simulated inductor using gic and its application in the...
TRANSCRIPT
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CHAPTER 2
Simulated inductor using GIC and
its application in the synthesis of
Butterworth Analog filters*
* Partial contents of this Chapter has been published in
1. D.Susan, S.Jayalalitha, “Analog filters using Simulated Inductors”, IEEE
International Conference on Mechanical & Electrical Technology, ICMET
pp.(659 – 662), Sep (10-12), 2010, Singapore. (Scopus Indexed)
2. D.Susan, S.Jayalalitha, “Notch filter using simulated inductor”,
International Journal of Engineering Science and Technology (IJEST),
Vol.3, No.6, pp.(5126-5131), June, 2011.
3. S.Jayalalitha, D.Susan, “Realization of analog filters using simulated
inductor”, National Conference on Power Electronics and Drives
(NCPED’09), pp.(97-105), SRC Kumbakonam, March, 2009
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2. Simulated inductor using GIC and its
application in the synthesis of
Butterworth Analog filter
2.1 Introduction
The use of inductors at low frequency is very much limited because it contains
more number of turns, enlarging the size. It makes it bulky increasing the loss in quality due to
its high internal resistance. This can be substantiated by considering an example. For
designing a low pass filter having the cut-off frequency of = 10 Hz, assuming C = 0.1µF
and using the formula
LC
f2
10 ---------------------------------------------------- (2.1)
the value of L is found to be 2,535.59 H. Such a high value of inductor is difficult to realize
practically. Also increasing the size of the inductor decreases the quality factor as per the
equation for the parallel resonant circuit which is considered for realizing the analog filters
L
RQ
---------------------------------------------------- (2.2)
Thus increase in size of L decreases the quality factor.
Therefore the disadvantages of using inductor at low frequencies are summarized [49] as
The required inductors at low frequencies are bigger in size and heavy
Their characteristics are reasonably non-ideal
Such inductors are impractical to manufacture in monolithic form
It is irreconcilable with any of the present methods for assembling in electronic systems
17
An alternate solution of using GIC (Generalized Impedance Converter) for low
frequency applications is proposed in this research work to eliminate the above mentioned
disadvantages of using inductor for low frequency application. The proposed method makes
use of simulating the inductor using the GIC [50, 51].The inductors simulated have akin
characteristics as that of the original inductor over the wide range of its frequency application.
It is highly sensitive in the desired band [52, 53].
2.2 Inductor simulation circuit
The simulated inductor (LS) also called as simulated L is obtained from the GIC
and consists of the active component namely the operational amplifier and passive component
namely resistors and capacitors. The GIC invented by Antoniou is given in Figure 2.1. The
impedance of the circuit is obtained by analyzing the circuit using the basic assumption of op-
amp [54]. The assumptions made are
The op-amps are ideal
Virtual short circuit appears between the two terminals of op-amp so that the potential
drop across terminals is zero
The current drawn by the two terminals of the op-amp is zero
The impedance of the circuit is obtained by writing the nodal equations at the nodes V1 and V2
and I is given by
1
1
Z
VVI
03
2
2
1
Z
VV
Z
VV
18
00
54
2
Z
V
Z
VV
+_
_ +
VI
Z1 Z2 Z3 Z4
Z5
A1
A2
VV
V1 V2
42
531
ZZ
ZZZZ
I
V
Figure 2.1 Antoniou inductor simulation circuit using GIC
On solving these equation for input impedance with respect to ground gives
42
531
ZZ
ZZZZ -------------------------------------------------------------- (2.6)
By properly selecting the impedances as
5544332211 ,,,, RZRZRZXZRZ C
4
2531
R
CRRsRZ ------------------------------------------------------------- (2.7)
which is equivalent to an inductor. This gives the value of 4
2531
R
CRRRLS . The value of LS
is obtained by properly selecting the resistances and capacitances. If RRRRR 4531
and CC 2 then
2CRLS ---------------------------------------------------------- (2.8)
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In the Equation 2.6 2Z or 4Z can be a capacitor. The simulated inductor LS has the similar
characteristics as that of the original inductor over a widespread range of frequencies. This
inductor realization circuit is used for many analog circuit applications.
