Circuit model for Transverse EM waves
Carlo Andrea Gonano
1
, Riccardo Enrico Zich
2
(1)
,
(2)
Energy Department
Politecnico di Milano,
Milan, MI, Italy,
carloandrea.gonano@polimi.it, riccardo.zich@polimi.it
Abstract—Circuit models are often used to describe a system
in a simply way, lumping its properties in few variables and
allowing to solve ElectroMagnetic (EM) problems without the
need to deal with the complete set of Maxwell’s equations. Even
if circuit theory is a powerful design tool it has some drawbacks,
since the electric field
E is approximated as conservative and thus
the propagation of Transverse EM waves is neglected. However,
in this paper we present a circuit model suitably designed to
describe also those propagation effects. For sake of simplicity,
we consider the problem of the two radiating infinite parallel
wires and introduce the concept of “impedance coupling” in
order to treat transverse waves. Since EM fields propagate in
“empty” space, we are going to interpret the potential vector
A
as a current iA flowing in circuit elements.
I. I NTRODUCTION
The Circuit Theory[1] is based on the possibility of in-
troducing lumped or distributed element models instead of
full-wave ones. Usually an EM problem can be treated with
circuits just if the
E field is conservative, so that it’s possibile
to introduce a potential V . This condition is satifisfied if
the characteristic length l of the system is much smaller
than the operating wavelength λ, so that every point of the
system experience almost the same EM force (Max Abraham
condition).
l<
λ
2π
= ⇒
− →
∇∧
E ≈
0 = ⇒
E = −
− →
∇V (1)
Since
E is conservative, the usual Circuit Theory exhibits
strong limits when you are dealing with irradianting devices
(i.e., antennas) and with the propagation of Transverse Electro-
Magnetic (TEM) waves. However, in this paper we are going
to show how to model in terms of circuits also that class of
effects.
II. TEST PROBLEM: INFINITE PARALLEL WIRES
Let’s consider an xz-plane containing two infinite electric
wires parallel to the z-axis (Fig.1). At the instant t =0 a
uniform current i
E
(t) starts to flow into the wire 1, which
will so radiate towards the wire 2. The system has a traslation
symmetry along the z-axis, so:
∂
E
∂z
=
0;
∂
J
E
∂z
=
0;
∂
A
∂z
=
0 (2)
where:
•
E is the electric field
Fig. 1. Infinite parallel wires. A current flows in the wire 1, making it
to irradiate. The
E and
A waves are transversal because of the system’s
symmetry in the z direction.
•
J
E
is the flux of electric charge
•
A is the potential vector, such that
B =
− →
∇∧
A
If the wire 1 has a section S, then
J
E
=
i
E
/S and so:
u
z
//
i
E
//
J
E
//
E//
A (3)
where u
z
is the unitary versor parallel to z-axis.
Since all the vector fields are parallel to the z-axis and
perturbations cannot propagate in this direction (the traslation
symmetry holds), it yields that the produced wave will be
purely transversal. In other words, all the vector fields are
non-conservative (e.g.
− →
∇∧
E =
0) and have zero-divergence.
− →
∇
T
·
E = 0;
− →
∇
T
·
J
E
= 0;
− →
∇
T
·
A = 0; (4)
To remark the absence of “compression”, we rewrite the
Faraday’s Law with the potentials V and
A as:
∂A
z
∂t
+ E
z
u
z
= −
− →
∇V = ⇒ −
− →
∇V = −
∂V
∂z
u
z
(5)
= ⇒
∂A
z
∂t
+ E
z
= −
∂V
∂z
(6)
but the system is z-symmetric, so
∂V
∂z
=0 and therefore V is
constant in space.
III. CIRCUIT DISCRETIZATION
Reducing a system to an equivalent circuit implies some
kind of space discretization, because every circuit-net is made
of nodes and edges. For our problem we model the space
between the wires as a series of vertical lines (not directly
613 978-1-4799-3540-6/14/$31.00 ©2014 IEEE AP-S 2014