A systemized parameter set applicable to microwave absorption for ferrite based materials

Abstract

When electromagnetic waves are absorbed by materials, the most significant factor is the resistance to the induced current and the induced oscillations of electric and magnetic dipoles in the material. In this paper, a set of related parameters has been integrated into a theoretical system to characterize electromagnetic wave absorption from materials and to reveal the connection of the different physical processes for wave absorption. The parameters used are all associated with imaginary numbers. The system is general and can be applied to the important ferrite based materials. The system characterizes absorption that is common to a variety of physical processes. A problem found in the literature concerning the measurement of reflection loss has been corrected.

Appendices

Appendix 1

1.1 The series and parallel equivalent circuits

Microwave absorption of a material can be modeled in two equivalent ways, either by a capacitor of capacitance C _P connected with a resistor of resistance R _P in a parallel circuit, or by a capacitor of capacitance C _S connected with a resistor of resistance R _S in a series circuit. The characteristic impedance Z _P for the parallel circuit is

$$\frac{{\text{1}}}{{{Z_P}}}=j\omega {C_P}+\frac{{\text{1}}}{{{R_P}}}=\frac{{j\omega {C_P}{R_P}+{\text{1}}}}{{{R_P}}}$$

(99)

The characteristic impedance Z _S for the series circuit is

$${Z_S}=\frac{1}{{j\omega {C_S}}}+{R_S}=\frac{{1+j\omega {C_S}{R_S}}}{{j\omega {C_S}}}$$

(100)

In order for the two circuits to be equivalent, Eq. 101 must be satisfied.

$$\frac{{1+j\omega {C_S}{R_S}}}{{j\omega {C_S}}}=\frac{{{R_P}}}{{1+j\omega {C_P}{R_P}}}$$

(101)

or

$$j\omega {C_S}{R_P}=\left( {1+j\omega {C_S}{R_S}} \right)\left( {1+j\omega {C_P}{R_P}} \right)=\left[ {1 - \left( {\omega {C_S}{R_S}} \right)\left( {\omega {C_P}{R_P}} \right)} \right]+j\left( {\omega {C_S}{R_S}+\omega {C_P}{R_P}} \right)$$

(102)

Equations 103 and 104 are obtained from Eq. 102. They must both be satisfied for the two circuits to be equivalent.

$$\left\{ {\begin{array}{*{20}{c}} {\omega {C_S}{R_P}=\omega {C_S}{R_S}+\omega {C_P}{R_P}} \\ {1 - \left( {\omega {C_S}{R_S}} \right)\left( {\omega {C_P}{R_P}} \right)=0} \end{array}} \right.$$

(103)

$$\frac{1}{{\omega {C_P}{R_P}}}=\omega {C_S}{R_S}$$

(104)

Thus, the tangent loss is given by

$$tg\delta =\frac{{{I_{RP}}}}{{{I_{CP}}}}=\frac{{V/{R_P}}}{{V/(1/\omega {C_P})}}=\frac{1}{{\omega {C_P}{R_P}}}=\omega {C_S}{R_S}=\frac{{{R_S}}}{{(1/\omega {C_S})}}=\frac{{{Z_{RS}}}}{{{Z_{CS}}}}=\frac{{{V_{RS}}}}{{{V_{CS}}}}$$

(105)

Equation 107 is obtained by inserting Eq. 106 into Eq. 105.

$$\begin{gathered} {R_P}=\frac{1}{{\omega {C_0}{{\varepsilon ^{\prime\prime}}_r}}} \hfill \\ {C_P}={{\varepsilon ^{\prime}}_r}{C_0} \hfill \\ {Z_{CP}}=\frac{1}{{\omega {{\varepsilon ^{\prime}}_r}{C_0}}} \hfill \\ \end{gathered}$$

(106)

$$tg\delta =\frac{{{{\varepsilon ^{\prime\prime}}_r}}}{{{{\varepsilon ^{\prime}}_r}}}$$

(107)

Appendix 2

2.1 A mnemonic for the left and right hand conventions in electromagnetism

We have previously introduced mnemonics for the easy assignment of R and S, and L and D conventions with chiral molecules [43]. The right R and left L hands shown in Fig. 4a, b can also be used via a mnemonic to establish the relationship between directions that is often difficult to remember. The mnemonic can also be used to remember the order of v and B in Eq. 108 and that of I and B in Eq. 109. The mnemonic is not only useful for describing the left and right hand conventions in electromagnetism, but it also reveals the connection and difference between Eqs. 108 and 109.

