electromagnetic methods (theory)

7/30/2019 Electromagnetic Methods (Theory)

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Chapter 6

Electromagnetic Methods

EM methods use time-varying magnetic fields that drive currents in the subsurface. The subsurface currents

are detected and used to model the area under investigation. There is no need for direct physical contact with

the ground, so these are inductive methods. Inductionis the general property of an electrical circuit to generate

a magnetic field in the surrounding medium, as well as the complementary property that a time varying magnetic

field will generate an electromotive force (and hence a current) in a surrounding conductor. The main advantage

of EM systems over resistivity methods is that since there is no need for physical contact (no electrodes) the

EM methods can be used in non-conductive conditions.

EM methods can be either passive (if they use naturally occurring time varying magnetic fields Mag-

netoTellurics) or active (if they use an artificial transmitter near field = Slingram, Turam, Sundberg; far

field = Very Low Frequency). Measurements can be made in either the time domain or frequency domain.

EM methods are used in mineral exploration, groundwater surveys, environmental studies (e.g. contamination

plumes), geotechnical surveys (e.g. pipe detection), geothermal studies and geological mapping of faulting or

cavities.

6.1 Principles of EM Theory

A time varying magnetic field can be generated by driving an alternating current through a loop of wire,

or through a wire grounded at both ends. If there is any conductive material present within the generated

magnetic field, induced or eddy currents will flow within the conductor in closed paths normal to the direction

of the magnetic field. These eddy currents will generate their own magnetic field so that at any point in space

the total magnetic field consists of two parts. A primary field due to the initial current source, and a secondary

or disturbing field due to the eddy currents induced in the conductor. The resultant magnetic field is usually

measured in terms of the voltage induced in a loop of wire used as a receiver (figure 6.1).

Some care needs to be taken with the terminology associated with magnetic effects. We will refer to B,

which is measured in Tesla (1 T = 1 N/(A m)) as the magnetic field, as is most natural. However, it is also

common to refer to this quantity as the magnetic flux density or the magnetic induction, terms which also have

other meanings. B can be generated by a material with a magnetization (M the magnetic dipole moment

per unit volume) arising due to the alignment of atomic dipoles. B can also be generated by free currents

(the ordinary currents with which we are familiar, magnetization produces so called bound currents in that the

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EM 55

ReceiverLoop

TargetConductor

Transmitter

Loop

Figure 6.1: Schematic illustration of the eddy currents, primary and secondary magnetic fields of an EM survey.

electrons are attached to a specific atom and thus are not free to move throughout the material). The field

generated by free currents is called called the auxiliary field (H), also known as the magnetic field intensity.

Both H and M have units of A/m, and the overall relationship between these three quantities is

B = 0(H+M) (6.1)

where 0 = 4 107 N/A2 is the permeability of free space.

Some materials are naturally magnetized, all materials will show some amount of induced magnetization,

i.e. the atoms have a non-zero magnetic moment which become aligned when exposed to an external magnetic

field. The measure of induced magnetization strength is M, the magnetic susceptibility. In linear media the

susceptibility is independent of the applied field strength and

M = MH (6.2)

such that

B = 0(1 + M)H = H (6.3)

where is the permeability of the material. In most common materials the magnetic susceptibility is very small

(on the order of 105) and 0.

6.1.1 Steady Fields

In exploration EM the magnetic field is generally created by running a current through a wire. The strength

of the generated magnetic field will be proportional to the current and inversely proportional to the distance

from the wire. The direction of the magnetic field will be in circular loops perpendicular to the wire and obeythe right-hand rule (see figure 6.2).

In free space, the magnetic field due to a steady current can be calculated using the Biot-Savart Law

B =04

Id r

r2(6.4)

For an infinite line current, the associated magnetic field is B = 0I/2a. If we consider the line integral around

a loop of the field (the circle C with associated tangent vector c) we have

CB dc =

0I

2a2a

C

B dc = 0 Ienc (6.5)

(6.6)


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We can turn the left-hand side of this equation into a surface integral using Stokes Theorem and the right-hand

side of the equation in to an integral over the same surface by using the definition of the current density, J

(with units A/m2).

