# LEAFAGE: Example-based and Feature importance-based Explanations for Black-
box ML models

Ajaya Adhikari1, David M. J. Tax2, Riccardo Satta1, Matthias Faeth2 1TNO, The
Hague, The Netherlands2Delft University of Technology, Delft, The Netherlands

###### Abstract

Explainable Artificial Intelligence (XAI) is an emergent research field which
tries to cope with the lack of transparency of AI systems, by providing human
understandable explanations for the underlying Machine Learning models. This
work presents a new explanation extraction method called LEAFAGE. Explanations
are provided both in terms of feature importance and of similar classification
examples. The latter is a well known strategy for problem solving and
justification in social science. LEAFAGE leverages on the fact that the
reasoning behind a single decision/prediction for a single data point is
generally simpler to understand than the complete model; it produces
explanations by generating simpler yet locally accurate approximations of the
original model. LEAFAGE performs overall better than the current state of the
art in terms of fidelity of the model approximation, in particular when
Machine Learning models with non-linear decision boundaries are analysed.
LEAFAGE was also tested in terms of usefulness for the user, an aspect still
largely overlooked in the scientific literature. Results show interesting and
partly counter-intuitive findings, such as the fact that providing no
explanation is sometimes better than providing certain kinds of explanation.

##  I Introduction

In the context of Artificial Intelligence, Machine Learning (ML) is a rapidly
growing field. There has been a surge of high-performance models for
classification and prediction. Still, the application of these models in high-
risk domains is more stagnant due to lack of transparency and trust: there is
a disconnect between the black-box character of these models and the needs of
the users. Explainable Artificial Intelligence (XAI) has recently emerged to
provide solutions to this issue by attempting to create understandable
explanations for the reasoning of a black-box model.

Example-Based Reasoning (EBR), i.e., motivating a decision by providing
examples of similar situations, is widely recognized as an effective way to
provide explanations [1], as it bears a close resemblance to the way humans
think. As a result, it is commonly used e.g. in the health-care sector for
decision-support systems [2, 3] and in law for justifying arguments, positions
and decisions [4]. However, the usage of EBR to explain black-box ML models
(i.e., models whose inner mechanisms are either unknown by the user, or too
complex to be practically comprehensible by a human), has been largely
overlooked so far in the scientific literature. This is partly because of the
difficulty of finding examples according to the inner reasoning of such a
model. Notably, most of the scientific literature focuses instead on
evaluating the relative importance of features (feature importance-based
explanations, see e.g. LIME [5]).

In this paper, we propose a new method for providing both feature importance-
based and EBR explanations of the local reasoning of black-box models. Here,
local refers to the ability of tailoring the explanation to a single
prediction taken by the ML model, as opposed to providing a global explanation
of the whole model logic. We named the method LEAFAGE \- Local Example and
Feature importance-based model AGnostic Explanations. LEAFAGE approximates the
local reasoning of the black-box model by a (transparent) linear model. As a
byproduct, LEAFAGE is also able to provide the importance of each feature for
a prediction.

We evaluate LEAFAGE both in terms of fidelity, and of usefulness to the user.
Fidelity refers to whether the extracted explanation reflects the true
reasoning of the underlying black-box ML model. The usefulness to the user is
evaluated by conducting a user-study in terms of perceived aid in decision-
making and objective transparency.

The remainder of this paper is structured as follows. In Chapter II, we
provide background information and related work on XAI, and explore approaches
to provide explanations that leverage on social research. Chapter III
describes LEAFAGE, which is then evaluated in terms of fidelity and usefulness
to the user in Chapters IV and V. Finally, Chapter VI draws conclusions and
suggests future research directions.

##  II Background

This Section surveys current literature on XAI for ML models, and on the
user’s perspective on an explanation.

An explanation about a ML model can be of global or local scope. A global
explanation clarifies the inner workings of the whole ML model, i.e., how the
relationship between input and output spaces is modeled [6]. Local
explanations look instead at the reasoning behind a decision/prediction over a
single input data point (test sample), thus targeting a sub-region of the
input space. As the complexity of the ML model grows, it becomes harder to
generate an understandable global explanation. However, it is likely that the
logic of the ML model in the neighbourhood of a single test sample will be
much simpler, thus allowing to generate understandable local explanations.

Three main strategies for extracting human-understandable explanations from ML
models can be found in the literature: transparent-by-design, model-oriented
and model-agnostic. In the first strategy, the ML model is designed from the
start to be globally transparent and possibly simple enough to be
understandable by humans (e.g. a small decision tree). The latter two
strategies deal instead with an existing model that has not been made
transparent by design. In the model-oriented strategy, certain parts of the
model are used to extract an explanation (e.g., see [7]). In case when the ML
model is too complex, or when internal workings of the model are not
accessible, a model-agnostic strategy is used. This strategy views the ML
model as a black-box, and queries it using a set of instances from the input
space in order to gain insights in the behaviour of the model.

The proposed method, LEAFAGE, falls into the latter category. One of the most
recent methods on the same category is LIME (Local Interpretable Model-
agnostic Explanations) [5], from which LEAFAGE borrows its main ideas. LIME
provides a local explanation by linearly approximating the decision boundary
of the ML model in the neighbourhood of the test sample. Figure 1 shows an
example of how LIME works in a binary classification problem. The two classes
are Red and Blue, respectively represented by ’+’ and full circles. The
decision boundary of the ML model is between yellow/blue areas. The point
marked with a bold ’+’ is the test sample $\mathbf{z}$. In order to generate
an explanation as for why the model ML classified $\mathbf{z}$ as Red, other
Red and Blue synthetic data points are sampled from the input space.

A linear model is learned on the synthetic data points; higher importance is
given in correctly classifying the synthetic instances that are close to
$\mathbf{z}$. In Figure 1 their size represents their proximity to
$\mathbf{z}$. This proximity from an synthetic instance $\mathbf{x}$ to
$\mathbf{z}$ is defined by an exponential kernel
$\pi(\mathbf{z},\mathbf{x})=e^{-D(\mathbf{x},\mathbf{z})^{2}/\sigma^{2}}$. In
[5], $\sigma$ is fixed to $0.75*\sqrt{dimension}$.

The parameter $\sigma$ of the proximity kernel plays an important role: a
fixed $\sigma$ can lead to neighbourhoods that do not include a decision
boundary, or that include a too big part of the decision boundary which cannot
be approximated linearly. Laugel et al. [8] spotted this problem and suggested
to sample instances close to the nearest decision boundary to $\mathbf{z}$
within a fixed hyper-sphere. However, the fixed hyper-sphere can also lead to
too small or too big neighbourhoods.

Figure 1: The dashed line approximates the blue/red decision boundary in the
neighbourhood of the bold ‘+’ point [5].

XAI research has been criticized for overlooking the viewpoint of the end-
user, i.e., if he/she is satisfied with provided explanations [9]. To address
this, the present work focuses on providing explanations according to Example-
Based Reasoning (EBR), a paradigm where explanations are related to previous
experience [4, 10]. This type of reasoning lies very close to how humans think
[3, 1].

EBR applications can be divided into two types: problem-solving and decision-
justification [4]. In problem-solving, previous similar situations are used as
aid to decide how to proceed with the current situation. In decision-
justification, previous similar situations are leveraged to support or dismiss
certain arguments and decisions. Worth noting, Common Law, which is used in
most English-speaking countries, is based on the same principle (judicial
decision are made on similar cases from the past [10]).

The ultimate goal of an ML explanation system is to provide valuable insights
on an automated prediction/decision to the user. Such aspect can be evaluated
by conducting user-studies. In recommendation systems, extensive research has
been conducted in designing user-studies which evaluate explanations that
clarify why a certain item is recommended, from the user’s point of view [11,
12, 13]. Tintarev [11] defines seven goals for an explanation system, namely
transparency, scrutability, trust, effectiveness, efficiency, persuasiveness
and satisfaction. All of them can be evaluated subjectively by asking
questions to the user [11]. However, while trust, satisfaction and
effectiveness are subjective by nature, transparency, efficiency and
persuasiveness can also be measured objectively. E.g., one can test whether
users have understood the reasoning behind the recommendations [13], measure
the interaction time [12] or check whether the user agrees to buy a
recommended item.

##  III LEAFAGE

This Section describes our proposed method, LEAFAGE (Local Example and Feature
importance-based model AGnostic Explanation). It provides explanations in the
form of examples drawn from the training set, that are similar to the test
sample according to the ML model logic, and shows the importance of each
feature for the prediction.

Let $f:\mathcal{X}\rightarrow\mathcal{Y}$ be a black-box ML model that solves
a binary classification problem with $\mathcal{X}=\mathbb{R}^{d}$ and
$\mathcal{Y}=\\{c_{1},c_{2}\\}$, and $\mathbf{z}\in\mathcal{X}$ be the test
sample, an instance of the input space with $f(\mathbf{z})=c_{z}$,
$c_{z}\in\mathcal{Y}$.

Furthermore, let $X=[\mathbf{x}_{1},..,\mathbf{x}_{n}]$ with the corresponding
true labels $y_{true}=[y_{1},..,y_{n}]$ be the training set used to train $f$,
and $y_{predicted}=\\{f(\mathbf{x}_{i})|\mathbf{x}_{i}\in X\\}$ be the
predicted labels of the training set. Next, let $\\{\mathbf{x}\in\mathcal{X}\
|f(\mathbf{x})=c_{z}\\}$ and $\\{\mathbf{x}\in\mathcal{X}\ |f(\mathbf{x})\neq
c_{z}\\}$ be defined as the ally and the enemy instances of $\mathbf{z}$ [14],
respectively.

LEAFAGE uses $X$, $y_{predicted}$, $\mathbf{z}$ and $c_{z}$ to explain why
$\mathbf{z}$ was predicted as $c_{z}$. It works as follows:

  * •

A subset of the training set in the neighbourhood of $\mathbf{z}$ is used to
build a local linear model. The coefficients of this model provide a measure
of importance of each feature locally.

  * •

These coefficients are used to define a local dissimilarity measure between
any instance $\mathbf{x_{i}}\in\mathcal{X}$ and $\mathbf{z}$. In turn, this
measure is used to retrieve examples similar to $\mathbf{z}$ from the training
set.

