INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS
Int. J. Circ. Theor. Appl. (2015)
Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cta.2064
Fractional-order mutual inductance: analysis and design
Ahmed Soltan1, Ahmed G. Radwan2,3,*,† and Ahmed M. Soliman4
1
School of Electrical and Electronic Engineering, Newcastle University, UK
Department of Engineering Mathematics and Physics, Cairo University, Cairo, Egypt
3
Nanoelectronics Integrated Systems Center (NISC), Nile University, Cairo, Egypt
4
Department of Electronics and Communications Engineering, Cairo University, Cairo, Egypt
2
SUMMARY
This paper introduces for the first time the generalized concept of the mutual inductance in the
fractional-order domain where the symmetrical and unsymmetrical behaviors of the fractional-order mutual
inductance are studied. To use the fractional mutual inductance in circuit design and simulation, an
equivalent circuit is presented with its different conditions of operation. Also, simulations for the impedance
matrix parameters of the fractional mutual inductance equivalent circuit using Advanced Design System and
MATLAB are illustrated. The Advanced Design System and MATLAB simulations of the double-tuned
filter based on the fractional mutual inductance are discussed. A great matching between the numerical
analysis and the circuit simulation appears, which confirms the reliability of the concept of the fractional
mutual inductance. Also, the analysis of the impedance matching using the fractional-order mutual
inductance is introduced. Copyright © 2015 John Wiley & Sons, Ltd.
Received 13 July 2014; Revised 9 October 2014; Accepted 6 January 2015
KEY WORDS:
mutual inductance; fractional elements; double-tuned filter; equivalent circuit
1. INTRODUCTION
In recent years, fractional calculus has been widely used in modeling the dynamics of many real life
phenomena because of the fact that it has higher capability of providing accurate description than
integer dynamical systems. This added flexibility is mainly due to the fact that fractional-order
systems can be characterized by infinite memory, whereas integer-order systems are characterized by
finite memory [1]. Moreover, because of the extra fractional-order parameters, more flexibility is
added in the modeling, analysis, and control of many applications such as determining voltage–
current relationship in a non-ideal capacitor [2, 3], fractal behavior of a metal insulator solution
interface [4], electromagnetic waves [5], and recently in electrical circuits such as filters [6–11] and
oscillators [12–14]. Furthermore, applications of fractional calculus have been reported in many
areas such as physics [15], nonlinear oscillation of earthquakes [16], and mathematical biology [17].
The Caputo definition of the fractional derivative of order α is written as follows [18]:
α
a D t f ðt Þ
:¼
8
>
>
>
<
t
1
f ð m Þ ðτ Þ
dτ
∫
Γðm αÞ 0 ðt τ Þðαþ1mÞ
m
>
>
>
: d f ðt Þ
dt m
m1<α<m
(1)
α¼m
*Correspondence to: Ahmed G. Radwan, Department of Engineering Mathematics and Physics, Cairo University,
Cairo, Egypt.
†
E-mail: agradwan@ieee.org
Copyright © 2015 John Wiley & Sons, Ltd.
A. SOLTAN, A. G. RADWAN AND A. M. SOLIMAN
where a and t are the initial and the required time of calculation. Applying the Laplace transform to the
general fractional derivative of (1) with zero initial conditions yields:
L 0 Dαt f ðt Þ ¼ sα F ðsÞ
(2)
Therefore, it becomes possible to define a general fractance device with impedance proportional to
sα [19], where the traditional circuit elements—capacitor, resistor, and inductor—are special cases of
this fractional-order element when the order is 1, 0, and 1, respectively. During the last 10 years,
several promising trials have been introduced for the realizations of the fractional element and based
on different techniques such as chemical reactions [20], fractal shapes [4], and graphene material
[21]. Moreover, many finite circuit approximations were suggested to model fractional-order
elements, for example, a finite element approximation of the special case Z = 1/(Cs0.5) was reported
in [22]. This finite element approximation relies on the possibility of emulating a fractional-order
capacitor via semi-infinite resistor–capacitor trees. The technique was later developed by the authors
of [23–25] for any order.
