1)^{2}}.$

When this negative friction outweighs the electromagnetic damping with the
coefficient (12), the stationary dynamo state becomes unstable giving rise to
periodic oscillations as in the original Bullard dynamo model. This
instability is illustrated in Fig. 8 where the oscillations can be seen to
cease decaying when the motor is driven by a fixed frequency $\Omega_{d}=1.2$
and the frequency at which the torque attains maximum is increased from
$\Omega_{m}=1$ to $1.1.$

(a)(b)

Figure 8: Temporal evolution of the rotation rate $\Omega$ and the magnetic
flux $\Phi$ in the presence of a background field with
$\Phi_{0}=0.01\Gamma_{\text{max}}^{1/2}$ and $\lambda=1$ when the disc is
driven by an electric motor with synchronous frequency $\Omega_{d}=1.2$ and
the maximal torque $\Gamma_{\text{max}}=0.01\,\text{(a)},0.1\,\text{(b)}$
developed at $\Omega_{m}=1,\,1.1.$

##  5 Discussion

The measured coil voltage allows us to assess the electric current and the
associated ohmic heating. Firstly, using the coil resistance [17]

$R_{c}=\frac{1+\beta^{2}}{2\pi
d\sigma_{\text{Cu}}}\ln\frac{r_{o}}{r_{i}}\approx 061.\mathrm{\mu\Omega,}$

and the Ohm’s law, we can estimate the current generated by the dynamo at the
coil voltage $V=10\,\mathrm{mV}$ as $I=V/R_{c}\approx 16\,\mathrm{kA}.$ Then
the associated power of ohmic heating in the coil and disc with the
resistivity $R_{d}=R_{c}(1+\beta^{2})^{-1}$ is

$P=I^{2}(R_{c}+R_{d})=\frac{V^{2}}{R_{c}}\left(1+(1+\beta^{2})^{-1}\right)\approx
200\,\mathrm{W.}$ (15)

This power is significant relative to the maximal power the motor is expected
to produce when operating outside its optimal regime with a large slip.
Additional, but presumably insignificant, ohmic as well as viscous power
dissipation is expected in the liquid metal contacts.

On the other hand, using the current estimate above, the average magnetic flux
density produced by the helical coil can be evaluated as

$\bar{B}_{c}=\frac{\beta\text{$\mu_{0}$}Ir_{o}\bar{\phi}(\tilde{r}_{i}/r_{o})}{4\pi
S}\approx 4\,\mathrm{mT.}$ (16)

where $S=\pi(r_{o}^{2}-\tilde{r}_{i}^{2})$ is the area of the helical part of
the coil, $\beta=\tan(58^{\circ})\approx 1.6$ is the helicity of the
logarithmic spiral arms, $\tilde{r}_{i}/r_{o}\approx 1/4$ is the ratio of the
outer and inner radii of the coil and $\bar{\phi}(1/4)\approx 1.38$ is the
respective dimensionless flux [17]. It is interesting to note that this
estimate is significantly lower than the magnetic field strength measured on
the surface of the coil. There may be several reasons for this difference.
Firstly, the distribution of the magnetic field along the surface of the coil
is highly non-uniform. In the current-sheet approximation, the vertical flux
density $B_{z}=r^{-1}\partial(rA_{\phi})$, which is plotted in Fig. 9 using
the analytical solution for the vector potential $A_{\phi}$ obtained in [17],
becomes unbounded at the inner and outer radii of the coil, where it has
logarithmic singularities. Besides that the magnetic flux density can be seen
in Fig. 9 to change direction close to the outer edge of the coil. As a
result, the average flux density is much lower than the local field strength.
Nevertheless, the field strength at the centre of the coil, which represents a
well-defined characteristic value for this system, is comparable to the field
strength measured in the vicinity of the inner radius of the coil. Secondly,
regardless of the current-sheet approximation, there are logarithmic
singularities in the field strength at the sharp edges of the coil which may
affect the readings of the Hall sensor when placed directly atop the coil.

Figure 9: Radial distribution of the vertical magnetic field $B(r)$ along the
top surface of the coil computed using a quasi-stationary current-sheet
approximation [17] for the coil voltage $V=10\,\mathrm{mV}$ which corresponds
to the dynamo current $I\approx 16\,\mathrm{kA}.$ The two horizontal lines
show the average values for the coil and disc, respectively.