  * •

The importance of each feature and the most similar examples are given as
explanation of the classification.

Section III-A and III-B illustrate respectively the adopted dissimilarity
measure, and the strategy to build the local linear model. Next, Section III-C
explains how LEAFAGE explanations can be presented to the user.

###  III-A Defining a local dissimilarity measure

Consider a binary classification problem where an ML model predicts whether a
house has a high or low value according to two features, area and age, as
shown in Figure 2(a) (in green, the decision boundary of a simple linear
classifier). A test house $\mathbf{z}$ is predicted as value high. To find
similar houses in the training set, one could use the Euclidean distance
(Figure 2(a)). However, this choice does not reflect the reasoning of the
classifier, which only looks at the feature area; in fact, according to the
classifier, $\mathbf{z}$ is more similar to $\mathbf{x}_{2}$ than
$\mathbf{x}_{1}$ (Figure 2(b))

(a) Euclidean distance: $\mathbf{z}$ is more similar to $\mathbf{x}_{1}$ than
$\mathbf{x}_{2}$

(b) Black-box classifier: $\mathbf{z}$ is more similar to $\mathbf{x}_{2}$
than $\mathbf{x}_{2}$

(c) Approximate the decision boundary with a linear model.

Figure 2: Illustration of different types of distances.

A way to compute a dissimilarity measure that takes into account the reasoning
of the classifier is to use feature weights derived from a local linear
approximation $\hat{f}_{z}(\mathbf{x})=\mathbf{w}_{z}\mathbf{x}+c$ with
$\mathbf{w}_{z}=(w_{z1},...,w_{zd})^{T}$ of the decision boundary (the blue
line in Figure 2(c)). Then, $\mathbf{w}_{z}$ will denote the most
discriminative direction for the classification of $\mathbf{z}$.

In the simple example of Figure 2 the whole decision boundary can be
approximated accurately by a linear model. ML models are usually much more
complex, see e.g. Figure 3. However, we assume that locally the closest
fragment of the global decision boundary to $\mathbf{z}$ is smooth enough to
be linearly approximated (see the blue line in Figure 3).

The following definitions describe the local behaviour of the ML model around
$\mathbf{z}$. These definitions applied to the housing example are illustrated
in Figure 3.

Figure 3: A complex decision boundary that cannot be accurately approximated
by a linear model.

######  Definition 1.

Let _the local decision boundary_ of $\mathbf{z}$ be defined as the closest
fragment (according to a distance measure $D(\mathbf{x}_{1},\mathbf{x}_{2})$,
e.g. the Euclidean distance) of the global decision boundary to $\mathbf{z}$.

######  Definition 2.

Let _the local linear model_ of $\mathbf{z}$ be the model that approximates
the local decision boundary of $\mathbf{z}$.

######  Definition 3.

Given the local linear model
$\hat{f}_{z}(\mathbf{x})=\mathbf{w}_{z}\cdot\mathbf{x}+c$ of $\mathbf{z}$ let
the _the black-box dissimilarity measure_ between $\mathbf{z}$ and an instance
$\mathbf{t}\in\mathcal{X}$ be defined as the following:

$b(\mathbf{t})=D(\mathbf{w}_{z}^{T}\mathbf{t},\mathbf{w}_{z}^{T}\mathbf{z})*D(\mathbf{t},\mathbf{z}),$

(a) First factor

(b) Second factor

(c) Whole formula

Figure 4: Contour-line visualization of the black-box dissimilarity measure.

If $D$ is the Euclidean distance, in the 2D case the black-box dissimilarity
has the form shown in Figure 4(c). In the first factor of the Definition 3,
$\mathbf{w}$ is used as weights to reflect features’ importance according to
$\hat{f}_{z}$ (Figure 4(a)). However, $\hat{f}_{z}$ is only valid in the
neighbourhood $N$ of $\mathbf{z}$, and it is not straightforward to define
$N$. To cope with that, we propose to leverage on the fact that closer
instances to $\mathbf{z}$ (according to the $D$ distance measure on the input
space) are more likely to be within $N$: therefore, in Definition 3) a second
factor is added, the distance on the input space (Figure 4(b)). Please note
that the dissimilarity measure defined as such, does not always satisfy the
condition of identity of indiscernibles
$b(\mathbf{z^{\prime}})=0\Leftrightarrow\mathbf{z}=\mathbf{z^{\prime}}$:
depending on $\mathbf{w}_{z}$, the features that differ between $\mathbf{z}$
and $\mathbf{z^{\prime}}$ could have no influence on $b$. Therefore this
dissimilarity measure cannot be properly considered a ‘metric’ in a
mathematical sense (instead, it is a ‘pseudometric’).

###  III-B Computation of the local linear model

The local linear model is computed from a neighborhood of $\mathbf{z}$ sampled
from the original training set. Let us denote it as the _local training set_
of $\mathbf{z}$.

Methods to sample this local training set have been proposed in LIME [5] and
LS [8] (relevant details have been provided in Section II). Both methods have
shortcomings related to the right choice of the size of the neighbourhood from
which the local training set was sampled.

Taking into account the issues of LIME and LS, we suggest two desired
characteristics that a local training set of $\mathbf{z}$ should adhere to:

  1. 1.

The convex hull of the local training set of $\mathbf{z}$ should contain the
local decision boundary of $\mathbf{z}$.

  2. 2.

There should be enough instances to represent all classes.

We propose a novel sampling strategy that covers both aspects. Its steps are:

  1. 1.

The local training set of $\mathbf{z}$ is sampled around the local decision
boundary of $\mathbf{z}$ (similar to the idea of LS [8]). This makes it
possible to sample enough instances from both classes. We assume that the
closest enemy $\mathbf{x}_{border}$ of $\mathbf{z}$ from the training set lies
close to the local decision boundary of $\mathbf{z}$ and sample around
$\mathbf{x}_{border}$.

  2. 2.

$i_{small}\cdot d$ samples of each class from the training set are sampled,
that lie the closest to $\mathbf{x}_{border}$ according to the distance
measure $D$. $d$ instances per class are the minimum amount of examples needed
for a good linear approximation, assuming that these $d$ instances lie along
the closest decision boundary of $\mathbf{z}$. Since these $d$ instances might
not lie exactly along the decision boundary, the amount is increased with
$i_{small}$ which is a small integer greater than 1.

An example of this strategy applied on a 2D case with $i_{small}=10$ is shown
on Figure 5. The green and red shapes are instances sampled from the training
set to build the local linear model of $\mathbf{z}$.

Given the local training set of $\mathbf{z}$, a linear classification
algorithm can be used to build the local linear model of $\mathbf{z}$.

Figure 5: Sampling of the local training set of $\mathbf{z}$.

###  III-C Explanation extraction

Given the local linear model
$\hat{f}_{z}(\mathbf{x})=w_{0}+w_{1}x_{1}+w_{2}x_{2}+...+w_{d}x_{d}$ and an
instance $\mathbf{z}=[z_{1},..,z_{d}]$, the importance of each feature $z_{i}$
can be evaluated as $abs(w_{i}*z_{i})$, and can be provided as an explanation
to the user as for which features the original model deems as relevant for its
decision on $\mathbf{z}$. We refer to it as feature importance-based
explanation.

As discussed in Section II, a way to provide explanations that are closer to
how humans think is to use Example-Based Reasoning, i.e. to provide examples
that (according to the logic of the black-box model) are related to the test
point $\mathbf{z}$. As the logic of the black-box model is locally represented
by the black-box dissimilarity measure, the latter can be used to find
training examples similar to $\mathbf{z}$ to motivate the decision.
Furthermore, one can provide both examples belonging to the predicted class
$c_{z}$ and to the opposite class, which provides insights on the differences
between classes according to the black-box model. We refer to them as example-
based explanations. Feature importance-based and example-based explanations
can be also combined to provide better insights.

An example of a test house predicted as high value by a black-box model, and a
LEAFAGE explanation for this prediction, are shown in Figures 6(a) and 6(b),
respectively. The left graph of Figure 6(b) shows the relative importance of
each feature. The two tables on the right show the top 5 similar (according to
the black-box dissimilarity measure) houses from the training set, belonging
to the same class (high value) and from the opposite class low value. From
these explanations, the user can spot insights on the classification logic,
e.g., similar low value houses have smaller living area than similar high
value houses.

(a) A house predicted as value low by a black-box model.

(b) LEAFAGE explanation for the house above.

Figure 6: Example of a LEAFAGE explanation.

##  IV Quantitative Evaluation

This Section evaluates the ability of LEAFAGE to reflect the true local
reasoning of a black-box ML model (faithfulness of the local approximation).

Four different datasets with different number of features, data points and
complexity are used: wine[15], breast cancer[15], banknote[15] and one
artificial dataset. The latter is a set of 2D data points from two highly non-
separable classes. Instances of each class are sampled from two bi-variate
normal distributions with different means ($[0,0]$ and $[0,1]$, respectively)
and the same covariance matrix ($\begin{bmatrix}2&0\\\ 0&2\end{bmatrix}$). The
multi-class datasets are converted to binary datasets of one-vs-rest fashion.
A combination of a binary dataset and a classifier is referred to as a
_setting_ in the following. Each dataset is randomly split into train (70$\%$)
and test set. The train set is used to train six classifiers, namely _Logistic
Regression_ (LR), _Support Vector Machine_ with linear kernel (SVM), _Linear
Discriminant Analysis_ (LDA), _Random Forest_ (RF), _Decision Tree_ (DT) and
KNN with $K=1$111 _scikit-learn 0.19.2_ (http://scikit-learn.org/stable/) was
used to build these models with their default parameters unless stated
otherwise.. In total 36 different settings are tested.