On the other hand, the conventional mutually coupled circuits (MCCs) have a wide range of
applications in instrumentation, communications, control systems, signal processing, and modeling
[26–28]. They can also be used for analog filters, particularly for replacing the magnetic transformer
in stagger-tuned filters [29, 30]. Using the MCC in the previous blocks has improved the design
flexibility and the system performance. The MCC is characterized by primary inductance, secondary
inductance, mutual inductance, and the coupling factor. The MCC is characterized by the following
impedance matrix [31]:
V1
V2
¼s
L11 ± M
M
M
L22 ± M
I1
I2
(3)
where L11, M, and L22 are the primary inductance, mutual inductance, and the secondary inductance,
respectively. From the MCC T-model illustrated in Figure 1(a), the MCC depends on using the
inductors to fulfill the mutual inductance. Although, there are very good inductors or capacitors,
they are considered a fractional-order behavior with fractional order 0.999 ≤ α ≤ 0.9999 [32, 33]. In
addition, this fractional-order value could be less in Radio Frequency (RF) applications [34].
Physical prototypes of the fractional-order capacitors and inductors are presented in [3,35]. Thus, it
becomes necessary to propose the analysis of the MCC based on the fractional-order model of the
inductors. In this case, a new concept arises, which can be referred to as fractional-order mutual
coupled circuits (FMCCs). The FMCC could be used instead of the integer-order mutual inductance
because it increases the design degree of freedom because of the increased parameters in the design.
This paper is organized as follows: Section 2 discusses the idea of the fractional mutual inductance
(FMI). After that, an FMCC is presented in Section 3. Applications based on the FMI and the FMCC
like the double-tuned filters and impedance matching are introduced in Section 4. Finally, the
conclusion of the paper is presented in Section 5.
(a)
(b)
Figure 1. (a) T-model of the integer-order mutual inductance and (b) T-model of the proposed fractional mutual inductance.
Copyright © 2015 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. (2015)
DOI: 10.1002/cta
FRACTIONAL MUTUAL INDUCTANCE ANALYSIS AND DESIGN
2. PROPOSED FRACTIONAL MUTUAL INDUCTANCE
The conventional MCC is a lossless network, which is not practical; thus, fractional parameters can be
used to add the losses terms, which were proven to be frequency dependent as mentioned by the
Coilcraft report [34,36]. For generalization, assume the primary and secondary inductors are of
different fractional orders α, β, respectively. Consequently, in this case the mutual inductance is not
symmetric, and the mutual inductance could be considered of the fractional-order γ. Hence, the
induced emf at each fractional-order inductor should be given by
emf 1 ¼ v1 ¼ L11
d α i1
d γ12 i2
±
M
12
dt α
dt γ12
(4a)
d γ21 i1
d β i2
±
L
22
dt γ21
dt β
(4b)
emf 2 ¼ v2 ¼ M 21
Then, by taking the Laplace transform of (4), the FMI can be represented by the following
impedance matrix equation:
V1
V2
¼
sα L11
sγ12 M 12
sγ21 M 21
sβ L22
I1
I2
(5)
For the traditional case α = β = γ12 = γ21 = 1, the matrix of (5) represents the matrix equation of the
integer-order MCC presented in (3). The impedance matrix of (5) represents the behavior of the
proposed FMI, which is the general case of the traditional mutual inductance. The FMI has unequal
phase response for the primary and the secondary inductors. In addition, the coupling between the
primary and the secondary inductors is unsymmetrical. The FMI modeled by (5) could be
represented by the equivalent T-circuit shown in Figure 1(b) for only the case of symmetrical
coupling γ12 = γ21 = γ and M12 = M21. Now, while a resistor–capacitor ladder can be used to
approximate a fractional-order capacitor, this same topology is difficult to realize a fractional-order
inductor as it requires many inductors [37].Therefore, the fractional-order capacitor could be used
with the general impedance converter circuit (GIC) to implement the grounded fractional-order
inductors as shown in Figure 2 [12,38]. On the other hand, to implement a floating fractional-order
inductor, two cascaded GICs are used [39, 40]. Hence, the input impedance of the GIC of Figure 2
is given as follows:
Figure 2. General impedance converter circuit used to simulate a grounded fractional-order inductor using a
fractional-order capacitor.
