Research Journal of Applied Sciences, Engineering and Technology 4(17): 2988-2994, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: December 22, 2011 Accepted: March 10, 2012 Published: September 01, 2012
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2
1
2
Abstract: Inductors are bulky and costly but, reducing the size of the inductors reduces the quality factor. There is also a fundamental limitation of using inductor that it cannot be suitable for the micro miniature structure and integrated circuits applications. Because of these limitations, inductors cannot be used in most of the analog circuit applications. Hence simulated inductors are used as an alternative. In case the simulated inductor becomes floating, the simulated circuit for replacement of inductor will be different. In this paper the various floating inductors used in the literature are reviewed and some ideas are proposed which make use of linear transformation for converting floating inductor into grounded inductor. Such transformation results in the formation of a new D element called the Frequency Dependent Negative Resistance (FDNR). Also, this paper presents the basic circuits for simulating the Frequency Dependent Negative Resistance and its applications in analog circuits in the place of floating inductor.
Keywords: Analog filters, current conveyor, frequency dependent negative resistor, generalised impedance converter, simulated inductor
INTRODUCTION
Inductor is the one of the most important passive components used in all analog circuits. But there are serious limitations in using it. Some of them are summarized. UMESH KUMAR and SUSHIL KUMAR
SHUKLA have highlighted many disadvantages of physical inductor (Umesh and Sushil, 1989; Roy and
Shail, 1991) and few are listed below.
C The magnetic material forming the core and the conductors forming the windings is placed on the semi-conducting material surface which does not reveal the ferromagnetism. This arrangement results in inductors of very low inductance L and reduced quality factor Q.
C Lessening the size of the inductor shrinks the quality factor
C
The best achievable value of the quality factor with the inductor is 1000 since the large inductors are lossy. But even higher values of 5,000 to 10,000 can be accomplished with the capacitor.
C
At low frequencies, inductors are seldom used because the inductors are unwieldy.
C As the inductors made of ferromagnetic materials which are nonlinear in nature, they create harmonic distortion if ,the amplitude of the signal is large and direct currents are not avoided
C
Inductors have a tendency to act as miniature antennas, radiating as well as picking up electromagnetic waves. This can result in unwanted noise and mixing of signals in circuits containing inductors.
It is relatively simple to reduce the size of the resistor and capacitor but it is impractical to reduce the size of the inductor. There also exists another serious and fundamental limitations that it cannot be fabricated on IC chip where there is a need for the miniature size (Roy and
Shail, 1991). Hence it is not well-suited with the any of today’s IC Technology.
The use of inductors at low frequency also has the following limitations (Umesh and Sushil, 1989). This includes
C
The size and weight of the inductors are large and physically bulky
C The quality factor becomes very low
C Their characteristics are also rather non-ideal
Many methods are mentioned in the literature to have an equivalent circuit for inductor like Riordans’s circuit,
Antoniou’s circuit, current conveyor, CFOA (Current
Feedback Operational Amplifier) etc. This study presents the overview of the design of another inductor simulator circuit namely the Frequency Dependent Negative
Resistance which is widely used in replacing the floating physical inductor and its application in analog circuits like filters, oscillators etc. Experimental result shows that the performance of the FDNR is same as that of the physical inductor over the complete range of frequencies.