A local linear model $\hat{f}_{x_{i}}$ is built using LEAFAGE for each
instance of the test set. Laugel. et al [8] suggested to test the performance
of $\hat{f}_{z}$ on the test instances that fall into a hyper-sphere with a
fixed radius and $\mathbf{z}$ as center. Having a fixed radius has a
disadvantage that the sphere may include only instances of the same class.
Therefore, we propose to use a custom radius by expanding it until the
corresponding hyper-sphere includes $p$ percentage of instances that do not
have the same predicted label as $c_{\mathbf{z}}$. $p$ should be smaller than,
and close to, one ($p$ = 0.95 is used in the experiments), such that the
closest testing instances of the opposite class of $\mathbf{z}$ are included
and to make the evaluation local, respectively. The scores given by
$\hat{f}_{z}$ are compared with the scores given by the black-box classifier
on all the test instances that fall into this hyper-sphere, using the Area
Under the ROC (AUC). We define the _average fidelity score_ as the average AUC
score over the whole test set.

We then compare the average fidelity scores of LEAFAGE (with $i_{small}=10$)
with LIME, in the various experimental settings as shown in Table I.

Table I: Average local fidelity per setting (the standard deviation in brackets). The strategy with the highest mean along with other strategy that are statistically not significantly different are denoted in bold. |  | Wine | BreastCa. | BankNote | AD
---|---|---|---|---|---
| Classifier
---
Name
Strategy |  | Class 0
---
vs rest
| Class 1
---
vs rest
| Class 2
---
vs rest
| Benign vs
---
Malignant
0 vs 1 | 0 vs 1
LDA | LIME | 100 (0.0) | 100 (0.0) | 100 (0.0) | 99.5 (1.0) | 100 (0.0) | 100 (0.0)
LEAFAGE | 100 (0.0) | 96.0 (9.4) | 100 (0.0) | 99.9 (0.3) | 99.9 (1.7) | 98.6 (4.0)
LR | LIME | 100 (0.0) | 100 (0.0) | 100 (0.0) | 99.9 (0.6) | 100 (0.0) | 100 (0.0)
LEAFAGE | 100 (0.0) | 97.1 (14.2) | 100 (0.0) | 98.6 (7.8) | 99.8 (0.9) | 98.6 (4.0)
SVM | LIME | 100 (0.0) | 100 (0.0) | 100 (0.0) | 99.9 (0.6) | 100 (0.0) | 100 (0.0)
LEAFAGE | 100 (0.0) | 100 (0.0) | 100 (0.0) | 98.6 (7.8) | 99.8 (0.9) | 99.4 (1.2)
DT | LIME | 91.9 (14.9) | 87.9 (22.4) | 91.9 (14.7) | 85.0 (16.2) | 99.0 (2.6) | 59.5 (32.7)
LEAFAGE | 92.9 (16.0) | 85.8 (24.1) | 100 (0.0) | 86.5 (18.7) | 98.7 (4.2) | 65.0 (33.0)
RF | LIME | 100 (0.0) | 99.9 (0.5) | 100 (0.0) | 99.9 (0.3) | 99.1 (2.5) | 61.4 (36.2)
LEAFAGE | 100 (0.0) | 99.2 (3.7) | 100 (0.0) | 99.9 (0.8) | 98.7 (3.8) | 67.4 (32.9)
KNN | LIME | 98.0 (13.6) | 62.8 (37.1) | 60.5 (35.7) | 95.8 (8.2) | 100 (0.0) | 65.6 (34.3)
LEAFAGE | 91.3 (15.9) | 62.9 (36.1) | 60.3 (36.1) | 97.3 (6.0) | 99.9 (0.5) | 65.5 (36.8)

Both LIME and LEAFAGE methods perform better than a baseline model (which
predicts the majority class), in all settings. Further, both methods work
better with linear ML models (SVM, LDA, LR) as opposed as non-linear ones (DT,
RF, KNN), especially with the artificial dataset. This was expected, as
LEAFAGE and LIME are based on linear approximations. On linear models, LIME
scores significantly better than LEAFAGE in 11 out of 18 settings, while on
non-linear models LEAFAGE performs better 5 out of 18 times.

The better performance of LIME on linear ML models could be explained by
taking into account that LIME uses a high amount of samples over the whole
input space to fit the local linear model. LEAFAGE on the other hand, samples
around the closest decision boundary and limits the sampling amount to a
minimum (in a sense, it is more local). This also explains the better
performance of LEAFAGE over LIME on non-linear models.

In conclusion, overall LIME performs better than LEAFAGE on linear ML models,
while LEAFAGE performs better on non-linear models.

##  V Empirical Evaluation

In order to assess the usefulness of LEAFAGE from the user perspective, we
performed a user-study. The target group for this study was the general
public. 114 participants with a well spread demographics in term of gender,
age and education (but mostly from the Americas) were recruited from Amazon
Mechanical Turk, and asked to imagine they were looking for a house to buy,
and that they could use an AI application to estimate the value of a house as
low or high. The AI application could also provide an explanation for its
estimation.

The IOWA housing dataset [16] was used, from which 5 interpretable features
(i.e., features whose meaning can be directly and easily interpreted by
humans) of a house were chosen, as shown in Figure 6(a). We investigated 4
types of explanations for the prediction of a house, namely feature
importance-based (Figure 6(b) left), example-based (Figure 6(b) right), a
combination of example and feature importance-based (Figure 6(b)) and no
explanation, as a baseline.

The evaluation was split into a subjective and an objective part: perceived
aid in decision-making and objective transparency. In the first part, the
participants were asked to rate how much they agree to the given explanation
from 1 to 5 in terms of: transparency (I understand how LEAFAGE made the
prediction); information sufficiency (the explanation provided has sufficient
information to make an informed decision), competence (the explanation
corresponds to my own decision making) and confidence (the explanation made me
more confident about my decision). Next, the objective transparency was
measured by testing participants as follows: the participants were shown
another house, similar to the test one; he/she had to indicate what the system
would predict as the sale value of this new house.

Attention checks were implemented; Results were gathered from the 86
participants that passed the checks.

An SVM model with a RBF kernel was trained on the a training set from the IOWA
dataset (70%) to predict the binary class (low or high value). All
participants saw forty houses randomly chosen from the test set (30% of the
IOWA dataset), with the corresponding predicted value, and one of the four
explanation types. All participants saw the same explanations in a randomized
order. Finally, the perceived aid in decision making and objective
transparency was measured.

Table II shows the results per explanation type and dependent variable. The
median score of the explanation types differ significantly over all dependent
variables according to Kruskal-Wallis H-tests [17] with $p<0.001$ and H
statistic equal to 124, 202, 55, 125 and 52 (in the left to right order of
table II, respectively). The participants perceived getting explanation as
more helpful than providing no explanation. Dunn’s post-hoc test with
Bonferroni correction [18] revealed that in terms of transparency, information
sufficiency, competence and confidence both example-based and combination
explanation perform significantly better than no explanation and feature
importance-based explanation, while no significant differences were found
between example-based and combination explanations. Moreover, feature
importance-based explanation performed significantly better than no
explanation regarding transparency, information sufficiency and confidence but
not in terms of competence. However, in terms of objective transparency,
feature importance-based explanation performed significantly worse than the
rest of the explanation types including no explanation. Example-based
explanation has the highest average objective transparency score, however no
statistically significant difference was measured between pairs of no
explanation, example-based and combination explanation.

Finally, the participants provided general remarks for each explanation type.
When no explanation was provided, they indicated that they could still
understand the prediction, but that they needed “complete trust in the system
to find it helpful”. The participants liked the simplicity and visual aspect
of the feature importance-based explanation, but they did not find it detailed
enough to perform well on the objective transparency part. Moreover, they
found it hard to estimate what value of a certain feature changes the
prediction, and how the importance really relates to the prediction of a
house. Regarding example-based explanation, participants appreciated that they
could compare similar houses with different sale values. However, some
participants disliked this explanation type because of the amount of
information present in the tables. Finally, the combination of example-based
and feature importance-based explanation received a mixed reaction: some
participants liked to get a detailed explanation while others were overwhelmed
and focused on one chart and ignored the other.

| Transparency | Info. Suff. | Competence | Confidence | Objec. Trans.
---|---|---|---|---|---
No Explanation | 3.66 (1.03) | 3.43 (1.17) | 3.70 (0.96) | 3.52 (1.1) | 8.40 (1.48)
Feature importance | 3.92 (0.85) | 3.76 (0.97) | 3.78 (0.91) | 3.68 (1.04) | 7.20 (1.66)
Example-based | 4.07 (0.76) | 4.02 (0.84) |  3.96 (0.86) |  3.98 (0.86) |  8.83 (1.40)
Ex. and Feat. |  4.13 (0.8) |  4.10 (0.83) | 3.93 (0.93) |  3.98 (0.9) | 8.56 (1.68)
Table II: Results of perceived aid in decision making and objective
transparency per explanation type.

##  VI Conclusion

In this paper, we presented LEAFAGE, a novel method to provide local
explanations for the predictions of a black-box ML model. LEAFAGE explanations
performed overall better than the state of the art on non-linear models in
terms of local fidelity. We also evaluated LEAFAGE empirically, by engaging
people in a user study. This aspect has been largely overlooked by the
scientific literature on XAI.

The empirical evaluation showed that overall the participants perceived having
explanations behind a prediction as more helpful than having no explanation
for the goal of decision making. Interestingly, when participants were tested
about their gained knowledge after seeing an explanation, no significant
advantage was found compared to providing no explanation. We suspect that this
is due to the simplicity of the test, in future work a more comprehensive test
could be used to measure the actual transparency. The user study also showed
that, with regards to objective transparency, feature importance-based
explanations are less effective than providing no explanation at all. This is
an important result, which suggests that feature importance-based explanation
confuses users more about the prediction than providing no explanation.

Further, example-based explanations performed significantly better than
feature-importance based explanation in terms of perceived aid in decision
making. Showing both example-based and feature-importance-based explanation
did not increase the perceived aid in decision making significantly. This
could be due to the overload of information as the participants described.
Interestingly, some participants indicated that a tabular view of the example-
based explanation was hard to read. In our future work, we will focus on
designing them in a more readable and intuitive manner.

## Acknowledgement

This research is supported by the Hybrid AI Explainability and VP AI &
Robotics programs at the Netherlands Organisation for Applied Scientific
Research (TNO).