Copyright © 2015 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. (2015)
DOI: 10.1002/cta
A. SOLTAN, A. G. RADWAN AND A. M. SOLIMAN
Z in ¼ sα
CR1 R3 R5
R2
(6)
From (6), the inductance of the fractional-order inductor of order α is given as L = CR1R3R5/R2.
In addition, the proposed FMI symbol for the special case of symmetrical coupling is depicted in
Figure 3(a). For simulation purposes, the T-model of Figure 1(b) can be replaced with the circuit of
Figure 3(b). Then, the impedance matrix becomes as follows:
V1
V2
¼
sα L11 þ sγ M
sγ M
sγ M
sβ L22 þ sγ M
I1
I2
(7)
Subsequently, for the case of α = β = γ, the model represents a symmetric fractional-order mutual
inductance. In this case, the phase of all the parameters of the impedance matrix is απ/2. Then, the
phase value depends on the fractional-order α, which increases the design degree of freedom. The
circuit simulation for the FMCC of Figure 3(b) is depicted in Figure 4 at different values of α. The
phase response for the traditional case is π/2, but when the inductor elements are replaced with the
fractional-order elements, the phase changes depending on the value of α as illustrated in Figure 4.
The phase error between the simulated and the ideal phase responses (the dashed lines in Figure 4)
is ± 3o for α = 0.8 and ± 4ofor α = 0.7 during the simulated frequency range. This small error
indicates that the FMCC is suitable for a wide bandwidth of applications.
Another important case arises when α = β ≠ γ, the phase of the impedance matrix parameters in this
case can be calculated using the formula of [36] as follows:
(a)
(b)
Figure 3. (a) Proposed symbol of the fractional mutual inductance and (b) equivalent T-model of the
fractional mutual inductance.
Figure 4. Circuit simulation for the fractional mutual inductance for α = β = γ and L11 = L22 = M = 10μH.
Copyright © 2015 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. (2015)
DOI: 10.1002/cta
FRACTIONAL MUTUAL INDUCTANCE ANALYSIS AND DESIGN
∠Z 11;22 ¼
8
y
>
>
q11;22
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2tan1
>
>
<
x11;22 þ x2
þ y2
11;22
>
>
>
>
:
x > 0 or y ≠ 0
11;22
π
x < 0 and y ¼ 0
Undefined
x ¼ 0 and y ¼ 0
∠Z 12;21 ¼
where
γπ
2
(8a)
(8b)
y11;22 ¼ ωα L11;22 sinð0:5απ Þ þ ωγ Msinð0:5γπ Þ
(9a)
x11;22 ¼ ωα L11;22 cosð0:5απ Þ þ ωγ Mcosð0:5γπ Þ
(9b)
where L11,22 are the primary and secondary inductances of the FMI model. Actually, the phase
response of the impedance parameters Z11 and Z22 is the same as the phase response of the practical
inductor model [41, 42]. This should be expected, because these impedance parameters represent the
impedance of the mutual inductance inductors. On the other hand, the terms Z12 and Z21 represent
the coupling between the two inductors and do not represent a real inductor. From this simple
analysis, the impedance matrix of (7) represents the behavior of FMCC even if the fractional orders
are different. In addition, the fractional-order model gives the ability to control the parasitic
components of the FMCC by changing the value of the fractional orders (α, γ). Also, the phase is a
function of the inductance values (L11,22 and M) and the frequency of operation that increases the
design degree of freedom as shown in Figure 5(a) and (b) for different frequency points. From (8a),
the effect of the fractional orders (α, γ) is symmetric as shown in Figure 5. For the special case of
α = γ = 0, the elements tend to work as a resistance because the phase in this case equals zero as
(a)
(b)
Figure 5. Phase response for the impedance parameters Z11 and Z22 with respect to α and γ at different frequencies at L11 = L22 = M = 10μH (a) ωo = 1 rad/s and (b) ωo = 1 krad/s.