A

( B) da = 0 A

J da

B = 0 J (6.7)

Where A is the surface bounded by C. Equations (6.5, 6.7) are the integral and differential forms of Amperes

Lawfor the case of steady currents and fields.

wire carryingcurrent I

circle ! wire, radius a

B, out ofpage at top

B, into pageat bottom

r

d

Figure 6.2: Magnetic field about a line current

6.1.2 Time Varying Fields

For non-steady currents the use of Stokes theorem is complicated by the ability to choose arbitrary surfaces as

there are an infinite number of surfaces that will bounded by C. Charge can accumulate when we allow non-

steady currents, therefore we can find equally valid choices of A which will have differing amounts of current

passing through them. In the time-varying case a second term must be added to Amperes Law to give

B = 0J+ 00E

t(6.8)

This new term is known as the displacement current although it does not correspond to a physical current; it

accounts for the magnetic field that is generated by a time varying electric field. In matter this equation takes

the form

H = Jf +D

t(6.9)

where Jf is the free current and D = E is known as the electric displacement. The displacement current is

negligible compared to the induced, free current in a good conductor. However in a poor conductor it can

be the primary cause of magnetic field generation. In the absence of free current the displacement current is

the primary source of magnetic field generation and is key to the generation of EM waves, as we saw when

considering GPR.

Similarly, a time varying magnetic field generates an electric field. If we consider the magnetic fluxthrough

a loop of wire

=

A

B da (6.10)

where A is a surface bounded by the loop, then the motional emf (electromotive force) generated in the loop as

it is moved through a spatially varying magnetic field, or if there is a time-varying magnetic field, isE dl = E =

d

dt(6.11)


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The time varying emf corresponds to a time varying electric field within the loop and hence a time varying

current. This is the integral form of Faradays Law, the differential form is

E = B

t(6.12)

6.1.3 Inductive coupling

Equations (6.8 and 6.12) can be combined to show that a time varying current in one loop will induce a time

varying magnetic field in the surrounding volume, which in turn will induce an emf that will drive current in a

second nearby conductor (figure 6.3). Let the current in the transmitter loop be I(t) = I0 sin(t). The magnetic

field strength at the centre of this circular loop is

B =0I(t)

4

dl r

r2=

0I02a

sin(t) (6.13)

The field is approximately, spatially constant inside the loop so

=

B dA a2B =

a0I02

sint (6.14)

If the second, receiver loop is sufficiently close by, this will also be the flux in that loop. This time varying flux

will create an emf in the receiver loop

Er =

t=

a0I02

cost = a0I0

2sin(t /2) (6.15)

The induced emf lags the primary current by a quarter cycle (/2).

Transmitter

loop

Receiver

loop

B

(t

)

Figure 6.3: Time varying current in the transmitter loop (bottom) generates a time varying magnetic field (B)and hence a time varying flux in the receiver loop (top).

If we consider only the peak amplitudes we see that the flux in the receiver loop is proportional to the

current in the transmitter loop. That is = MI where M is the mutual inductance of the two loops. The unit

of inductance is the henry (1 H = 1 V s/A). In case above M= a0/2; however, it is clear that M will depend

on the shape, position and orientation of both loops (figure 6.4). It can be difficult to calculate the mutual

inductance for a general set up; however, it can be shown that the inductance in loop 2 due to a current in loop

1 is

M2,1 =04

dl1 dl2

r(6.16)

an expression known as the Neumann formula. This formula shows that M is a purely geometric quantity. Also,

we are free to number the loops however we want, so clearly M1,2 = M2,1 = M, hence mutual inductance.

Whatever the shapes and positions of the loops, the flux through loop 2 when we run a given current around

loop 1 is exactly the same as the flux through loop 1 if we run the same current around loop 2. In general, if


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we run a current I(t) = I0eit through our transmitter loop, then we will have an emf Er = iMI(t) in the

receiver loop.

r

dl1

dl2

Figure 6.4: Arbitrary loops

Additionally, we have seen that there is a time-varying flux through the transmitter loop. Therefore, an

emf is induced in the transmitter loop as well. This is the self inductance of the current loop which leads to

E = LI

t(6.17)

where L is the inductance of the loop. This emf acts to oppose any attempts to change the current around the

loop, a property known as Lenzs Law.