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# Tradeoff between Efficiency and Melting for a High-Performance
Electromagnetic Rail Gun

William C. McCorkle and Thomas B. Bahder
Army Aviation and Missile Research, Development, and Engineering Center,
Redstone Arsenal, AL 35898 USA
Email<EMAIL_ADDRESS>
###### Abstract

We estimate the temperature distribution in the rails of an electromagnetic
rail gun (EMG) due to the confinement of the current in a narrow surface layer
resulting from the skin effect. In order to obtain analytic results, we assume
a simple geometry for the rails, an electromagnetic skin effect boundary edge
that propagates with the accelerating armature, and a current carrying channel
controlled by magnetic field diffusion into the rails. We compute the
temperature distribution in the rails at the time that the armature leaves the
rails. For the range of exit velocities, from 1500 m/s to 5000 m/s, we find
the highest temperatures are near the gun breech. After a single gun firing,
the temperature reaches the melting temperature of the metal rails in a layer
of finite thickness near the surface of the rails, for rails made of copper or
tantalum. We plot the thickness of the melt layer as a function of position
along the rails. In all cases, the thickness of the melt layer increases with
gun velocity, making damage to the gun rails more likely at higher velocity.
We also calculate the efficiency of the EMG as a function of gun velocity and
find that the efficiency increases with increasing velocity, if the length of
the gun is sufficiently long. The thickness of the melted layer also decreases
with increasing rail length. Therefore, there is a tradeoff: for rails of
sufficient length, the gun efficiency increases with increasing velocity but
the melted layer thickness in the rails also increases.

##  I Introduction

Electromagnetic launch systems, such as the railgun, are based on transient
phenomena [1]. During launch, the transient involves the build-up and
penetration of a magnetic field into the surrounding metallic material. The
dynamics of magnetic field penetration into the metal rails is described by a
well-known diffusion equation [2]. The diffusion of the magnetic field leads
to a skin effect, where a large current is transported inside a narrow
channel. In a railgun that has a moving conducting armature, the effect is
called a velocity skin effect (VSE) and is believed to be one of the major
problems in limiting railgun performance [2, 3], because it leads to intense
Joule heating of the conducting materials, such as rails and armatures. To
what extent the VSE effect is responsible for limiting the performance of
solid armatures is still the subject of research [4, 5].

Recently, using a new generation of magnetic field sensors, the magnetic field
distributions caused by the VSE in the rails have been measured [6, 7]. These
experimental efforts are even more significant in light of the large
investments planned by the navy to develop EMGs and power sources for nuclear
and conventional warships [8, 9]. The motivation for this paper is the large
investment planned for EMG technology and the historical lack of understanding
of the reasons for the low endurance of the gun rails in service, sometimes
limited to one shot at maximum energies before replacement is needed.

For high-performance EMGs, in order to increase the armature velocity while
keeping the length of the rails fixed, the current pulse during firing must be
shorter and have a higher average amplitude, causing a stronger skin effect in
the rails, which leads to an increase in Joule heating of the rails. In this
paper, we show that the EMG efficiency is higher at higher velocity, but there
is increased melting of the rails, leading to a tradeoff between efficiency of
the EMG and melting of the rails due to Joule heating (during a single
firing). We do not discuss gun barrel erosion due to repeated firings [10,
11]. We also do not treat the interaction of the high temperature plasma in
the contact regions of the EMRG.

The article is organized as follows. We define our model for EMG heating in
Section II. We consider the problem of rail heating in two steps. First, in
Section III we assume a current-carrying channel described by a local skin
depth $\delta$ (along the rail) that has a simple time dependence due to
motion of the armature, which leads to a time and position-dependent current
density that causes the Joule heating. In Section IV, we use an improved
expression for the current density based on the diffusion of the magnetic
field into the rails. In Section V, we give a crude approximation to the
armature heating. In Section VI, we discuss the efficiency of the EMG and its
dependence on gun velocity and length. We show that the there is a tradeoff
between efficiency and melting the rails. Finally, we present our conclusions
in Section VII.

##  II Temperature Distribution in Rails

The dynamics of an EMG can be properly described in terms of a thermodynamic
free energy that expresses the coupling of the mechanical and electromagnetic
degrees of freedom [12]. For example, for the case of an electromagnetic gun
that can be described in terms of a lumped circuit model, which has a rotor
coil with self inductance $L_{R}$ carrying current $I_{R}$, and a stator coil,
rail, and armature circuit with self inductance $L$ with current $I$, the free
energy has the form [13]

$F(I_{R},I,x,\theta)=F_{0}(T)+\frac{1}{2}L_{R}I_{R}^{2}+\frac{1}{2}L(x)I^{2}+M(\theta)\,I_{R}\,I$
(1)

where $L(x)$ depends on the $x$-position of the armature,

$L(x)=L_{0}\,+\,L^{\prime}\,\,x$ (2)

and where $L^{\prime}=dL_{S}/dx$, and $L_{0}$ is the self inductance of the
stator, rail and armature circuit when the armature is at $x=0$. The
interaction of the stator and rotor circuits is specified in terms of their
mutual inductance, $M(\theta)$, where $\theta$ is the angle of the rotor coil
with respect to the stator coil. The term $F_{0}(T)$ depends only on
temperature. Derivatives of the free energy with respect to the coordinates,
$x$ and $\theta$, give the generalized forces on the system [12]. By Newton’s
law, the acceleration of the armature, $\ddot{x}$, is given in terms of the
derivative of the free energy with respect to the coordinate,

$m\,\ddot{x}=\left({\frac{{\partial{\kern
1.0pt}F}}{{\partial\,x}}}\right)_{I_{R},I,\theta}$ (3)

where $m$ is the mass of the armature (including the payload, comprising the
launch package), leading to the well-known dynamical equation

$m\,\ddot{x}=\frac{1}{2}{\kern 1.0pt}L^{\prime}{\kern 1.0pt}I^{2}$ (4)

Equation (4) shows that the acceleration is directly proportional to the
square of the instantaneous current, $I$. As a first approximation, we assume
that the current in the rails is constant during a shot, $I(t)=I_{o}$. If the
current is constant, and the armature starts at $t=0$ at $x=0$, and moves to
the end of the gun rails at $x=\ell$ at time $t=t_{f}$, then we have the
following relations between armature mass $m$, gun length $\ell$, self
inductance per unit length of armature travel $L^{\prime}$, and mass velocity
$v$,

$\begin{array}[]{l}I_{o}=\left({\frac{{m\,}}{{L^{\prime}\,\ell}}}\right)^{1/2}\,v\\\
t_{f}=\frac{2}{{I_{o}}}\left({\frac{{m\,\ell}}{{L^{\prime}}}}\right)^{1/2}\\\
t_{o}(x)=\frac{2}{{I_{o}}}\left({\frac{{x{\kern
1.0pt}m}}{{L^{\prime}}}}\right)^{1/2}\\\ \end{array}$ (5)

The function $t_{o}(x)$ gives the time at which the mass $m$ is at position
$x$ along the rails, see section III.

In order to calculate the Joule heating in the rails, a lumped circuit model
is not sufficiently detailed. Instead, we must use a more detailed model,
where the free energy is expressed in terms of electromagnetic fields with a
spatial distribution. The problem is complicated due to the coupling of the
electromagnetic and mechanical degrees of freedom. The problem is further
complicated by the fact that the dynamics of an EMG shot is a transient effect
in time. Consequently, when Long [14] and Nearing and Huerta [15] computed
current density (and the heating in EMG rails) they assumed that the
mechanical and electromagnetic degrees of freedom are decoupled. Furthermore,
they assumed the armature was moving at a constant speed so the problem became
translationally invariant in time, thereby avoiding the complexities
associated with the initial conditions and the resulting transient effects. In
particular, it is the transient nature of the EMG shot that gives rise to a
dependence of EMG performance on rail length, see for example our Eq. (14) and
(36) for dependence of rail temperature rise and EMG efficiency, respectively,
on rail length $\ell$. Using the above stated assumptions, Long [14] and
Nearing and Huerta [15] solved for the complicated distribution of the current
density using a simplified geometric model of the rails and armature [16].

In this paper, we use a simpler approach that addresses the transient nature
of the EMG shot and allows us to get approximate analytic results for the
temperature distribution in the rails and armature of an EMG. We model the
skin effect in the rails, which limits the channel through which current can
flow. For a given total current in the rails, a narrower channel (smaller skin
depth) leads to a higher current density and results in greater Joule heating
of the rails. In contrast to Nearing and Huerta [15], our approach allows us
to discuss transient effects dependent on the length of EMG rails and how they
impact EMG performance, see for example our Eq. (14) and (36). We use the same
2-dimensional simplified geometry for the rails and armature as Nearing and
Huerta. However, we assume that the conducting armature is arbitrarily thin,
which allows us to get simple results. Corrections to such an assumption are
expected to be of order $O(a/\ell)$, where $a$ is the armature width and
$\ell$ is the length of the rails. We take the coordinate $x$ running down the
length of the rails, and we assume that the top rail occupies $y>0$ and the
bottom rail occupies $y<-b$, where $b$ is the rail separation. We take the
length of the gun from breech to muzzle to be $\ell$. We assume that the rails
have an arbitrary large thickness $w$ in the $z$-direction, see Figure 1. When
an EMG is fired, the current flows down one rail, through the conducting
armature, and up the other rail to complete the electrical circuit. The
transient response leads to a complicated distribution of eddy currents in the
rails and the armature. We assume that we can describe this effect by the
local skin effect that depends on position and time.

We take the temperature rise due to the Joule heating to be

$T(x,y,t_{f})-T_{0}=\frac{{1\,}}{{\rho\,\,{\kern
1.0pt}C}}\int\limits_{0}^{t_{f}}{\,\frac{{J^{2}(x,y,t)}}{\sigma}{\kern
1.0pt}\,dt}$ (6)

where $T(x,y,t_{f})$ is the temperature at position $(x,y)$ at time $t_{f}$,
when the armature leaves the rails, assuming the current starts at $t=0$. The
quantity $T_{0}=T(x,y,0)$ is the initial temperature at position $(x,y)$ at
time $t=0$ before the shot, $C$ is the specific heat (assumed constant up to
the melting point) of the metal rails, and $\rho$ is the density of the metal
rail. We assume the electric field $E(x,y,t)$ is linearly related to the
current density, $J(x,y,t)=\sigma\,E(x,y,t)$, where $\sigma$ is the electrical
conductivity that is independent of temperature up to the melting point. In
Eq. (6), we have neglected the heat of melting, so the temperature rise is
only valid up to the melting point of the metal rail. If latent heat of
melting $L_{Q}$ is included, then the term $-L_{Q}/C$ must be added to the
right side of Eq. (6). This additional term subtracts from the temperature
rise that may be expected when additional Joule heat is created beyond what is
required to reach the melting temperature. For Cu or Ta, this term is
significant, with value $L_{Q}/C=465\,^{\circ}{\rm C}$ and $1141\,^{\circ}{\rm
C}$, respectively. Below, we do not consider the temperature rise above the
melting point of the metal rails.