METHODOLOGY
Methods of simulating FDNR-theory:
Generalised impedance converter-Method I: The
Impedance converter is an active RC circuit which makes
Corresponding Author: D. Susan, School of EEE, SASTRA University, Thanjavur, Tamil Nadu, India
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Fig. 1: Generalized impedance converter
C3
R4
C1
R2
R5
Res. J. Appl. Sci. Eng. Technol.,(17): 2988-2994, 2012
_
A
1
+
A
Z=
Z
1
Z
3
Z
5
Z
2
Z
4
Z
1
Z
2
Z
3
Z
4
Z
5
+
A
2
_
R3
R4
C1
R2
C5
C3
R4
R1
R2
C5
Fig. 2: Three different ways of forming FDNR from GIC use of two operational amplifiers and passive components namely resistors and capacitors (Sergio, 2007).They are designed to simulate the frequency dependent elements such as inductance for use in the synthesis of active filters. Out of many configurations, the Generalized
Impedance Converter (GIC) (Bhattacharyya et al ., 1974) is the most prominent one used for realizing not only the inductor but also the FDNR. The general circuit of GIC
(Bruton, 1978) is given in the Fig. 1. The impedance of the circuit is given by:
Z
=
Z Z Z
5
Z Z
4
( 1 )
Z
3 =
By choosing the impedances as: Z
1
R
3
, Z
4
= R
4
and Z
5
= 1 / j
T
C
5
gives
= 1/
T
C
1
,Z
2
= R
2
,
Z
=
R
3
(
ω
1
ω
1
)( j C
1 j C
5
)
= −
1
ω 2
D where, D
=
R R C C
5
R
3
Thus the circuit simulated is the FDNR circuit. Since the capacitance generates a voltage proportional to the integral of the current, the FDNR is an element that integrates the current twice. The value of D is adjusted by varying one of the resistances. There are different ways of forming the simulated inductor. Fig. 2 shows three different ways of forming FDNR (Jim and Francis, 1981;
Lech, 1988; Susan and Jayalalitha, 2010a; Bruton, 1978;
Susan and Jayalalitha, 2011a; Susan and Jayalalitha,
2011b; Senani, 1979a, b).
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Res. J. Appl. Sci. Eng. Technol.,(17): 2988-2994, 2012
A
C
1
OA2
R
2
C
3
V
1
I
C
1 y x z w
R
2
C
2
OA2
R
1
V
1
I
R
5
R
4
Fig. 3: Alternate method of forming FDNR
Fig. 4: Symbol of FDNR y x z w
R
C
2
C
1
Fig. 5: The first proposed FDNR simulator
Generalised impedance converter-Method II: An alternate method of realizing the D element is shown in the Fig. 3. The value of the D element is given by:
D
=
C R C R
4
R
5
The symbol of the D element is shown in Fig. 4.
Using single feed back op-amp-Method III: This method of realization makes use of a single Current
Feedback Operational Amplifier (CFOA) as the active element and three or four passive components to realize
FDNR. The Current Feedback Operational Amplifier
(CFOA) (Firat and Kuntman, 2009; Nandi et al ., 1984) is employed as an exact current-mode dynamic in the current mode signal processing circuits (Bruton, 1978).
This is because it offers ample signal bandwidth and
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Fig. 6: The second proposed FDNR simulator linearity far above the ground than the conventional operational a mp lifier configuration. Many implementations of CFOA based circuits have also been developed by different researchers (Bruton, 1969; Jim and
Francis, 1981; Firat and Kuntman, 2009; Nandi et al .,
1984; Lech, 1988; Bhattacharyya et al ., 1974; Sedra and
Smith, 2002). Numerous methods for realizing frequency dependent negative resistors using different types of active elements are available in the literature (Lech, 1988;
Bhattacharyya et al ., 1974; Sedra and Smith, 2002; Susan and Jayalalitha, 2010b). The two circuits proposed for the synthesis of FDNR is shown in Fig. 5 and 6. These circuits are characterized by:
[ ]
= s D eq
⎢
⎡
⎣
+
1
−
1
−
1
+
1
⎥
⎤
⎦
CMOS current feedback op-amp is used at high frequencies for various applications (Ismail and Soliman,
2000).
Realization of FDNR using current conveyor-Method-
IV: A fresh active current conveyor (CC-II) (Senani,
1984; Nandi et al ., 1983; Soliman, 1978; Higashimura and Fukui, 1986; Higashimura and Fukui, 1987)-RC realization of a floating admittance function Y(s) = s 2 D
(Bruton, 1969) corresponding to an ideal FDNR element can be implemented. It has various advantages like
C Employs only two capacitors and a resistor
C
Extremely less active sensitivities
C Single-resistor adjustability of D
C No design limitation involving component matching
C
Implementation of ideal FDNR even with non ideal
CC II devices
Another choice for realizing ideal floating FDNR is to use RC: CR transformation to a floating inductor circuit which makes use of three CCII and one CCI circuit
(Senani, 1982). Symmetrical novel circuits can also be used to implement floating FDNR with less passive elements (Ahmed et al ., 2010).