2.3 Experimental validation of simulated L
The magnitude of simulated L (LS) is validated experimentally and the value of
simulated L (LS) is compared with its designed values and theoretical values. Table 2.1 shows
the different designed values of simulated L (LS) using the equation 2.6 by properly choosing
the parameters as RZZZ 231 , 4
4
1
sCZ , 55 RZ so that 45CRRLS
The experimental and theoretical values of simulated L (LS) are given in the Table 2.2
and the average error is found to be only-1.95. The comparison is shown graphically in
Figure 2.2 .
Table 2.1 Design values of Simulated L Table 2.2 Comparision between the Theoritical and
Simulated Value of L
C4=1μf,R=1KΩ
R5 L=C4RR5
1K 1H
2K 2H
3K 3H
4K 4H
5K 5H
6K 6H
7K 7H
8K 8H
Theoretical value
of simulated L
Experimental value of
Simulated L
% Error
1H 1H 0
2H 2.21H -10.5
3H 2.947H 1.77
4H 4.03H -0.75
5H 5.05H -1
6H 6.12H -2
7H 7.08H -1.14
8H 8.16H -2
20
Figure 2.2 Comparison of experimentally simulated L with theoretical value of L and error calibration curve
2.4 Analog filters
Frequency selective circuits that pass electrical signals of specified band of
frequencies and attenuate the signals of other frequencies are called analog filters. The oldest
technology for realizing such filters makes use of resistors, inductors and capacitors. The
resulting filters are passive filters which work well at high frequencies that is at radio
frequencies (RF). However, at low frequencies (dc to 100 KHz) the use of inductors is
problematic as the required inductors are physically hefty and bulky due to more number of
turns which in turn adds to the series resistance degrading the inductors performance. It lowers
the Q resulting in higher power dissipation [55]. Their characteristics are pretty non-linear.
Also they are discordant with any of the modern techniques for assembling in electronic
systems. In this chapter, the research highlights the applications of the simulated inductor LS
(simulated L) in filters at low frequencies.
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An ideal filter will have an amplitude response that is unity (or at a fixed
gain) for the frequencies of interest (called the pass band) or else zero in all other instance
(called the stop band). The frequency at which the response changes from pass band to stop
band is referred to as the cutoff frequency. The filter transfer functions T(s) or H(s) is the ratio
of output voltage Vo(s) to the input voltage Vi(s) given by)(
)()(
sV
sVsT
i
o . The filters are
characterized by the some parameters namely pass band frequencies, stop band frequencies,
cut off frequency, quality factor and the damping ratio. The filters designed with the simulated
L in this chapter is the butter worth filter
2.4.1 Butterworth filters
The Butterworth filters give a maximally flat response. The transfer function of
the Nth order filter is given by N
O
pSpSpS
KST
21
where K is a constant
equal to the required gain of the filter. This filter finds wide applications at low frequencies
[56, 57]. The Butterworth filters are constructed from basic LCR resonator circuit. But the use
of L causes many disadvantages as it has been mentioned already. Hence the L is replaced by
the simulated L (LS) in the basic LCR resonator and is used for constructing different types of
Butterworth filters.
2.4.2 Basic LCR resonator
The basic LCR resonator of second order is shown in Figure 2.3 can be used to
realize different filter types
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R C
Vi
x y z
L
Figure 2.3 Basic LCR resonator circuit
The resonator is excited with a current source connected in parallel. The response
function in terms of impedance is given by
LCRCss
Cs
YI
Vi
1)1(
12
---------------------------------------------- (2.9)
Equating the denominator to the standard form 2
00
2 )/( Qss leads to
LC
12
0 ----------------------------------------------------------------------- (2.10)
CRQ
10
----------------------------------------------------------------------- (2.11)
Which gives
LC
10 and CRQ 0 ----------------------------------------------- (2.12)
2.4.3 Realization of Butterworth filters using LCR resonator
circuit
Using the basic LCR resonator circuit, the nodes x, y and z can be connected in
different ways to obtain all types of filters. The Table 2.3 shows how different types of filters
are realized.
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Table 2.3. Formation of different types of filters using LCR resonator
2.4.4 Realization of low pass filter
The low pass filter obtained from the basic LCR circuit is shown in Figure 2.4
(a). Figure 2.4 (b) shows the low pass filter with the actual L replaced by the Simulated
Inductor (LS). The complete circuit of the low pass filter with simulated inductor (LS) is shown
in Figure 2.5 and the PSPICE simulation circuit with simulated inductor (LS) is given in the
Appendix A.