##  III Current Channel with Constant Current in Rails

The moving armature carries all the current of the rails. As the armature
moves, it exposes a new plane region on the rail that carries current. Due to
the diffusive nature of the magnetic field ${\bf H}$, and the relation

${\bf J}=\textrm{curl}\,\,{\bf H}$ (7)

current does not flow uniformly in the rail [12, 17], instead the current
flows through a layer of thickness $\delta$ that increases with time $t$,
starting with $\delta=0$ as the leading edge of the armature passes a point on
the rail. Assuming a plane geometry for the rail, we can approximate the time
dependence of the skin depth to be [17]

$\delta(t)=\left({\frac{{4\,t}}{{\mu\,\sigma}}}\right)^{1/2}$ (8)

where $\mu$ is the magnetic permeability and $\sigma$ is the electrical
conductivity, and $t$ is the time elapsed since the armature has passed a
given element on the rail. The exact factor inside the square root (here we
take it to be 4) is somewhat arbitrary in defining a skin depth. (In Section
IV we remove this arbitrariness by using the solution of the time-dependent
magnetic field diffusion equation to compute the current density
distribution.) As stated in the second to last paragraph in Section II, we
assume an idealized armature that is arbitrarily thin in the $x$-direction and
we assume that the skin depth $\delta$ in the rail starts at zero thickness at
the position where the armature contacts the rails. As remarked above, this
drastic assumption is expected to have corrections of order $O(a/\ell)$ where
$a$ is the width of the armature and $\ell$ is the length of the rail. This
assumption allows us to obtain analytic results and see the dependence on a
number of parameters. As a start, we model the conducting channel by assuming
that the current density in the rail inside the skin depth $\delta$ is a
function of $x$ and $t$ but not $y$, and that the current density is zero
outside the skin depth $\delta$, see Figure 1 [18]. Furthermore, we assume
that the skin depth has zero thickness on the leading edge of the armature,
since all the current in the rails has to flow through the armature. Assuming
a current carrying channel of finite width $\delta$, at time $t$ and position
$x$ along the rail, the $y$ coordinate of the boundary of the current carrying
channel is

$y_{c}(x,t)=\left\\{{\begin{array}[]{*{20}c}{\delta(t-t_{o}(x))\,,\quad
t>t_{o}(x)}\\\ {0\,,\quad\quad\quad\quad\quad t\leq t_{o}(x)}\\\
\end{array}}\right.$ (9)

where $t_{o}(x)$ is the time the leading edge of the armature passes the
position $x$. In other words,

$\frac{{d\,t_{o}(x)}}{{d\,x}}=\frac{1}{{v(x)}}$ (10)

where $v(x)$ is the velocity of the armature when it is at point $x$, see
Figure (1).

Consider an element of volume $dV=wdx\,dy$ at position $(x,y)$. Define
$\tau(x,y)$ as the time at which the boundary of the current carrying channel
intersects this volume element. From Eq. (9) for the time-dependent boundary
of the current carrying channel, we find

$\tau(x,y)=t_{o}(x)+\frac{1}{4}\mu\sigma y^{2}$ (11)

For time $0<t\leq\tau(x,y)$ there is no current flowing through this volume
element. At time $t=\tau(x,y)$, the boundary of the current carrying channel
intersects the volume element at position $(x,y)$ and Joule heating starts.
See Figures 2, 3 and 4. Finally, at time $t=t_{f}$ the armature leaves the
rails, the circuit is broken, and there is no more Joule heating of the
element $dV$.

The current density in the rail can be written down by considering three
domain regions, see Figure 5. At position $(x,y)$ and time $t$, we take the
current density to have the form

$\begin{array}[]{l}J(x,y,t)=\\\
\quad\left\\{{\begin{array}[]{*{20}c}{\frac{{I(t)}}{{w\,\delta(t-t_{o}(x))}},}&{\quad
x<x_{o}(t)\quad{\rm{and}}\quad y\leq\delta(t-t_{o}(x))}\\\
{0,}&{x<x_{o}(t)\quad{\rm{and}}\quad y>\delta(t-t_{o}(x))}\\\
0&{x>x_{o}(t)}\\\ \end{array}}\right.\\\ \end{array}$ (12)

where $x_{o}(t)$ is the function that gives the $x$ coordinate of the armature
at time $t$, and $\delta(t)$ is given by Eq.(8).

Figure 1: The rails and conducting armature are shown for the electromagnetic
gun. At time $t=0$ the leading edge of the armature is at $x=0$. The trailing
edge of the armature leaves the rails at time $t=t_{f}$. Figure 2: At time
$t<\tau$ the skin depth boundary has not yet reached the element of volume
$dV=w\,dx\,dy$ at position $(x,y)$.  Figure 3: At time $t=\tau$ the skin depth
boundary overlaps the element of volume $dV=w\,dx\,dy$ at position $(x,y)$. At
this time, current starts to flow in the element $dV$ and the temperature
starts to rise due to Joule heating. Figure 4: After the skin depth boundary
passes the element of volume $dV$, heating of the element continues until the
armature exits the rails at the time $t_{f}$. For time $t>t_{f}$, we assume
that no energy is input into the element $dV$.  Figure 5: The three domain
regions are shown for the current density. The non-zero current density is
assumed to be inside the skin depth for $x<x_{o}(t)$ where $x_{o}(t)$ gives
the $x$-coordinate of the armature at time $t$.

Using Eq. (6) and assuming a constant current in the armature, $I(t)=I_{0}$,
from Eq. (6) and (12) we obtain

$\begin{array}[]{l}T(x,y,t_{f})-T_{0}=\\\
\;\frac{{\mu\,I_{0}^{2}}}{{4w^{2}\rho
C}}\;\left\\{{\begin{array}[]{*{20}c}{Log\left({\frac{{t_{f}-t_{o}(x)}}{{\tau(x,y)-t_{o}(x)}}}\right),}&{y\leq\left[{\frac{4}{{\mu\sigma}}\left({t_{f}-t_{o}(x)}\right)}\right]^{1/2}}\\\
{0,}&{y>\left[{\frac{4}{{\mu\sigma}}\left({t_{f}-t_{o}(x)}\right)}\right]^{1/2}}\\\
\end{array}}\right.\\\ \end{array}$ (13)

Equation (13) gives the temperature rise, $T(x,y,t_{f})-T_{0}$, of an element
of volume at position $(x,y)$ at time $t_{f}$ at which the armature leaves the
rails, for a constant gun current over the time interval $0<t<t_{f}$. For
realistic time-dependent currents, see Figure 3 in McCorkle [19].

Using a lump circuit model to describe the constant acceleration $\ddot{x}=a$
of the armature for a constant current $I_{0}$, we can write the position of
the armature as a function of time as $x=\frac{1}{2}at^{2}$. The time at which
the armature leaves the rails, $t_{f}$, is then related to the length of the
gun, $\ell=\frac{1}{2}at_{f}^{2}$. The time $t_{o}(x)$ at which the armature
passes coordinate $x$ is then $t_{o}(x)=(2x/a)^{1/2}$, see also Eq. (5). Using
these approximations in Eq. (13) gives the temperature rise at position
$(x,y)$

$\begin{array}[]{l}T(x,y,t_{f})-T_{0}=\frac{{\mu\,I_{0}^{2}}}{{4w^{2}\rho\,C}}\;\;\times\\\
\left\\{{\begin{array}[]{*{20}c}{Log\left[{\frac{{8\,\ell}}{{\mu\,\sigma\,v\,y^{2}}}\left({1-\left({\frac{x}{\ell}}\right)^{1/2}}\right)}\right],}&{y\,\leq\,\,\tilde{y}(x)}\\\
{0,}&{y>\,\tilde{y}(x)}\\\ \end{array}}\right.\\\ \end{array}$ (14)

where

$\tilde{y}(x)=\left({\frac{{8\,\ell}}{{\mu\,\sigma\,v\,}}}\right)^{1/2}\,\left({1-\left({\frac{x}{\ell}}\right)^{1/2}}\right)^{1/2}$
(15)

and $x$ is assumed to be in the interval $0<x\leq\ell$. Note that Eq. (14)
predicts that the temperature on the surface of the rail, at $y=0$, is
infinite. This feature of the solution is well-known and is not a problem
[17]. The singularity with respect to $y$ is integrable, and therefore the
energy deposited in a thin layer near the surface is finite. We use Eq. (14)
to locate the points $(x,y)$ of the surface that reaches the melting point.

Using the values for the gun parameters and material parameters in Tables I
and II, we plot the distribution of the temperature rise in the rail,
$T(x,y,t_{f})-T_{0}$, as a function of the $y-$coordinate, for several points
along the rail given by coordinate $x$. For comparison, we also plot the
melting point of Cu and Ta, which is 1083${}^{\circ}C$ and 2996${}^{\circ}C$,
respectively. The $y-$coordinate where the curve intersects the melting point
of each metal is the depth to which melting of the rails occurs, if the EMG
gun shot occurred with rails at initial temperature $T_{0}$. As mentioned
previously, we have not taken into account the heat of melting, so these
curves are not correct for temperatures above the melting point of the rails.

Figure 6: Plot of Eq. (14), giving the temperature rise, assuming rails made
from copper, plotted vs. position $x$ along the rail. Parameters used are
given in Tables I and II. The gun velocity is taken as 3000 m/s and the inital
temperature $T_{o}=0^{o}$C. Constant current is assumed.

Figure 6 and 7 show a plots of the thickness of the melted layer assuming the
rails are made of copper and tantalum, respectively, for a constant total gun
current, which corresponds to a unifomly accelerating armature.