Copyright © 2015 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. (2015)
DOI: 10.1002/cta
A. SOLTAN, A. G. RADWAN AND A. M. SOLIMAN
expected. On the other hand, for α = γ = 1, the elements act as a pure inductor, and the phase equals π/2
as illustrated in Figure 5. Yet, with the fractional-order elements, this is not the only condition to fulfill
the pure inductance behavior. From (8a), the condition to satisfy the pure inductance behavior is given
as follows:
αγ
¼
ω11;22
pure
Mcosð0:5γπ Þ
L11;22 cosð0:5απ Þ
(10a)
y11;22pure ¼ ωα11;22pure L11;22 sinð0:5απ Þ þ ωγ11;22pure Msinð0:5γπ Þ
(10b)
From (10a), there is an infinite number of conditions at which the pure inductance response can be
obtained. So, for a given value of α and at the required frequency, the value of the fractional-order γ
could be calculated to fulfill the condition of (10a) as shown in Figure 6(a). Then, the value of the
pure inductance is calculated using (10b) as depicted in Figure 6(b). Similarly, the condition of the
pure resistance response and its value is obtained by replacing the cosine function with the sine
function as follows:
αγ
¼
ω11;22
pure
Msinð0:5γπ Þ
L11;22 sinð0:5απ Þ
(11a)
x11;22pure ¼ ωα11;22pure L11;22 cosð0:5απ Þ þ ωγ11;22pure Mcosð0:5γπ Þ
(11b)
It is interesting to note here that the phase response is a function of the frequency of operation,
which matches with real response of the electrical circuit components [34].
Finally, the more general case of α ≠ β ≠ γ represents fractional-order mutual inductance, but the
phase of the impedance matrix parameters Z11 and Z22 is different. The mutual inductance can be
considered in this case unsymmetrical, which defines a new concept of the asymmetric mutual
inductance. The phase equations of the impedance parameters Z11, Z22 are similar to that of (8a) but
after using the proper fractional orders α, β and the proper inductance values L11, L22, respectively,
for each element.
Circuit simulations for the fractional-order mutual inductance model of Figure 3(b) are presented in
Figure 7 in the case of different fractional orders. The case of (α, β, γ) = (1, 1, 0.7) is presented in
Figure 7(a) where the phase response of the impedance parameters Z12,21 is close to 63o, and the
frequency effect on it is negligible as expected by (8b). Yet, the effect of the frequency on the phase
response of the impedance parameters and Z11,22 is larger as discussed before in (8a). On the other
hand, the simulation of the general case of different fractional orders (α, β, γ) = (0.5, 1.4, 0.4) is
(a)
(b)
Figure 6. (a) The value of γ that satisfy the condition of (10a) at different frequencies at L11,22 = M = 10μH
and (b) pure inductance value versus the fractional-order α at different values of the frequencies at
L11,22 = M = 10μH.
Copyright © 2015 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. (2015)
DOI: 10.1002/cta
FRACTIONAL MUTUAL INDUCTANCE ANALYSIS AND DESIGN
(a)
(b)
Figure 7. Phase response of the impedance parameters for fractional-order T-model at different values of the
fractional orders and at L11 = L22 = M = 10μH (a) α = β = 1 and γ = 0.7 and (b) α = 0.5, γ = 0.4, and β = 1.4.
depicted in Figure 6. Although, the phase value of the impedance parameters Z12,21 can be predicted
from (8a) and equals 36o, the phase response of Z11,22 is frequency dependant. Consequently, the
circuit simulations prove the previous discussion, where the output phase of the impedance matrix
parameters becomes variable and dependent on the order of the fractional-order element.
