Alex Protopopov

Correspondence: Alex Protopopov proto.alex@hotmail.com

**Author Affiliations**

Moscow Institute of Physics and Technology, Russia.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Quench of a superconducting magnet in MRI scanners may cause damage to the coil, but is unavoidable in some situations. Knowledge of the temporal evolution of current and the maximum temperature developed in the coil make it possible to determine the state the coil after the quench. Commonly, these parameters are computed using three-dimensional simulation programs. However, numerical models do not provide physical links between basic parameters of the magnet and quench dynamics. This paper presents analytical formulas that may serve as reasonable estimates, showing relationship between quench dynamics and parameters of a superconducting magnet.

**Keywords**: Superconducting coils, quench, magnet, MRI

In recent years, great attention has been paid to the problem of simulation and assessment of aftermaths of quench in MRI. When quench occurs in a superconducting magnet, it is not always clear if the coil requires repairs or not. If it was damaged, then the magnet cannot be ramped up, and lengthy and expensive repairs are in order. However, if the probability of coil damage is minimal, then re-initialization is the optimal, cost-effective choice. Due to that, numerous cumbersome 3D simulation programmes have been created that enable the computation of quench dynamics in time and space, and allow estimation of values of current and peak temperature. Such programmes depend on many numerical parameters, which need to be known before the program can be executed. Sometimes, even little errors in the values of these parameters may lead to significant deviations. Therefore, a simple qualitative theory with relatively easily computable formulas, that can be used to estimate quench dynamics, would present an advantage in determining the state of the coil. In the present short communication such formulas are derived with very reasonable simplifications.

In the simplified model, only lateral propagation of quench will be considered. Lateral propagation of quench is the spread of temperature increase along the superconducting wire. Thus, the effect that the rise of local temperature has on the surrounding wires is neglected. The reason for this assumption is the high velocity of quench propagation, about 2 m/s for 240 A coils, and the design of the superconducting coil, in which each superconducting wire is separated from the neighboring wires by layers of supporting and shielding materials.

Figure 1 explains the model. Non-linear thermal dependence
of the specific resistance *r* around critical temperature *T _{c}* is
the principle phenomenon of the quench. What is essential, that
it is practically zero below

(1)

Figure 1 : **Superconducting wire with specific resistance [Ohm.
cm], cross section [cm ^{2}], thermal conductivity, and specific
heat capacity.**

This approximation may be called the bar-step approximation.
Before the quench starts, the wire transmits initial current *I _{0}* . We
assume that the quench originates at

(2)

begins to develop within this element of wire. Thus, variation
of the heat during time increment *dt* is

(3)

As the temperature at *x* = *x _{0}* rises due to this heat, heat flux
spreads out of the volume

(4)

where *k(x)* is thermal conductivity and *ρ *- density of
the superconducting wire. Note that the second term in the
left-hand side acts negatively because the second derivative
of temperature around *x _{0}* is negative.

**Solution for the current**

Equation (4) has to be completed by the electrical equation,
which follows from the energy conservation law: the sum of
the magnetic energy *E* stored in the magnet and the power
transferred into heat *Q* is constant:

(5)

Here *δ Q* is the total dissipation along the entire length of
the superconducting wire:

(6)

According to phenomenology of quench, we introduce the
longitudinal velocity of quench propagation *v _{0}* [cm/s]. With
it, the length

(7)

Since other sections remain superconducting with *r* = 0 ,
the integral in the equation above is equal to *r* . *v _{0} t* :

(8)

Magnetic energy is

(9)

where L is the magnet inductance, and

(10)

which makes the equation

(11)

It gives simple analytical formula for evolution of the current during quench:

(12)

We introduced parameter *τ* that may be called the quench
time - the time interval during which the current drops *e* times from the value of stable operation. The resistance per
unit length

(13)

is always known: it is one of the design parameters of the magnet wire that is always carefully characterized. Typically, η ≈ 0.15 Ohm/m [1-4]. With this, the quench time transforms to

(14)

Dimension of the right-hand side is time since inductance L measured in Henry is equal to

where C stands for the Coulomb, Wb - for the Weber (magnetic flux), F - for the Farad, W - for the Ohm. Formula (14) for the quench time clearly shows that the smaller the magnet inductance the shorter the quench. Alternatively, the smaller the resistance above superconducting threshold the longer the quench.

Now, we shall compare the formula for the current (12)
with experimental results. Qualitatively, it is consistent with
all the experimental results on quench published so far. For
instance, Figure 2 shows the Gaussian curve (12) fitted into
the results published in [5], with the fitting parameters *I _{0}*
= 200 A and τ = 7 s.

Whatever good the qualitative agreement is, does our theorydescribe the phenomenon quantitatively? The answer to this question can only be given with the complete set of data, characterizing the superconducting magnet. One of the few publications that provide necessary data is [6]. Technical specifications of the magnet described in this article are reproduced in the Table 1 below.

It uses NbTi superconducting filaments evenly spread within the Nb wire, which is embedded into a bulk cupper strip.

Figure 3 compares experimental data for quench current
from [6] with our theory. Analytical formula (12) is fitted into
the experimental data with the value *τ* = 12 s. The immediate
conclusion that follows from this figure is that, qualitatively, the
analytical formula is again in agreement with the experiment.
Next, consider quantitative details, specifically the quench
velocity *v _{0}* . From (12) it follows that

(15)

The table gives the value of magnetic energy stored in the magnet

and the current *I* = 232 A. This is the single-coil magnet, so
that we find *L *= 15.4 H. For η we do not have necessary
data, but we may take the typical value cited above: *η* = 0.15 Ohm/m. With all these data, formula (15) gives *v _{0}* = 1.4 m/s. Quench velocity can hardly be measured, but
luckily, we can compare this value to the value numerically
simulated in [6] and presented in Figure 4 in the form of
lines. Our value, computed for the particular current 232 A,
is shown by the open circle.