The magnetic force F on a charge q moving with velocity v, exerted by a magnetic field B is

$${\mathbf{F}}=q{\mathbf{v}} \times {\mathbf{B}}$$

(108)

The directions of v, B, and F are shown in Fig. 4a. The right hand should be used to predict the direction of F as shown on the left side of Fig. 4a. A mnemonic to remember this can be derived from Fig. 4b. Suppose the magnetic line cannot be cut. The movement of the wire in Fig. 4b would make the magnetic lines denser on the right side of the wire than on the left side. This result would be equivalent to inducing a field B′ if the wire had not moved, whereas the addition of B and B′ established more lines on the right side of the wire and some lines on the left would be canceled. To induce such a field B′, an inward force resulting from the change should be needed to produce such an inward current, which is predicted by the right hand R in Fig. 4a. The directions of v, B, and F obtained from Fig. 4b can be represented by the right hand R in Fig. 4a but not by the left hand.

Another way of writing Eq. 108 is

$${\mathbf{F}}=I{\ell _1} \times {\mathbf{B}}$$

(109)

Although Eqs. 108 and109are different, they are related and their relationship is represented by Eq. 110.

$${\mathbf{F}}=q{\mathbf{v}} \times {\mathbf{B}}=q\frac{{d{\ell _1}}}{{dt}} \times {\mathbf{B}}=\frac{{dq}}{{dt}}{\ell _1} \times {\mathbf{B}}=I{\ell _1} \times {\mathbf{B}}$$

(110)

here in Eq. 109 the movement of the charge itself is a current I. The directions of I, B, and F are shown in Fig. 5a. Since Eqs. 108 and 109 are essentially the same, the right hand R in Fig. 5a can be used in the same way as established above in Fig. 4. However, Eq. 108 can be used for a current induced by a wire movement while Eq. 109 is used for a wire movement induced by a current. Thus, left hand L in Fig. 5a is conventionally used in conjunction with Eq. 109 to predict the direction of F, i.e. if the magnetic lines are perpendicular to the palm, and the four fingers are in the direction of I, then the thumb is in the direction of the force. The wire movements indicated by Eqs. 108 and 109 are opposite which is consistent with a general reflection the law of the action and reaction.

A mnemonic consistent with Fig. 4b to remember the different hand conventions can be derived from Fig. 5b. The current in the wire shown by Fig. 5b induces a field B′. The addition of B and B′ creates more lines on the right side of the wire and some lines on the left are canceled. The wire would move toward the side with less populated magnetic lines, indicating the direction of the force exerted on the wire. The directions of I, B, and F obtained from Fig. 5b can be represented by the left hand L in Fig. 5a but not the right hand. It should be noticed that Figs. 4b and 5b are only designed to remember the hand conventions and have no independent significance.

Appendix 3

3.1 Background information on complex permeability

A current loop in Fig. 6 experiences a torque M

$${\mathbf{M}}={\ell _2} \times {\mathbf{F}}={\ell _2} \times (I{\ell _1} \times {\mathbf{B}})=IS \times {\mathbf{B}}={\mathbf{\mu }} \times {\mathbf{B}}$$

(111)

If the current loop is formed by an electron moving in a circle with radius r

$$\left| {\mathbf{\mu }} \right|=IS=\frac{{e{\mathbf{v}}}}{{2\pi \left| {\mathbf{r}} \right|}}\pi {\left| {\mathbf{r}} \right|^2}=\left| {\frac{e}{{2m}}m{\mathbf{r}} \times {\mathbf{v}}} \right|$$

(112)

More generally

$${\mathbf{\mu }}=g\frac{e}{{2m}}m{\mathbf{r}} \times {\mathbf{v}}=\gamma m{\mathbf{r}} \times {\mathbf{v}}$$

(113)

g is the Lande factor [20]. M is the driving force to align µ with B. Judged from the mnemonic or from the movement of µ, the torque always tends to increase the magnetic flux within the current loop. From Eqs. 111 and 113, we obtain Eq. 114 if r is fixed.