According to the previous analysis, the fractional-order mutual inductance can be considered the
generalization of the traditional mutual inductance. Also, to model the behavior of the real
transformers, many resistors and capacitors are added to represent the dependence of the phase on
the frequency and the inductors values [41, 42]. Yet in the case of fractional-order mutual
inductance, the addition of these extra modeling elements is not necessary. Because the phase
response depends on the frequency and the inductance values (as shown in (8)) without adding any
extra modeling elements, the FMI is closer to the real transformer behavior than the integer mutual
inductance as mentioned before in the analysis.
3. EQUIVALENT CIRCUIT
In this section, an equivalent circuit for the FMI based on the differential voltage current-controlled
conveyor transconductance amplifier (DVCCCTA) [29] is presented in Figure 8. The FMI is floating
in nature. The equivalent circuit consists of two DVCCCTAs and three grounded fractional-order
capacitors of orders (αc, γc, βc). The port relationships of the DVCCCTA can be written as follows:
2
IY1
3
2
0
0
6I 7 60 0
6 Y2 7 6
6
7 6
6 V X 7 ¼ 6 1 1
6
7 6
6
7 6
4 IZ 5 4 0 0
IO
0
0
0
32
V Y1
3
0
0
0
0
RX
0
1
0
6
7
07
76 V Y 2 7
76
7
6
7
07
76 I X 7
76
7
0 54 V Z 5
0
gm
0
(12)
VO
Figure 8. Fractional-order mutual coupled circuit version of the mutually coupled circuit presented in [29].
DVCCCTA, differential voltage current-controlled conveyor transconductance amplifier.
Copyright © 2015 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. (2015)
DOI: 10.1002/cta
A. SOLTAN, A. G. RADWAN AND A. M. SOLIMAN
where RX is the intrinsic resistance at the X terminal and gm is the transconductance from the Z terminal
to the O terminal of the DVCCCTA. The circuit realization of the DVCCCTA used in this work is
presented in [29]. Routine analysis of the FMCC equivalent circuit illustrated in Figure 8 gives the
impedance matrix of (13), which is similar to the impedance matrix of the general case of the FMI
given in (5). Consequently, the circuit of Figure 8 works as an unsymmetrical coupling fractionalorder mutual inductance. The relations between the parameters of the impedance matrix of (5) and
the components of the equivalent circuit of Figure 8 are tabulated in Table I. The circuit of Figure 8
fulfills different behaviors for the fractional-order mutual inductance at different conditions. A
summary of these responses and their required conditions are summarized in Table II where the
traditional mutual inductance behavior happens at αc = βc = 1 and γc = 0 and also C3 = C1. The phase
response for the impedance matrix parameters in this case is π/2 as shown in Figure 9(a). In
addition, the case of symmetrical FMI behavior is fulfilled at αc = βc, γc = 0 and C3 = C1. Phase
response of the impedance matrix parameters for the case of symmetrical FMI is depicted in
Figure 9(b).
V1
V2
2
6
¼6
4
sαc
C 1 RX 1
C1
þ sαc γc
gm1
gm1 C 2
C
1
sαc γc
gm1 C 2
sβc γc
sβ c
C3
gm2 C 2
C 3 RX 2
C3
þ sβc γc
gm2
gm2 C 2
3
7 I1
7
5 I
2
(13)
Finally, a capacitance effect appears at the coupling points (Z12, Z21) when the fractional-order αc or
βc is less than the fractional-order γc. This property could be used in the impedance matching circuit to
compensate the inductance of the mutual inductance circuit.
4. APPLICATIONS
Now, it is important to prove the reliability of the FMCC. So, the goal of this section is to use the
FMCC in different applications.
Table I. Summary of the relation between the impedance
matrix parameters and the circuit elements.
Parameter
α
β
γ12
γ21
L11
L22
M12
M21
Relation to the equivalent circuit components
αc
βc
αc γc
βc γc
C 1 RX 1
gm1
C 3 RX 2
gm2
C1
gm1 C 2
C3
gm2 C 2
Table II. Required conditions to satisfy the proposed fractional mutual inductance.
Behavior description
Traditional mutual inductance
Symmetrical fractional-order mutual inductance
Unsymmetrical fractional-order mutual inductance
Fractional mutual inductance with capacitive effect
Fractional mutual inductance with resistive effect
Copyright © 2015 John Wiley & Sons, Ltd.