Current in the receiver loop will be determined by the emf ( Er(t)) due to mutual inductance with the

transmitter (Mrt), some level of resistance (Rr) and a self inductance (Lr). The current generated in the

receiver loop due to inductive coupling with the transmitter (Irt(t)) is described by a modified version of Ohms

law

Ert LrIrtt

= IrtRr (6.18)

If we assume that the current in the transmitter coil has the form I0eit, then the emf in the receiver due to

the transmitter current can be written as Ert = iMrtI0eit and the solution to equation (6.18) is

Irt(t) = iMrtI0Rr + iLr

eit (6.19)

To this point we have considered our two loops in isolation. What if there is also a large conductive body

nearby (figure 6.1)? The time varying magnetic field from the transmitter will produce an emf and a current

directly in the receiver loop (Irt) as described by equation (6.19). However, the transmitter will also induce anemf and hence eddy currents (Ist) in this secondary, conductive body, with

Ist(t) = iMstI0Rs + iLs

eit (6.20)

where Mst is the mutual inductance between the secondary body and the transmitter loop and Rs and Ls are

the internal resistance and self inductance of the secondary body.

These eddy currents will generate the so-called secondary magnetic field which will also be inductively

coupled to the receiver loop generating a secondary emf and hence a current. The current arising due to mutual

inductance between the receiver and the secondary body will be

Irs(t) = iMrs

Rr + iLrIst(t) (6.21)


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The total current measured in the receiver loop will therefore be

Ir(t) = Irt + Irs

= iMrtI0Rr + iLr

eit Mrs

Rr + iLr

MstI0Rs + iLs

eit

= iMrtI0

Rr + iLr

1 iM

stMrsMrt(Rs + iLs)

eit (6.22)

6.1.4 Induction Amplitude and Phase Lag

The two sources of current induction in the receiver loop differ in both amplitude and phase. We can investigate

the phase lag of the secondary field relative to the primary field by considering the ratio

IrsIrt

= MstMrsMrt

i

Rs + iLs

=MstMrs

Mrt

R2s +

2

L2s

(Ls iRs)

=MstMrsMrt

R2s +

2L2s

LsR2s +

2L2s i

RsR2s +

2L2s

(6.23)

which is a complex number of the form A(cos + i sin ). The phase lag of the secondary field relative to the

primary field is

=

2+ tan1

LsRs

(6.24)

Re

Im

AIQ

IIP

Figure 6.5: Plot of the current ratio in the complex plane

When the conductivity of the secondary body is (very) low we find that Rs /2 and the

secondary field lags the primary by 90 and is, therefore, out of phase with the primary field. Conversely when

the conductivity is (very) high we have Rs

0

and the secondary field is considered to be in phasewith the primary (albeit with a reversal of sign). As a result, the components of the current ratio are known as

the in-phase component (for the real) and the out-of-phase or quadrature component (for the imaginary). We

can therefore express the current ratio as Irs/Irt = IIP + iIQ and comparison with equation (6.23) gives

A =MstMrsMrt

R2s + L

2s

2=

I2IP + I

2Q (6.25)

Note that although the ratio of secondary to primary current may be large for a highly conductive target (A is

large) the quadrature component will be small. Therefore, if only the imaginary component of the current in

the receiver loop is measured, good conductors will produce very small signals. In frequency domain EM we can

measure the receiver response relative to the known transmitter forcing and use plots of the quadrature against

the in-phase component to interpret the properties of the subsurface. (Note that it is common to speak of the Q


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and IP components of the signal, these should not be confused with other terms having the same abbreviations;

in this context IP is NOT induced polarization!).

L

R

C

E(t)

Figure 6.6: RLC series circuit

We can gain further insight into the phase lag between the primary and secondary fields by considering that

the ground under investigation acts like an electrical circuit with resistive, inductive and capacitive components

to which we are applying a time varying emfE(t) = E0 sint (figure 6.6). In such a circuit the resulting current

is

I(t) = E0

[L (1/C)]2 + R2

1/2

sin(t ) (6.26)

where

= tan1L (1/C)

R

(6.27)

is the phase lag of the current with respect to the applied voltage. Note that the term which depends on

capacitance is inversely proportional to frequency, thus at high frequencies the effects of subsurface capacitance

can b e ignored and we are in the purely inductive regime. Recall that induced polarization measurements

involved frequencies of 10 Hz or less so that the capacitance was important but not the inductance. Frequency

domain EM methods generally use frequencies of hundreds to thousands of hertz, even up to several tens of

kilohertz.

In EM exploration the primary field (HP = H0 sint) is in phase with the generated current (figure 6.7a).