R
1
C
1
Res. J. Appl. Sci. Eng. Technol.,(17): 2988-2994, 2012
R
2
C
2
R
3 R
3
C
2
R
1
C
1
R
2
(a) (b)
R
1
C
1
C
2
R
2
R
1
R
3
C
1
R
2
C
2
R
3
(c)
Fig. 7: Four different circuits for generating frequency dependent negative resistance
A resonant bridge with frequency dependent negative resistance: Lech Tomawski has introduced four different circuits for the practical applications of FDNR in his paper and is shown in Fig. 7.
Real FDNR input impedance is given by:
Z in
= s E
+ sL
+ r
(d)
Simulated L
V i
R
This contains the inductor L, resistor r and the negative resistance-
T 2 E where E is given by E =
C
1
C
2
R
1
R
2
R
3
. The use of FDNR with the resonant bridge offers new possibilities for measurement of inductance with high losses, as these losses do not damp oscillations.
APPLICATIONS OF FDNR
Synthesis of low pass filter: The simulated L used in analog circuits (Lech, 1988; Bhattacharyya et al ., 1974;
Sedra and Smith, 2002; Susan and Jayalalitha, 2010a) for replacing the physical inductor L should be of grounded inductor or floating inductor. But most of the simulated L is of grounded one. In case, the L is a floating one, FDNR
C
+
_
V
0
Fig. 8: Circuit showing the floating L in the low pass filter
R L
R
-1 L
-
+
V i
+
C
Vo
+
V i
-
+
D
Vo
-
Fig. 9: Circuits for low pass filter with L and with D element is used for the replacement. Now a days, the LC filters are implemented using the simulated inductor. The restriction
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Res. J. Appl. Sci. Eng. Technol.,(17): 2988-2994, 2012
L
R
-1 r
V o
+
V i
-
C
1
R
1
OA2
R
2
OA1
R
3
C
2
Fig. 10: Circuit for lowpass filter using FDNR- Method I
R
-1
L
+
V i
-
C
1
OA2
R
2
C
3
OA2
V o
R
5
R
4
Fig. 11: Circuit for lowpass filter using FDNR- Method II is that, when the simulated L is used for implementing
(say) a low pass filter (Sedra and Smith, 2002), the simulated L becomes floating as shown in Fig. 8. Hence transformation is used and the new element formed is the
FDNR (Susan and Jayalalitha, 2010a). The transformations can also be applied to band pass filter
(Antoniou, 1971).
Each value in the original network in Fig. 8 is divided by j
T
(Bruton, 1978, Massara and Al-Najjar, 1981). This transforms resistances into capacitances, inductances in to resistances and capacitances into D element as shown below:
R/j
T
------->1/ j
T
R -1 (Capacitance of value R -1 ) j
T
L/j
T
------> .L
(Resistance of value L )
(1/j
T
C)/jw------>(
!
1/
T
2 C) (D element of value C)
The transformed network is given in the Fig. 9.
Dividing all the impedances of the network by the same factor yields a transfer function which is same as that of the original one.
12V
8V
4V
0V
Cut off frequency = 100Hz
0V (C 1:2) Frequency
Fig. 12: Response of lowpass filter using FDNR
A low pass filter designed using the D element is shown in Fig. 10. The alternate method of implementing low pass filter using D element is given in Fig. 11. The frequency response of the low pass filter for the cut-off frequency of 100 Hz obtained using the simulation software PSPICE is given in the Fig. 12. The design values of various elements of the circuit are given below.
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Components data: Method-I: f o
= 1 kHz; Q = 5; R
1
= R
2 =
10 nF; r = 1 M
S
R
3
= 31.83 k
S
; C
1
= C
2
=
Components data: Method-II: C
1 k
S
, C
3
= 0.01 µF, R
4
= 0.01 µF, R
= 31.8 Kohm, R
5
2
= 31.8
= 31.8 k
S
, R -1
(equivalent to a capacitor) = 0.1 µF, L (equivalent to R)
= 8.06 k
S
The FDNR circuit is used for different analog circuits
(Nandi et al ., 1983) like switched capacitor filter, oscillators (Senani, 1984), anti aliasing filters etc.
CONCLUSION
FDNR circuits are used for replacing the floating simulated inductor. It can be used for realizing any analog circuits which requires a grounded FDNR (Ashish et al .,
2011). Implementation in integrated circuits is also made possible. The circuits described so far are not suitable for floating FDNR.
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