Required condition
αc = βc = 1 and γc = 0 and C1 = C3
αc = βc and γc = 0
αc = βc > γc
αc or βc < γc
αc or βc = γc
Int. J. Circ. Theor. Appl. (2015)
DOI: 10.1002/cta
FRACTIONAL MUTUAL INDUCTANCE ANALYSIS AND DESIGN
(a)
(b)
Figure 9. Circuit simulation of the equivalent circuit of the fractional-order mutual coupled circuit at different cases at C1 = C3 = 10nF and C2 = 1/1330 (a) αc = βc = 1 and γc = 0 and (b) αc = βc = 0.4 and γc = 0.
4.1. Double-tuned filter
Double-tuned filters are considered one of the applications that are mostly based on the mutual
inductance. The conventional double-tuned filter is composed of a series resonance circuit and a
parallel resonance circuit. So, the double-tuned filter can achieve a wider bandwidth band-pass filter
than the traditional filters. So, the goal of this section is to use the fractional-order mutual
inductance to build a fractional-order double-tuned filter. The circuit diagram of the double-tuned
filter is illustrated in Figure 10(a), and its transfer function is given as follows:
T ðsÞ ¼
M
sγc
V out
1
¼ α þ1 R C L1C L α
1
β
þ1
1
c
c
V in
s
þRC s þC L
s c þ R C sβc þ C 1L
P
P P
P
(14)
S S
P
P P
S
S
S S
Although the transfer function in (14) represents a band-pass filter of order αc + βc + 2, the circuit
does not give this response. Yet, the circuit works as a two band-pass filters operating at different
half power frequencies. So, this circuit is called double-tuned circuit. Then, the transfer function can
be represented as follows:
T ðsÞ ¼ sαc þ1
a1 sγ1
a 2 sγ 2
1
α
1
β
þ1
þRC s c þC L
s c þ R 1C sβc þ C 1L
P
P
P P
S
S
(15)
S S
Assuming for simplicity that γ1 = αc, γ2 = βc, and γ1 + γ2 = γc and also a1 a2 ¼ R C ML C L . Then, the
transfer function can be rewritten as follows:
P
T ðsÞ ¼ sαc þ1
a1 sαc
a 2 sβ c
1
α
1
β
þ1
þRC s c þC L
s c þ R 1C sβc þ C 1L
P
P
P P
S
S
P P
S S
(16)
S S
Accordingly, the transfer function of (16) represents two cascaded fractional-order band-pass
filters. So, the critical frequency points (cut-off frequency, maximum and minimum frequency
points, and the right phase frequency) of this filter can be determined using the algorithm
presented in [7–9, 43]. Hence, the critical frequency points become functions of the fractional
orders α, β, which increase the design degree of freedom [7–9, 43]. Actually, the frequency
response of the double-tuned filter based on the FMI has two resonance frequencies as shown in
the numerical analysis of Figure 10(b). Then, the filter bandwidth is increased as expected. In
addition, Figure 10(b) presents the frequency response of the filter using the same element values
but at different fractional orders. It is clear that the filter cut-off frequency changes with the
fractional orders as mentioned before.
Copyright © 2015 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. (2015)
DOI: 10.1002/cta
A. SOLTAN, A. G. RADWAN AND A. M. SOLIMAN
(a)
(b)
(c)
Figure 10. (a) Double-tuned filter using the fractional-order mutual coupled circuit equivalent circuit, (b)
numerical analysis of the double-tuned filter with RP = Rs = 11kΩ and Cs = 20nF, CP = 10nF, and (c) circuit
simulation for the double-tuned filter RP = Rs = 11kΩ and Cs = 20nF, CP = 10nF.
Finally, the circuit simulation of the double-tuned filter based on the FMI is depicted in Figure 10(c)
for the same cases of the numerical analysis. There is a great matching between the numerical analysis
and the circuit simulations of Figure 10(b) and (c), respectively. This matching confirms the reliability
of the FMI.