Thirdly, and most importantly, the magnetic field distribution is strongly
affected by the iron frame of the set-up, especially by the four crossed arms
holding the coil. . As the Hall sensors were placed on the coil next to these
beams, they could be affected by the field concentration caused by these
ferromagnetic elements. The effect of the iron frame is evident in the
background magnetic field which was found to be $\sim 1\,\mathrm{mT,}$ _i.e_.,
more than an order of magnitude stronger than Earth’s field. This fact implies
that the iron frame can have a similar magnitude effect also on the magnetic
field generated by the coil.

Given the relative smallness of the coil area covered by the frame, the latter
is expected to have a comparably small effect on the average magnetic flux
density. This is confirmed by the effective magnetic flux density through the
disc $B_{\text{eff}}$ which can be estimated by comparing the e.m.f generated
by the disc, $\Omega B_{\text{eff}}(r_{o}^{2}-r_{i}^{2})/2,$ and the total
voltage drop over the disc and coil. The latter can be estimated using the
ohmic heating power $P$ (15) as $P/I,$ which results in

$B_{\text{eff}}=2V\frac{1+(1+\beta^{2})^{-1}}{\Omega(r_{o}^{2}-r_{i}^{2})}.$
(17)

As seen in Fig. 10, $B_{\text{eff}}$ varies nearly linearly with the coil
voltage as predicted by the quasi-stationary approximation well outside this
regime. The best fit of this variation for $V<6\,\mathrm{mV}$ produces an
effective flux density of the background magnetic field $\bar{B}_{0}=0.29\pm
0.03\,\mathrm{mT},$ which is about a factor of three lower than the strength
of the background field measured under one of the four iron beams holding the
coil.

On the other hand, the effective magnetic flux density, which reaches
$B_{\text{eff}}\approx 7\,\mathrm{mT}$ at the coil voltage
$V=10\,\mathrm{mV}$, is somewhat higher than $\bar{B}_{c}$ defined by Eq.
(16). This difference may be due to the inner radius of the disc
$r_{i}=4.5\,\mathrm{cm}$ being smaller than the respective radius of the
helical part of the coil $\tilde{r}_{i}=7.5\,\mathrm{cm}.$ As a result, the
inner part of the disc with $r<\tilde{r}_{i}$ is exposed to a relatively
strong magnetic field which is generated by the coil in this region (see Fig.
9). On the other hand, the respective magnetic flux density averaged over the
whole area of the disc, $\bar{B}_{d}\approx 6\,\mathrm{mT,}$ is relatively
close to $B_{\text{eff}}.$

Figure 10: Effective magnetic flux density $B_{\text{eff}}$ (17) against the
coil voltage $V$ for three runs along with the best linear fit
$B_{\text{eff}}=B_{0}+\kappa V$ for $V<6\,\mathrm{mV}$ which yields
$B_{0}=0.29\pm 0.03\,\mathrm{mT}.$

The critical frequency $\Omega_{c}$ can be determined by fitting the quasi-
stationary solution (13), which can be written in terms of the respective
quantities as

$\displaystyle V(\Omega)$ $\displaystyle=$
$\displaystyle\frac{V_{0}\Omega}{\Omega_{c}-\Omega},$ (18) $\displaystyle
B(\Omega)$ $\displaystyle=$
$\displaystyle\frac{B_{0}\Omega_{c}}{\Omega_{c}-\Omega},$ (19)

to the variation of the coil voltage and magnetic field with the rotation
rate, which is plotted in Fig. 11. Limiting the fit to
$\Omega<6\,\mathrm{Hz},$ where the variation is expected to be sufficiently
close to quasi-stationary, and using the coil voltage, we find
$\Omega_{c}\approx 7.2,6.9,7.2\,\mathrm{Hz}$ for the respective run (see Fig.
11a). The best fit of the magnetic field yields $\Omega_{c}\approx
7.0,6.7,6.9\,\mathrm{Hz},$ which are somewhat lower than the respective
previous values. The last value is also slightly higher than the apparent
equilibrium rotation rate in the third run which may be seen in Fig. 2 to
reach $\approx 6.8\,\mathrm{Hz}.$