$$\frac{{d{\mathbf{\mu }}}}{{dt}}=\gamma \frac{d}{{dt}}m{\mathbf{r}} \times {\mathbf{v}}=\gamma {\mathbf{r}} \times m\frac{d}{{dt}}{\mathbf{v}}=\gamma {\mathbf{M}}=\gamma {\mathbf{\mu }} \times {\mathbf{B}}$$

(114)

The individual components of the vector µ obey

$$\frac{{d{\mu _x}}}{{dt}}=\gamma \left( {{\mu _y}{B_z} - {\mu _{_{z}}}{B_y}} \right)$$

(115)

$$\frac{{d{\mu _y}}}{{dt}}=\gamma \left( {{\mu _z}{B_x} - {\mu _{_{x}}}{B_z}} \right)$$

(116)

$$\frac{{d{\mu _z}}}{{dt}}=\gamma \left( {{\mu _x}{B_y} - {\mu _{_{y}}}{B_x}} \right)$$

(117)

If the applied magnetic field is along the z axis, B _x = B _y =0.

$$\frac{{d{\mu _x}}}{{dt}}=\gamma {\mu _y}B$$

(118)

$$\frac{{d{\mu _y}}}{{dt}}= - \gamma {\mu _{_{x}}}B$$

(119)

$$\frac{{d{\mu _z}}}{{dt}}=0$$

(120)

By differentiating both sides of Eq. 118 with respect to time and substituting $\frac{{d{\mu _y}}}{{dt}}$ in the form given in Eq. 119, we obtain the free oscillation equation for µ _x.

$$\frac{{{d^2}{\mu _x}}}{{d{t^2}}}= - {(\gamma B)^2}{\mu _x}$$

(121)

Introducing the magnetic driving force and resistance for the oscillation, we obtain an equation similar to Eq. 1 or 5. When a magnetization vector is used instead of a single magnetic moment µ, the complex susceptibility χ can be introduced.

$$\chi =\chi ^{\prime} - j\chi ^{\prime\prime}$$

(122)

Appendix 4

4.1 Symbols used

α :

Polarizability

β :

Optical polarisability

χ :

Susceptibility

δ :

The loss angle or δ function

ε ₀ :

The permittivity of the vacuum

ε′, ε″ :

Real and imaginary parts of complex permittivity

φ :

Cylindrical coordinate

Γ :

Return loss

η′:

The damping factor

µ :

Magnetic moment

µ′, µ″:

Real and imaginary parts of complex permeability

ν:

The frequency of the microwave

ρ :

Cylindrical coordinate or resistivity depending on text

σ :

Conductivity or surface charge density dependent on the text

τ :

The relaxation time or element of volume dependent on the text

ω :

Circular frequency of radiation

ω ₀ :

The characteristic circular resonance frequency of material

Ω:

A range for integration

a(t):

Attenuation factor

a _P :

The power absorption coefficient

A :

A vector for an area with a direction

B :

Magnetic flux density

c :

The speed of microwave in vacuum

C :

Distributive capacitance

D :

The displacement vector

e :

Charge of electron

E :

Electric field

F :

Force

g :

The Lande factor

H :

Magnetic field

I :

Current

I ₀ :

The maximum amplitudes of current

J :

The current density

k :

The restoring constant or imaginary part of the refraction index

L :

Distributive inductance

m :

Mass of electron

m _H :

Magnitude of magnetic dipole moment

M :

Torque

M :

Molar mass

n :

Refraction index

N :

The number of dipoles in unit volume

N _A :

Avogadro constant

N :

Magnetic poles

p :

Electric dipole moment for a molecule

P :

Polarization vector

q :

Charge

r :

Coordinate vector

R :

Resistance or radius of a circle depending on text

R _M :

The reflection loss of material

s _ij, i, j = 1, 2:

Scattering parameter

S :

Magnetic poles

S :

A vector for an area with a direction

t :

Time

v:

The velocity of a wave

V :

Voltage or volume depending on the text

V ₀ :

The maximum amplitudes of voltage

x, y, z :

Cartesian coordinate

X :

The maximum amplitude of x

z :

May also indicates a complex number

Z ₀ :

The characteristic impedance

Z _l :

The characteristic impedance for transmission lines

Z _M :

The characteristic impedance for material measured