4.2. Fractional-order mutual coupled circuit in impedance matching
The problem of impedance matching is an important one, which must be addressed in most microwave
designs [44, 45, 31]. One of the most common techniques used in the impedance matching is the
mutual inductance. Hence, impedance transformation using the FMCC is presented here. The
equivalent T-model of the FMCC illustrated in Figure 3(b) is used for the impedance matching
analysis as shown in Figure 11(a).To simplify the analysis, the case of the symmetrical FMI that has
α = β = γ is discussed here. Then, from a simple routine analysis, the load impedance of the circuit
can be calculated from (17).
Copyright © 2015 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. (2015)
DOI: 10.1002/cta
FRACTIONAL MUTUAL INDUCTANCE ANALYSIS AND DESIGN
(a)
(b)
Figure 11. (a) Impedance matching circuit using the fractional-order mutual coupled circuit T-model and (b)
numerical analysis for ZLoad with respect to Zsource and α at L11 = 1mH and L22 = M = 10μH.
Z Load ¼
ωα cosðαπ ÞðL11 L22 þ M ðL22 þ L11 ÞÞ þ Z source cosð0:5απ ÞðL22 þ M Þ
L11 cosð0:5απ Þ þ Z source ωα þ Mcosð0:5απ Þ
(17a)
Z Load ¼
ωα sinðαπ ÞðL11 L22 þ M ðL22 þ L11 ÞÞ þ Z source sinð0:5απ ÞðL22 þ M Þ
ðL11 þ M Þsinð0:5απ Þ
(17b)
For the traditional case α = β = γ = 1, the relation between the input impedance and the load
impedance is as follows:
Z Load
L22 þ M
¼
Z source L11 þ M
(18a)
Z Load Z source ¼ ω2 ðL11 L22 þ M ðL22 þ L11 ÞÞ
(18b)
The relation in (18a) is the same as the well-known relation of the integer-order mutual inductance
[31], which confirms (17). Yet, the value of the resonance frequency in this case is negative as given in
(18b). This means that the system is unstable and requires a compensation capacitor to eliminate the
inductance effect. On the other hand, from (17) for the fractional-order mutual inductance, this
capacitor can be ignored because as mentioned before the FMI can have self-compensation
behavior. Consequently, matching using the fractional-order mutual inductance is simpler and hence
cheaper from the circuit implementation point of view because it requires fewer components.
From (17a), the load impedance is a function of the fractional-order α besides the circuit components
{L11, L22, M, Zin}, which increases the design flexibility. The effect of the input impedance on the
Copyright © 2015 John Wiley & Sons, Ltd.
Int. J. Circ. Theor. Appl. (2015)
DOI: 10.1002/cta
A. SOLTAN, A. G. RADWAN AND A. M. SOLIMAN
output impedance can be minimized at large values of α as shown in Figure 11(b). Then, the system
can be designed for fixed load impedance although the input impedance changes. On the other hand,
for small values of α, the effect of the input impedance on the load impedance is very large as
illustrated in Figure 11(b). For the case of equal orders (α = β = γ) and L11 = 1mH, L22 = M = 10μH, the
system is stable without using a compensation capacitor in the range demonstrated in Figure 11(b).
Consequently, the system can be designed for matching without using the compensation capacitor
but at a specific frequency range. This frequency range could be changed by changing the value of
the fractional order and the circuit component values.
5. CONCLUSION
Fractional-order mutual inductance analysis is discussed. To use the FMI, an equivalent circuit is
presented. It has been found that the phase response of the impedance matrix parameters of the
equivalent circuit can be controlled by the value of the fractional orders as shown in the Advanced
Design System and MATLAB simulations. Different applications based on the proposed FMCC
have been discussed like the double-tuned filter and impedance matching. For the double-tuned
filter, a good matching is found between the numerical analysis and the Advanced Design System
simulations. In addition, for the impedance matching based on the proposed FMCC, the design
equations are derived, and the design degree of freedom is increased because of the increased design
variables.
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