Figure 7: Plot of Eq. (14), giving the temperature rise, assuming rails made
from tantalum, plotted vs. position $x$ along the rail. Parameters used are
given in Tables I and II. The gun velocity is taken as 3000 m/s and the inital
temperature $T_{o}=0^{o}$C. Constant current is assumed.

Our model predicts that the thickness of the melted layer is largest at $x=0$,
at the breech of the gun rail. This is reasonable because the current at the
breech of the gun flows for the longest time, causing maximum Joule heating at
$x=0$.

The intersection of the curves in Figure 6 and 7 with the melting point of
metal (horizontal line) gives the thickness of the melted layer. The surface
inside the rails that reaches the melting temperature, $T_{melt}$, is given by
points $(x,y)$ that satisfy

$T(x,y,t_{f})-T_{0}=T_{melt}-T_{0}$ (16)

Alternatively, we can say that at position $x$ the thickness of the rail that
reaches the melting point, $y_{melt}(x)$, is given implicitly by
$T(x,y_{melt},t_{f})-T_{0}=T_{melt}-T_{0}$. For any element of volume, the
temperature rise is due to the length of time that the current was flowing
through that element. For an element at $(x,y)$, current starts to flow at
time $\tau(x,y)$. We assume that for all elements the current stops flowing at
time $t_{f}$, when the armature leaves the rails.

From Eq. (16) we solve for the thickness of the melted layer, $y_{melt}(x)$,
as a function of $x$ position along the rail

$y_{melt}(x)=\left({\frac{{8\,\ell}}{{\mu\,\sigma\,v}}}\right)^{1/2}\,\;e^{-\frac{{2\,{\kern
1.0pt}w^{2}\rho{\kern 1.0pt}{\kern 1.0pt}\,L^{\prime}\,\ell\,C{\kern
1.0pt}\left({T_{{\rm{melt}}}-T_{o}}\right)}}{{\mu\,{\kern
1.0pt}m\,v^{2}}}}\,\,\left[{1-\left({\frac{x}{\ell}}\right)^{1/2}}\right]^{1/2}$
(17)

Figure 8 shows a plot of the thickness of the melted layer assuming the rails
are made of aluminum, copper, and tantalum. We used numerical values given in
Tables I and II.

Figure 8: Plot of Eq. (17), giving the melt layer thickness vs. x along the
rails for rails made from copper, tantalum and aluminum. The initial
temperature of rails is assumed to be $T_{o}=0^{o}$C. Constant current is
assumed and the gun velocity is taken to be 3000 m/s.

The thickness of the melted layer has a strong dependence on gun velocity,
which is the velocity of the armature at $x=\ell$. Figure 9 shows plots of the
melted layer thickness at $x=0$ for Cu, Ta, and Al rails vs. gun velocity.

Figure 9: Plot of Eq.(17) giving the melted layer thickness for Cu, Ta, and Al
rails at the gun breech (at x=0) vs. gun velocity, $v$. The initial
temperature of rails is assumed to be $T_{o}=0^{o}$C. Constant current is
assumed. This plot also gives an estimate of the melted layer thickness in the
armature, see discussion in Section VI.

The plots in Figure 9 show that for gun velocities above 1500 m/s, using
materials such a copper, aluminum, and tantalum, the thickness of the melted
layer in the gun rails increases rapidly with gun velocity.

TABLE I: Gun parameters. quantity | symbol | value
---|---|---
width of rails | $w$ | 0.10 m
length of rails (gun length) | $\ell$ | 10.0 m
mass of armature | $m$ | 20 kg
derivative of self inductance of rail | $L^{\prime}$ | 0.46$\times$10-6 H/m
magnetic permeability | $\mu$ |  $4\pi\times 10^{-7}$ H/m
TABLE II: Material parameters for the rails. symbol | $T_{melt}$ | $\sigma$ | $\rho$ | $C$
---|---|---|---|---
|  melting temp. ${}^{o}C$ |  conductivity (Ohm m)-1 |  density kg/m-3 |  specific heat J/kg ${}^{o}C$
aluminum | 660 |  3.82 $\times$ 107 | 2720. | 950.
copper | 1084 |  5.8 $\times$ 107 | 8960. | 440.
tantalum | 2996 |  7.40$\times$106 | 16600. | 150.62

##  IV Current Distribution due to Magnetic Field Diffusion

In this section we obtain the current density in the rail, $J(x,y,t)$ (to be
used in Eq. (6)), by considering the magnetic field diffusion into the rails.
We treat the diffusion of the magnetic field into the rails as diffusion into
a plane surface, with a time dependent boundary condition on the field, given
by the armature passing a surface element of the rail. Consider an arbitrarily
thin armature at position $x_{o}$ at some time $t$. For the geometry in Figure
1, the magnetic field $H_{z}$ is in the $z$-direction. For $x>x_{o}$, which is
outside the rail-armature circuit, the magnetic field $H_{z}=0$. For
$0<x<x_{o}$, which is inside the rail-armature circuit, the magnetic field has
some constant value $H_{0}$. As the (arbitrarily thin) armature passes the
point $x$ on the surface of the rail, the surface magnetic field changes from
$H=0$ to some finite value $H_{0}$, and the field starts to diffuse into the
rail. Essentially, as the armature passes the point $x$ the boundary condition
on the field $H_{z}$ on the rail surface changes from $H_{z}=0$ to
$H_{z}=H_{0}$. Considering the rail to be a plane surface, the magnetic field
diffuses into the rail according to the equation [12, 17]

$\frac{{\partial^{2}H_{z}}}{{\partial y^{2}}}-\frac{1}{\kappa}\frac{{\partial
H_{z}}}{{\partial t}}=0$ (18)

For a plane surface with magnetic field boundary condition $H_{z}=0$ for $t<0$
and $H_{z}=H_{0}$ for $t>0$, the solution for $y>0$ is

$H_{z}=H_{0}\,{\rm{erfc}}\left({\frac{y}{{2\sqrt{\kappa{\kern
1.0pt}t}}}}\right)$ (19)

where $\kappa=1/(\mu\sigma)$ is the magnetic diffusion length,
${\rm{erfc}}\left(\xi\right)$ is the complimentary error function,
${\rm{erfc}}\left(\xi\right)=1-{\rm{erf}}\left(\xi\right)$ and
${\rm{erf}}\left(\xi\right)$ is the error function given by

$erf(z)=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}dt$

The current density associated with this magnetic field can be obtained from
Eq. (7), leading to

${\bf{J}}={\bar{J}}(y,t)\,\;{\bf{e}}_{x}=\frac{{H_{0}}}{{\sqrt{\pi{\kern
1.0pt}\kappa{\kern 1.0pt}t}}}\,\exp\left({-\frac{{y^{2}}}{{4\,\kappa\,{\kern
1.0pt}t}}}\right)\,\,{\bf{e}}_{x}$ (20)

where ${\bf{e}}_{x}$ is the unit vector in the $x$-direction, and the
auxiliary function, ${\bar{J}}(y,t)$, is defined by Eq. (20). The value of the
magnetic field is related to the total current $I_{o}$ in the rails by

$I_{o}=\int{\int{dy\,dz\,}}{\bar{J}}(y,t)=wH_{0}$ (21)

leading to $H_{0}=I_{o}/w$ where $w$ is the width of the rail in the
$z$-direction.

We use the plane surface solution to approximate the magnetic field diffusion
and resulting current distribution in the rail. As the armature sweeps past a
surface element in the rail at position $x$, the field starts diffusing into
the rail at time $t_{o}(x)$, where $t_{o}(x)$ gives the time the armature
passes point $x$. We take the current density to have the form

${\bf{J}}(x,y,z,t)=\bar{J}(y,t-t_{o}(x))\,{\bf{e}}_{x}$ (22)

when the coordinates $(x,y,z)$, and time $t$ satisfy the conditions

$0<x<x_{o}(t)\,\,\>,\,\,y>0\,\>,\,\,\,\>|z|<\frac{w}{2}\,\,\,,\,\,\;0<t<t_{f}$
(23)

and otherwise we take ${\bf{J}}(x,y,z,t)=0$. Here, $x_{o}(t)$ is the
$x$-position of the armature at time $t$, and ${\bar{J}}(y,t)$ is given by Eq.
(20). The quantity ${\bf J}(x,y,z,t))$ is an approximation to the time-
dependent current density in the rails during the EMG shot. Note that ${\bf
J}(x,y,t)$ does not depend on $z$ because we assume a large (essentially
infinite) extent in the $z$-direction.

The temperature rise is obtained from Eq. (6) and we neglect the effect of re-
distribution of heat during the time of the shot, since it takes a long time
on the time scale of a shot (which is $t_{f}$), The temperature rise
immediately after the shot is

$T(x,y,t_{f})-T_{0}=\frac{{\mu\,I_{o}^{2}}}{{\pi{\kern 1.0pt}{\kern
1.0pt}w^{2}{\kern 1.0pt}\rho{\kern 1.0pt}{\kern
1.0pt}C}}\,\,\,\Gamma\left({0,\frac{{\mu{\kern
1.0pt}\sigma}}{2}\frac{{y^{2}}}{{t_{f}-t_{o}(x)}}}\right)$ (24)

where $\Gamma(a,z)$ is the incomplete Gamma function, given by

$\Gamma(a,z)=\int_{z}^{\infty}t^{a-1}e^{-t}dt$

Equation (24) is our basic result for the temperature distribution in the
rails, and is expressed in terms of the current in the rails $I_{o}$. It is
useful to express the temperature rise in terms of the gun velocity $v$.

In order to get simple results, we assume that the acceleration is constant,
using the relations in Eq. (5) leads to the temperature rise

$T(x,y,t_{f})-T_{0}=\frac{{\mu{\kern 1.0pt}{\kern 1.0pt}m{\kern 1.0pt}{\kern
1.0pt}v^{2}}}{{\pi\,w^{2}\,\rho\,C\,L^{\prime}\,\ell}}\;\Gamma\left({0,\frac{{\mu\,\sigma\,v}}{{4\,\sqrt{\ell}}}\,\frac{{y^{2}}}{{\sqrt{\ell}-\sqrt{x}}}}\right)$
(25)

Figure 12 shows a plot of the temperature rise given by Eq. (25), for the case
of copper rails, using the parameters in Tables I and II.