Figure 4 : **Quench velocity simulated by numerical models [6]
and the value given by analytical formula (15).**

Thus, analytical formula (15) gives the result consistent with the numerical models.

**Solution for the temperature**

Quench current and propagation velocity are, of course,
important parameters, but even more important is evaluation
of the maximum spot temperature that is developed during
quench. If this temperature exceeds critical value, then the
superconducting coil is probably damaged. Consider the
basic equation (4) again. In it, specific heat capacity *C* is
a complicated function of temperature. Copper is the main
heat absorbing material of the wire, and for it *C(T )* is
shown in Figure 5 [4]. With temperature being in the interval
of 4 K<*T* <100 K, specific heat capacity of copper can be
reasonably described by a second-order approximation
*C* = 0.03 .*T ^{ 2}*, or in general:

(16)

Figure 5 : **Copper specific heat (solid line) and its approximation
C = 0.03 T^{2} (dashed line).**

In such an approximation, dimension of *α *is J/(kgxK^{b}) and *β *is dimensionless.

The basic equation (4) is the partial differential equation of
the second order that cannot be solved analytically without
simplifying assumptions. First, assume thermal conductivity
*k* independent of the temperature, and as such, independent
of *x* . This assumption is supported by experimental
phenomenology, showing that around the critical temperature
* T_{c} , k* varies insignificantly [8].

Our main goal is to derive the formula for the maximum
spot temperature. This point is located at the quench origin
*x _{0}* = 0 , from which the quench wave spreads to the more
distant sections of the wire. Here, the term with thermal
conductivity

This leads us to the equation

(17)

where again *η* = *r */ *S* is the resistance of the wire per unit
length, *µ* = *ρ* .*S * and is the mass of the wire per unit length.
Substituting (12) and (16) and integrating over temperature,
we obtain:

(18)

where *T*_{0} = 4.2 K is the initial temperature for the superconducting
wire. This gives the solution:

(19)

Here

is the well-known error-function, and

(20)

is an important parameter of a superconducting wire, determining
dynamics of the temperature. Dimension of this parameter
is the same as α, i.e., J/(kgxK^{b}).

With t→∞, temperature stabilizes at the maximum value *T _{max }*. Since

(21)

This is the very important result, showing how the maximum temperature depends on the parameters of the super-conducting wire, current, and quench time. It is not obvious that the dimension of this result is equal to Kelvin, therefore we show this explicitly:

Compare (21) to the results, computed in [6], using the 3D
model (Figure 6). High temperature specific resistance of a
superconducting material itself is very high. For instance, for
NbTi *r* ~ 5.10^{-7} Ohm.m at *T* ~ 100 K, whereas for the copper
cladding *r* ~ 10^{-10} Ohm.m [4]. Therefore, high-resistance
section of the superconducting filament will be bypassed
by surrounding copper, so that the values for *η* and *µ* in
(21) should be those for the copper. We used the following
set of parameters, appropriate to the result described in [6]:

*I _{0}* = 232 A;

Direct substitution of the above parameters into (21) gives T_{max }= 51 K, which is not very far from the experimental
value T_{max} ≈ 85 K [6]. The largest part of uncertainty lies in
the ratio *η /µ* . The above values of *η* and µ give *η* /*µ* =0.003 in SI units. Correcting it to 0.015, we obtain necessary
T_{max} = 85 K. With this only correction, theoretical curve of
formula (19) is shown in Figure 6.

Although entire qualitative behavior of the analytical formula
preserves the essential features of the spot temperature dynamics,
like steep initial trend and smooth finish to a stationary value
of T_{max }, it differs from numerical computations in the middle
part of the interval. Nonetheless, analytical formulas (19)-(21)
combine all the parameters together, explicitly showing the
influence of each of them - a feature that cannot be obtained
by numerical computations.

Analytical formula derived for temporal evolution of the quench current is in good quantitative agreement with experimental data. This formula establishes relation between the quench time and very few basic parameters of the superconducting magnet: wire resistance per unit length, magnet inductance, and longitudinal quench speed. The strength of this solution is not the simplicity of the Gaussian function that describes evolution of the current, but that it establishes quantitatively correct general analytical relationship between the current and only three basic measurable parameters of the superconducting magnet.

Using the formula for the quench current, longitudinal quench velocity can be estimated from the recorded log-file of the quench event. For that, the Gaussian function should be fitted into the experimental curve with only one fitting parameter: the quench time. Quench velocity turned out to be inversely proportional to the quench time squared.

Analytical formula for temperature dynamics is in qualitative agreement with experimental data. From it, the maximum value of the temperature increase during quench can be estimated. The estimate of the maximum temperature determines probability of whether or not the superconducting coil has been damaged during quench. This very initial assessment, done quickly and without applying cumbersome numerical computations, may be important for making initial decision after quench has happened.

The overall conclusion is that even very simple analytical analysis, accounting for only few driving physical considerations, may provide valuable initial insight into rather complicated problem.

The author declares that he has no competing interests.

EIC: Robert A. Lodder, University of Kentucky, USA.

Domenico Rubello, Padova University, Italy.

Received: 06-Feb-2014 Final Revised: 17-May-2014

Accepted: 17-Jul-2014 Published: 24-Jul-2014

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Volume 2

Protopopov A. **Analytical formulas for current dynamics and peak temperature during quench in MRI**. *Med Instrum*. 2014; **2**:3. http://dx.doi.org/10.7243/2052-6962-2-3

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