The induced voltage in the secondary conductor lags behind the primary field by /2, as in equation (6.15)

(figure 6.7b). Eddy currents in the secondary conductor take a finite time to generate and thus lag behind

the induced emf by a phase lag (equation 6.27) which depends on the electrical properties of the conductor

(figure 6.7c). The secondary magnetic field is in phase with the induced current in the conductor and thus lags

the primary field by /2 + (equation 6.24). The sum of the primary and secondary fields gives the resultant

field at the receiver coil, which will lag behind the primary field by some phase angle () (figure 6.7d).

We can plot the fields as vectors in a phase space consisting of their in-phase and quadrature components

(figure 6.8). By definition the primary field has only an in-phase component. The secondary field has both

in-phase and quadrature components and lags the primary field by /2 + . The resultant field, the vectorial

summation of the primary and secondary fields, has a phase lag .

6.1.5 Polarization

Thus far we have considered the temporal shift ( = [/2 + ]) of the secondary field with respect to the

primary. The secondary field will, in general, also point in a different direction than the primary field, i.e. there

will be an angle () between them. In order to determine the resultant field we need to resolve the primary and


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/2

a) Primary Current or Magnetic Field

Time

p

Secondary Current or Magnetic Fieldc) Hs

b) Voltage in Secondary Conductor Es

Resultant Magnetic Fieldd)

Hr = Hp +Hs

Figure 6.7: The various fields in the inductive coupling problem plotted against time. (After Beck (1981) viaReynolds (1997).)

/2

Hs

Hp

Hr

IP

Q

Figure 6.8: Field vectors plotted in Q-IP space

secondary fields into their vertical (z) and horizontal (x) components (figure 6.9a).

Hr(t) = Hrx(t)x + Hrz(t)z (6.28)

Hrx(t) = Hpx cost + Hsx cos(t + ) = Rx cos(t + 1)

Hrz(t) = Hpz cost + Hsz cos(t + ) = Rz cos(t + 2)

with H2r (t) = H2rx(t) + H

2rz(t). Both the magnitude and orientation of the resultant field vary through time

with the tip of the vector tracing out an ellipse in the x-z plane (figure 6.9b) which is known as the ellipse of

polarization. This ellipse is inclined with respect to horizontal by a tilt angle

=1

2tan1

RxRz cos(2 1)

R2x R2z (6.29)


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z

s

p

x

z

Hsx cos(t+ )

Hpx cos(t)

a)

x

z

Rx

b)

Hpz

cos(t

Hsz

cos(t+

Figure 6.9: a) The vertical (z) and horizontal (x) components of the primary and secondary fields. b) Theellipse of polarization of the resultant field.

6.1.6 Depth of Penetration

When considering GPR we saw that, in non-conductive media, the electric and magnetic fields obey the wave

equation. We also considered the case of waves within weakly conductive media. We did not consider the case

of highly conductive media. Within a good conductor Maxwells equations can be manipulated to obtain a

propagation equation for the electric field

2E =

E

t+

2E

t2(6.30)

The magnetic field obeys an equivalent expression. When conductivity is negligible this expression simplifies to

the wave equation we considered before. On the other hand, when conductivity is large the equation becomes

2E

E

t(6.31)

which is a diffusion equation. The effective diffusivity of the medium is = 1/.

Consider a plane wave with angular frequency incident on the surface (z = 0) of a conductive medium.

From the diffusion equation we find that within the medium the electric field is of the form

E(z, t) = E0ez/e[i(tkz)] (6.32)

where k =

/2 is the wavenumber of the field within the medium and = 1/k =

2/ is the skin depth.

As in the GPR formulation the amplitude of the field falls to 1/e of its initial value when z = ; however, note

that the diffusive skin depth is not equal to the wave skin depth of the GPR problem. The diffusive skin depth

depends strongly on frequency.

The response of an EM system will depend on how the separation between the transmitter and receiver

loops (RT) compares to the skin depth. If (RT)/ 1 then the skin depth is effectively large, as would be

the case for poorly conducting ground. This is known as the low induction number limit. In the low induction

number limit we expect that the in-phase component of the receiver current to be very small compared to the

quadrature component (as we saw above with low conductivity, figure 6.5). Therefore, in our expression for the

amplitude of the current ratio of the receiver loop (equation 6.25) we have IIP IQ, which is equivalent to

saying Ls Rs, and we obtain1

Rs=

IQ

MrtMstMrs

(6.33)

electromagnetic methods (theory)

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