Figure 10: The temperature rise is plotted vs. $y$ coordinate into the rail
for various positions $x$ along the copper rail. The time dependence of the
current is determined by the time dependence of magnetic field diffusion into
the rails. Initial temperature of the rails was assumed to be $T_{0}$ =0 C.
Parameters used are given in Tables I and II. Figure 11: Plot of the thickness
of the melted layer vs. position x along the rail, assuming rails made from
copper. Curves for three different velocities are shown. Initial temperature
of the rails was assumed to be $T_{0}$ =0 C. Parameters are those in Tables I
and II. Figure 12: Plot of the thickness of the melted layer vs. position x
along the rail, assuming rails made from tantalum. Curves for three different
velocities are shown. Initial temperature of the rails was assumed to be
$T_{0}$ =0 C. Parameters are those in Tables I and II.

In Eq. (25), the argument of the $\Gamma(\xi)$ function is large for all
values of $x$ and $y$, except for $y$ near $y=0$. For the values in Tables I
and II, the argument of the $\Gamma$ function is

$\xi=\frac{vy^{2}\mu\sigma}{4\left(\ell-\sqrt{\ell}\sqrt{x}\right)}=\frac{54663.71\,\,y^{2}}{10.-3.1622\sqrt{x}}$
(26)

Therefore, the $\Gamma(\xi)$ function for $\xi>>1$ can be approximated by its
asymptotic expansion

$\Gamma(0,\xi)=e^{-\xi}\left(\frac{1}{\xi}-\frac{1}{\left(\xi\right)^{2}}+O\left[\frac{1}{\xi}\right]^{3}\right)$
(27)

Near y=0, the behavior of the $\Gamma(\xi)$ function for $\xi<<1$ is
logarithmic

$\Gamma(0,\xi)=(-\gamma-Log[\xi])+\xi-\frac{\xi^{2}}{4}+O[\xi]^{3}`$ (28)

where $\gamma$ is Euler’s constant, $\gamma\approx 0.5772$.

As before, see Eq. (16), we use Eq.(25) to solve for the thickness of the
melted layer in the $y$-direction by setting the temperature rise equal to the
melting temperature of the metal. Figures 11 and 12 show plots of the
thickness of the melted layers verses position $x$ along the rail, for several
velocities, for rails made of copper and tantalum, respectively. From these
figures, we see that even though Ta has a much higher melting temperature than
Cu, 2996oC compared to 1084oC, respectively, the electrical conductivity of Ta
is lower, which leads to a comparable thickness for the melted layer for Ta
and Cu rails.

##  V Armature Heating

We can estimate the heating of the armature from the calculations that we have
already done. We now imagine that the armature has a finite thickness.
Furthermore, depending on the EMG design, the armature material can be
different than that of the rails, and consequently, the electrical
conductivity $\sigma$ may be different. Unlike the rails, the armature
conducts current for the whole duration of the shot from $t=0$ to $t=t_{f}$.
In this sense, the whole length of the armature has current flowing in it like
the element of rail at coordinate $x=0$. The thickness of the melted layer of
the armature can be found from Eq.(17) by setting $x=0$,

$y_{melt}(0)=\left({\frac{{8\,\ell}}{{\mu\,\sigma\,v}}}\right)^{1/2}\,\;e^{-\frac{{2\,{\kern
1.0pt}w^{2}\rho{\kern 1.0pt}{\kern 1.0pt}\,L^{\prime}\,\ell\,C{\kern
1.0pt}\left({T_{{\rm{melt}}}-T_{0}}\right)}}{{\mu\,{\kern 1.0pt}m\,v^{2}}}}$
(29)

Figure 9 shows a plot of the melted layer thickness of the armature verses gun
velocity, as given by Eq. (29).

##  VI Efficiency and Melt Layer Thickness

During an EMG shot, the energy that is supplied to the gun appears as
projectile kinetic energy, Joule heating of the rails and armature, armature
and rail heating caused by frictional forces between rails and armature,
vibration of rails and armature, sound and light produced in surrounding air,
and electromagnetic radiation due to a time-dependent magnetic field. We
neglect all these effects except for the Joule heating of the rails and
kinetic energy of the projectile and armature.

From the current density in Eq. (22), we calculate the energy in one rail,
$Q$, that is dissipated over one shot of the EMG by integrating the Joule
heating over the time $t_{f}$ during which the armature is in contact with the
rails,

$Q=2\int{d^{3}x}\int\limits_{0}^{t_{f}}{dt}\,\frac{{J^{2}}}{\sigma}$ (30)

where the spatial integration is over all space inside one of the rails. The
factor of two is due to the fact that there are two rails, and by symmetry the
energy dissipated is twice that of one rail. Assuming constant current, for
which the time $t_{f}$ is defined in Eq. (5), assuming the conductivity
$\sigma$ is a constant, and using the expression given by Eq. (22) for the
current density derived by considering magnetic field diffusion into rails
each of length $\ell$, we find

$\begin{array}[]{c}Q=\frac{{32}}{{15}}\left({\frac{\mu}{{\pi\sigma}}}\right)^{1/2}\,\frac{{\ell^{5/4}\,\,m^{1/4}}}{{L^{\prime
1/4}\,\,w}}\,\,I_{o}^{3/2}\\\
=\,\frac{{32}}{{15}}\left({\frac{\mu}{{\pi\sigma}}}\right)^{1/2}\,\frac{m}{{w\,L^{\prime}}}\,\ell^{1/2}\,\,v^{3/2}\\\
\end{array}$ (31)

where we have assumed a constant current (armature acceleration) and used the
value of $t_{f}$ in Eq. (5). Note that $Q$ is the total amount of energy
dissipated in both rails. The two forms for $Q$ are obtained assuming the
relation between EMG velocity and current given in Eq. (5), which assumes a
constant current. Equation (31) gives the amount of energy $Q$ dissipated in
the rails due to Joule heating over the time $t_{f}$ of one shot. Equation
(31) contains some interesting physics because the total energy dissipated as
Joule heat is not proportional to $I_{o}^{2}$ as is the case for a resistor
carrying a constant current $I_{o}$ for a time $t_{f}$. If the rails acted as
a simple resistor with resistance $R$, then the amount of energy dissipated
due to Joule heating during the shot would be $I_{o}^{2}\,R\,t_{f}$. If
$t_{f}$ was a constant independent of current $I_{o}$, then the amount of heat
dissipated in the resistor would be proportional to $I_{o}^{2}$. However, in
an EMG, the time $t_{f}$ is inversely proportional to current $I_{o}$, as
given by Eq. (5), because the armature spends less time in contact with the
rails at higher current (due to higher average armature velocity at higher
current). So if the rails acted like a resistor with no skin effect, and the
time $t_{f}$ varied inversely with current $I_{o}$, then the amount of heat
dissipated in such a resistor would be proportional to $I_{o}$. However, in
the EMG model that we are considering, we have included the skin effect. With
increasing current $I_{o}$, the time $t_{f}$ is shorter and there is a
stronger skin effect at larger $I_{o}$ leading to an increased resistance $R$,
resulting in an increased Joule heat dissipation during the shot time $t_{f}$,
since all current has to flow through a thinner channel in the rails. The net
effect is that the skin effect causes an increased Joule heat over the shot,
so that for the EMG with skin effect we have $Q\sim I_{o}^{3/2}$, as given in
Eq. (31).

The ratio of armature kinetic energy divided by the Joule heat is a sort of
“thermal $q$” and is given by

$\begin{array}[]{c}q=\frac{{\frac{1}{2}mv^{2}}}{Q}\\\
=\frac{{15}}{{32}}\left({\frac{{\pi\sigma}}{\mu}}\right)^{1/2}\frac{{{\rm{w}}\,L^{\prime{\kern
1.0pt}{\kern 1.0pt}5/4}}}{{\left({m{\kern
1.0pt}\ell}\right)^{1/4}}}\,I_{o}^{1/2}\\\
=\frac{{15}}{{32}}\left({\frac{{\pi\sigma}}{\mu}}\right)^{1/2}\,{\rm{w}}\,L^{\prime}\,\left({\frac{v}{\ell}}\right)^{1/2}\\\
\end{array}$ (32)

The quantity $q$ shows that, in order to achieve a certain velocity $v$, a
larger fraction of input energy goes into the armature kinetic energy when
using shorter rails (smaller $\ell$). Figure 15 shows the dependence of $q$ on
velocity $v$ for parameters given in Table I and II for gun rails made of
copper.

Figure 13: The plot shows the thermal $q$ defined in Eq. (32) as a function of
velocity, using the parameters in Table I, assuming copper rails with
conductivity in Table II.

However, the quantity $q$ does not contain the whole story. We next consider
the EMG efficiency.