Alternatively, $\Omega_{c}$ can be estimated using the the short equilibrium
state which appears in the third run between $t\approx 60\,\mathrm{s}$ and
$70\,\mathrm{s,}$ where $\Omega\approx 6.8\,\mathrm{Hz},$$B\approx
45\,\mathrm{mT}$ and $V\approx 12.9\,\mathrm{mV.}$ Substituting these values
into equations (18) and (19) and using $B_{0}\approx 1.1\,\mathrm{mT}$ and
$V_{0}\approx 0.45\,\mathrm{mT},$ which follow from the best of $B$ and $V$
for this run, after a few rearrangements, we obtain
$\Omega_{c}=\Omega/(1-B_{0}/B)\approx 6.97\,\mathrm{Hz}$ and
$\Omega_{c}=\Omega(1+V_{0}/V)\approx 7.0\,\mathrm{Hz}.$ As seen both results
are practically identical and lying between the values obtained from the best
fit of quasi-stationary solutions for $V$ and $B.$ Note that the theoretical
dynamo threshold for our set-up, which is defined for negligible contact
resistance by the minimal critical magnetic Reynolds number
$\mathit{Rm}\approx 35$ [17], corresponds to $\Omega_{c}\approx
8.2\,\mathrm{Hz}.$ Although the background magnetic field makes the
equilibrium rotation rate somewhat lower than the self-excitation threshold,
the difference between the theoretical threshold and that extracted from the
best fit of experimental results may be due to the current sheet approximation
used in the theoretical model.

(a)(b)

Figure 11: The coil voltage (a) and the vertical component of the magnetic
field at the inner radius of the coil (b) versus the rotation rate along with
the best fits of the quasi-stationary solution (18,19) for
$\Omega<5\,\mathrm{Hz.}$

The effective background magnetic field $B_{0}$ can also be estimated using
the best-fit parameters $V_{0}$ and $\Omega_{c}$ of the coil voltage.
Substituting $V$ from Eq. (18) into Eq. (17) and taking $\Omega\rightarrow 0,$
it is easy see that $\bar{B}_{0}$ is obtained by replacing $V$ and $\Omega$ in
Eq. (17) with $V_{0}$ and $\Omega_{c}.$ For the respective run, this yields
$\bar{B}_{0}\approx 0.33,0.26,0.30\,\mathrm{mT,}$ which are consistent with
the previous estimate obtained from the best fit of the effective magnetic
flux variation with the coil voltage plotted in Fig. 10. The larger scatter in
the last estimate is because it is obtained by using low rotation rates only,
whereas the previous relation holds in a much larger range of $\Omega$ which
may be seen in Fig. 10 to extend well beyond the quasi-stationary regime.

##  6 Summary and conclusions

In this paper, we presented experimental results from three successful runs of
a Bullard-type homopolar disc dynamo. The realisation of such a dynamo was
commonly thought impossible because of the prohibitively high rotation rates
which are required when sliding electric contacts are made of graphite
brushes. We overcome this problem by using $\mathit{GaInSn}$ eutectic alloy,
which is liquid at room temperature, for sliding electric contacts. The set-up
consisted of a copper disc with a radius of $30\,\mathrm{cm}$ and thickness of
$3\,\mathrm{cm}$ which was placed co-axially beneath a flat, multi-arm spiral
coil of the same size and connected to it electrically at the centre and along
the perimeter by $\mathit{GaInSn}$ contacts [17]. The dynamo was effectively
axisymmetric but anisotropic because of the spiral slits which deflected the
current in the coil azimuthally so generating an axial magnetic field.

The use of liquid metal in sliding electrical contacts came with two
complications. Firstly, the liquid metal was expelled from the peripheral
contact by centrifugal force radially inwards over the top surface of the
coil. This problem was largely mitigated by reducing the annular gap between
the disc and the coil from $3\,\mathrm{mm}$ in the original design [20] to
$0.25\,\mathrm{mm}$ in this set-up. Secondly, as the liquid metal was exposed
to air, it quickly oxidized when the device ran. This limited the time of the
experiment to a few minutes after which the electric contact between the disc
and the coil usually failed.