Within our model, we define the efficiency of an EMG, $\eta$, as the armature
kinetic energy, divided by the total energy

$\eta=\frac{{\frac{1}{2}mv^{2}}}{{\frac{1}{2}mv^{2}+Q+E_{mag}(\ell)}}$ (33)

We consider the total energy as a sum of three terms: the kinetic energy of
the armature $\frac{1}{2}mv^{2}$, the energy dissipated by Joule heating $Q$
in Eq. (31), and the energy stored in the magnetic field $E_{mag}(\ell)$ when
the armature is at the end of the rails at $x=\ell$. In general, the magnetic
energy $E_{mag}(x)$ is the energy stored in the magnetic field in the rails
and surrounding space when the armature is at position $x$:

$E_{mag}(x)=\frac{1}{2}\,L(x)\,I_{o}^{2}$ (34)

As previously defined in Eq. (2), we write $L(x)$ in terms of $L_{0}$ and
$L^{\prime}$, where $L_{0}$ is the self inductance of the EMG that does not
depend on armature position (and corresponds to magnetic energy
$L_{0}\,I_{o}^{2}\,/2$ stored in the power supply, surrounding space, and in
electrical leads to the rails) and $L^{\prime}$, which is the derivative of
the self inductance with respect to armature position $x$. When the armature
reaches the end of the rails, the circuit is broken and the magnetic energy
$E_{mag}(\ell)$ is dissipated in the form of an electrical arc, sound, light,
and mechanical vibration. Assuming constant current and acceleration, when the
armature is at the end of the rails at position $x=\ell$, the stored magnetic
energy is

$E_{mag}(\ell)=\frac{1}{2}\,mv^{2}\left({1+\frac{{L_{0}}}{{L^{\prime}\;\ell}}}\right)$
(35)

where we have used Eqs. (4). From Eqs. (30), (33) and (35), the efficiency of
the EMG is then

$\eta=\frac{1}{{2+\frac{{L_{0}}}{{L^{\prime}\;\ell}}+\frac{{64}}{{15\,w{\kern
1.0pt}L^{\prime}}}\left(\frac{\mu\,\ell}{\pi\sigma\,v}\right)^{1/2}\,\;\,}}$
(36)

Equation (36) shows how the efficiency depends on a number of variables. The
efficiency of an EMG increases with increasing electrical conductivity
$\sigma$, velocity $v$, and rail width $w$. The efficiency depends in a
complicated way on the rail length $\ell$. In the limit of high velocity $v$,
the efficiency of the EMG has the limiting value

$\mathop{\lim}\limits_{v\to\infty}\;\eta=\frac{1}{{2+\frac{{L_{0}}}{{L^{\prime}\;\ell}}\,}}$
(37)

The high-velocity limit of EMG efficiency depends on the ratio of the
stationary self inductance, $L_{0}$, to the dynamic part of the self
inductance of the rails, ${L^{\prime}\;\ell}$. We call ${L^{\prime}\;\ell}$
the dynamic part of the self inductance because it depends on the length of
the rails. For a given gun design, the quantities $L_{0}$ and ${L^{\prime}}$
are constants, but in principle the length of the rails, $\ell$, can be
increased to make the term $\frac{L_{0}}{L^{\prime}\;\ell}<<1$, thereby
increasing the high-velocity limit of the efficiency. Longer rails (larger
$\ell$) lead to a higher EMG efficiency at high velocity. However, for all
velocities the EMG efficiency for this model is always less than 1/2.

The denominator of the efficiency in Eq. (36) has three terms. For the
parameters in Table I and II, the third (last) term in the denominator is
$24.35/\sqrt{(}v)$, where $v$ is in units of m/s. Therefore, it is clear that
the EMG is more efficient at higher velocities. The efficiency will increase
with velocity significantly when the stationary part of the self inductance,
$L_{0}$, is smaller than the dynamic part of the self inductance,
$L^{\prime}\;\ell$, so that the term $\frac{{L_{0}}}{{L^{\prime}\;\ell}}<<1$,
which may rarely be true in real designs unless the length of the rails $\ell$
can be made sufficiently large.

Stated in another way, a longer EMG is more efficient at higher velocities,
however, the thickness of the melted layer also increases at higher
velocities. Therefore, there is a tradeoff between efficiency and melting of
the rails. Increasing the length of the rails may prevent melting, however,
this may lead to rail length that is not practical for applications.

In our simple model of an EMG, and our definition of efficiency, we have
neglected many effects such as friction between armature and rails, plasma
contacts, and many other details. These effects would make the denominator of
Eq. (36) larger, and hence would reduce the efficiency.

##  VII Conclusion

We have constructed a model of the EMG based on the electrodynamics of the
launch rails and armature. Our model takes into account the skin effect in the
rails and armature, which is significant due to the short-duration and
extremely high current densities in high-performance EMGs. We used two
approaches. The first approach, in Section III we modeled the skin effect by a
constant-in-time current flowing through a channel equal to the local skin
depth along the rail, $\delta$, given by Eq. (7) at each point along the rail.
We find that a finite thickness layer of the rails, given by Eq. (17), reaches
the melting point of the metal. Although aluminum is lighter by a large
factor, we have assumed the rails are made from copper because of its
significantly higher melting point (1083 C for copper verses 660 C for
aluminum). Figures 6 and 7 show plots of the temperature rise (immediately
after a shot) versus depth into the rails, for different positions spaced one
meter apart along the rails, assuming 10 meter long rails, for copper and
tantalum rails, respectively. For all positions along the rails, immediately
after the shot and before thermal diffusion takes place, the temperature rise
is the highest at the rail surface and decreases into the interior of the
rails. For constant current, near the breech end of the gun the skin depth
exceeds 15 mm, and a layer approximately 2 mm or 3 mm thick melts, for
tantalum or copper rails, respectively, see Figure 8. The model predicts that,
immediately after a shot and before thermal energy is redistributed, the
temperature is largest near the rail surface and a thin surface melt layer
exists for any rail gun, see Eq. (17).

In the second approach, in Section IV, we obtained an approximate current
density by computing the time dependence of the diffusion of the magnetic
field into the rail surface during the EMG shot. From the magnetic field ${\bf
H}$, we obtain the current density from Eq. (7), which is used to compute the
temperature rise immediately after a EMG shot, see Figures 6, 7 and 10.
Figures 11 and 12 show the melted layer thickness we can expect for copper and
tantalum rails, for different gun velocities (due to different gun currents,
which are related to velocities by Eq. (4)). We expect this approach (Section
IV) to be our most accurate estimation of the temperature rise in the rails
and resultant melting. In both approaches, we find the thickness of the melted
layer of the rail increases rapidly with EMG velocity, see Eq. (29) and Figure
9.

Finally, in Section V, we estimated the temperature rise of the armature, and
the resulting thickness of the melted layer. The temperature rise in the
armature may be a major limiting factor in EMG design, see Figure 9.

In Section VI, we computed the efficiency of an EMG as a function of armature
velocity and gun length. In this efficiency calculation, we only considered
Joule heating, kinetic energy of the armature, and the stored magnetic energy
in the system, and we neglected all other energies, such as frictional heating
between armature and rails. We find that if the stationary part of the self
inductance is small compared to the dynamic part of the self inductance (that
of the rails), which is equivalent to long rails, the efficiency of an EMG
increases with velocity. Stated in another way, if the rails are long enough
the EMG will have an efficiency that increases with velocity. In all cases the
upper limit of efficiency is 1/2 at high velocity. However, with increasing
velocity, the thickness of the melted layer in the rails also increases, see
Figure 9. The thickness of the melted layer also decreases with rail length.
Therefore, there is a tradeoff: for sufficiently long rails, with increasing
gun velocity the EMG is more efficient, however, at higher velocity the
thickness of the melted layer is larger, which is likely to result in more
damage to the rails. Also, we may need rails that are too long for practical
applications. In our model of an EMG, we have neglected many practical
effects, such as heating and ablation of armature/projectile at high
velocities in the atmosphere.

The most important conclusion is that, given the available choice of materials
for rails and armatures, the gun has increased efficiency at higher velocity,
however, there is increased melting of the rails and damage is more likely
during each firing of high performance guns, where payload and range is
designed for (high-velocity) naval guns. This analysis was performed for
simple rail guns. Other geometries exist, although all appear to rely on the
sliding armature to carry the high currents and hence will generate a
characteristic skin effect that results in significant rail heating.

Finally, heating of the rails on a microscopic scale depends on
inhomogeneities of the metal, such as crystalline alignment and dislocations.
These inhomogeneities will lead to spatial fluctuations of the current
density. This will likely lead to a spatially inhomogeneous deposit of energy,
and local “hot spots”. Such spatial fluctuations in the case of an EMG are
likely to lead to the analogue of what is usually termed “gun barrel erosion”.
A comment made at the end of the chapter written by I. Ahmad in 1988, entitled
“The Problem of Gun Barrel Erosion: An Overview”, is still highly relevant:
“Other forms of erosion/corrosion problems might appear as a result of
advances made in the development of liquid propellant guns and electromagnetic
gun technologies. In fact, the erosion in the latter could be quite severe, as
it involves interaction of high-temperature plasma with the launch surface
causing it to partially melt at each firing event. It will be necessary to
identify materials and design methodologies to minimize these problems.” Our
paper does not include any effect of the high-temperature plasma, but it shows
the effect of Joule heating and geometric design considerations on the rails.

## Acknowledgment

This work was sponsored in part by ILIR at the AMRDEC.

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  * [12] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, “Electrodynamics of Continuous Media”, Pergamon Press, New York, 2nd Edition, 1984.
  * [13] T. B. Bahder and J. D. Bruno, “Transient Response of an Electromagnetic Rail Gun: A Pedagogical Model”, Army Research Laboratory report No. ARL-TR-1663, May 1998. See http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA345008 or http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA345008&Location=U2&doc=GetTRDoc.pdf or http://handle.dtic.mil/100.2/ADA345008.
  * [14] G.C. Long, “Railgun current density distributions”, IEEE Trans. Magn. vol. 22, no. 6, pp. 1597–1602, Nov. 1986.
  * [15] J. C. Nearing and M. A. Huerta, ”Skin and heating effects of railgun current”, IEEE Trans. Magn., vol. 25, no. 1, pp. 381–386, Jan. 1989.
  * [16] For a more recent treatment that uses a more complicated rail geometry, see , J. D. Powell and A. E. Zielinski, ”Two-Dimensional Current Diffusion in the Rails of a Railgun”, Army Research Laboratory Technical Report ARL-TR-4618, October 2008, see http://www.arl.army.mil/www/default.cfm?Action=17&Page=239&TRAction=GetAbstract&ReportID=1683&Topic=TechnicalReports.
  * [17] See sections 3.8 through 3.12 and 4.6 to 4.7 of H. Knoepfel, “Pulsed High magnetic Fields”, North-Holland Publishing Company, London, 1970.
  * [18] The first part of this assumption is conservative, since a deviation from ${\frac{{\partial{\kern 1.0pt}I}}{{\partial\,x}}}$ increases the total heating, somewhat compensating for the optimistic assumption that $\frac{{\partial{\kern 1.0pt}I}}{{\partial\,y}}=\infty$ or $I=0$ at the boundary, see later discussion.
  * [19] W. C. McCorkle, “Compensated Pulse Alternators to Power Electromagnetic Railguns”, p. 364, 12th IEEE International Pulsed Power Conference, Monterey, California, 1999.
  * [20] M. Abromowitz and I. A. Stegun, Dover Publications, Inc. New York, 9th printing, 1965.