The runs differed mainly by how the motor driving frequency $\Omega_{d}$ was
varied. In the first run, $\Omega_{d}$ was increased at a nearly constant rate
and the magnetic field was measured using only the low-field probe which was
placed on the coil in its central part close to the inner radius of the spiral
slits. When the disc rotation rate reached $\Omega\approx 6.5\,\mathrm{Hz},$
the vertical field component was found to exceed $9\,\mathrm{mT},$ which was
the upper detection limit of this probe. Although the driving frequency was
set at $\Omega_{d}=7.33\,\mathrm{Hz},$ the rotation rate started to fall after
reaching $\approx 6.7\,\mathrm{Hz.}$ It means that the electromagnetic torque
braking the disc had exceeded the breakdown torque of the electric motor,
which, thus, started to stall. Extrapolation using the voltage drop across the
coil, which was measured in addition to the magnetic field, indicated that the
vertical field strength in the first run had reached
$\approx$$25\,\mathrm{mT}.$ It is important to note that the magnetic field in
the vicinity of the inner radius of the coil may be significantly higher than
the average over the coil surface.

In the second run, the low-field probe was moved to the outer radius and the
medium-field probe was installed in its place. Besides that, the driving
frequency was ramped up through intermediate constant steps rather than
continuously as in the first run. We again observed a steep increase of the
magnetic field with $B_{y}$ at the inner radius reaching $26.5\,\mathrm{mT}$
when the rotation rate approached $6.4\,\mathrm{Hz}.$ At this point, the
rotation rate started to fall as in the first run. However, the second run was
sufficiently long for the disc to start spinning up again. The re-acceleration
was enhanced by ramping up $\Omega_{d}$ to $8.47\,\mathrm{Hz}.$ This resulted
in $\Omega$ reaching $\approx 6.5\,\mathrm{Hz}$ and $B_{y}$ shooting up to
$\approx 38\,\mathrm{mT.}$ At this point, the motor started to stall again and
the experiment was terminated because of the loss of electric contact.

Stalling was not observed in the third run, in which the driving frequency
$\Omega_{d}$ was set higher and initially ramped up faster. With
$\Omega_{d}\approx 8.9\,\mathrm{Hz}$ reached in $\approx 60\,\mathrm{s},$ the
rotation rate saturated at $\Omega\approx 6.6\pm 0.1\,\mathrm{Hz}$ after
$\approx 45\,\mathrm{s}.$ The magnetic field kept growing as long as
$\Omega_{d}$ increased saturating at $\approx 45\,\mathrm{mT.}$ After $\approx
30\,\mathrm{s,}$ when $\Omega_{d}$ was raised to $\approx 9.7\,\mathrm{Hz},$
$B_{y}$ increased to $\approx 48.6\,\mathrm{mT}$ without a noticeable change
in the rotation rate $\Omega.$ This is a characteristic behaviour of a fully
developed disc dynamo.

We also proposed an extended disc dynamo model which qualitatively reproduces
experimental results by taking into account the background magnetic field,
transient eddy currents in the disc as well as the non-linearity of the
electric motor. The background magnetic field, which was found to be an order
of magnitude stronger than Earth’s magnetic field, was obviously due to the
iron frame of the set-up. At sub-critical rotation rates, i.e., those below
the kinematic threshold, the dynamo works as a homopolar generator amplifying
the background magnetic field. In quasi-stationary approximation, the
amplification rate increases with the rotation frequency and tends to infinity
at the dynamo threshold. We used this fact to determine the dynamo threshold
from the best fit of the magnetic field and the coil voltage. In this way, we
found a critical rotation frequency $\Omega_{c}\approx 7\pm 0.2\,\mathrm{Hz},$
which is somewhat larger than the actual saturation frequency observed in the
third run. This is consistent with the numerical results which showed that the
transient effects result in the best-fit value being somewhat higher than the
actual dynamo threshold. On the other hand, this experimental result is
somewhat lower than the theoretical prediction $\Omega_{c}\approx
8.2\,\mathrm{Hz}$ corresponding to the minimal critical magnetic Reynolds
number $\mathit{Rm}\approx 35$ for a negligible contact resistance [17]. This
difference is likely due to the very approximate nature of the current sheet
model which was used to evaluate the dynamo threshold. In particular, due to
the strongly uniform radial magnetic flux distribution (see Fig. 9), the total
magnetic flux through the disc may be larger than theoretically predicted
because the effective inner radius of the disc is smaller than that of the
coil. The critical rotation rate can be reduced further by optimizing the
inner and outer disc radii which were originally chosen to be the same as
those of the coil.

We are grateful to Adrian P rez, Carla Bello and and Manuel Valencia for the
technical support. This work was supported by the National Council of Science
and Technology of Mexico (CONACYT) through grant CB-168850 and the National
Polytechnic Institute (IPN) through grants SIP-20211736 and SIP-20220849.

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