Simulated L
Vi R CV0
+
_
Figure 2.4 a) Low pass filter Figure 2.4 b) Low pass filter with simulated L
Node x Node z Node y Resultant filter
Vi Ground Ground LPF
Ground Vi Ground HPF
Ground Ground Vi BPF
Vi Vi Ground Notch at ωo
Vi
L
R C
24
R1 R2 R3 R5C4
A1
A2
R6C6
K Vo
Vi
Simulated L
Figure 2.5 Complete circuit of the low pass filter with simulated L
The transfer function of the low pass filter is given by
53164
2
66
2
53164
2
1)(
RRRCC
R
RCss
RRRCCKR
sT
---------------------------------------------------- (2.13)
where K is the dc gain of the filter.
2.4. 5 Design of low pass filter
The cut off frequency for LPF is given by LC
f o2
1
For Hzfo 100 and fC 1 ,the value of L is 2.536H
Simulated inductor 2
4531
R
CRRRLS
If RRRRR 5321 and CCC 64 then 2CRLS
25
KC
LR S 592.1
KC
QR
O
126.16
6
The frequency response of the low pass filter with cut off frequency 100Hz is
shown in Figure 2.6. The amplitude can be also normalized for the maximum gain of 0 dB.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz
V(U3:OUT)
0V
5V
10VA
m
p
l
i
t
u
d
e
Cut off frequency = 100Hz
Stop bandPass band
Figure 2.6 Frequency response of the low pass filter with simulated L
2.4.6 Realization of high pass filter
In the high pass filter shown in Figure 2.7 (a), L is replaced by the Simulated Inductor
(LS) and is shown in Figure 2.7(b). The complete circuit of the high pass filter with simulated
inductor (LS) is given in Figure 2.8. The PSPICE simulation circuit with simulated inductor
(LS) is given in the Appendix A.
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Vi
R
C
V0
+
_Sim
ula
ted
L
Figure 2.7 a)High pass filter Figure 2.7 b)High pass filter with simulated L
R1 R2 R3 R5C4
A1
A2
R6
K Vo
Vi
Simulated L
C6
Figure 2.8 Complete circuit of High pass filter with simulated L
The transfer function of the high pass filter is given by
LR
C
Vi
27
53164
2
66
2
2
1)(
RRRCC
R
RCss
KssT
------------------------------------------------ (2.14)
where K is the high frequency gain of the filter.
2.4.7 Design of high pass filter
The cut off frequency for HPF is given by LC
f o2
1
For Hzfo 100 and fC 1 ,the value of L is 2.536H
Simulated inductor 2
4531
R
CRRRLS ,If RRRRR 5321 and CCC 64 then
2CRLS , KC
LR S 592.1 a K
C
QR
O
126.16
6
The frequency response of the high pass filter with cut off frequency 100Hz is shown in Figure
2.9.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz
V(U3:OUT)
0V
5V
10VA
m
p
l
i
t
u
d
eCut off frequency = 100 Hz
Stop band Pass band
Figure 2.9 Frequency response of high pass filter with simulated L
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2.4.8 Realization of band pass filter
The LCR circuit is modified to obtain a band pass filter as shown in Figure
2.10 a). Figure 2.10 b) shows the L replaced by simulated L (LS). The complete circuit of the
band pass filter with simulated inductor (LS) is shown in the Figure 2.11. The PSPICE
simulation circuit with simulated inductor (LS) is given in the Appendix A.
Vi
R
CV0
+
_Sim
ulat
edL
Figure 2.10 a)Band pass filter Figure 2.10 b)Band pass filter with simulated L
R1 R2 R3 R5C4
A1
A2
K Vo
Vi
Simulated L
R6
C6
Figure 2.11 Complete circuit of band pass filter with simulated L
R
C LVi
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The transfer function of the band pass filter is given by
53164
2
66
2
66
1)(
RRRCC
R
RCss
RCKs
sT
----------------------------------------------- (2.15)
where K is the centre frequency gain of the filter.
2.4.9 Design of band pass filter
The cut off frequency for BPF is given by LC
f o2
1
For Hzfo 100 and fC 1 ,the value of L is 2.536H
Simulated inductor2
4531
R
CRRRLS , If RRRRR 5321 and CCC 64 then
2CRLS , KC
LR S 592.1 K
C
QR
O
126.16
6
The frequency response of the band pass filter for cut off frequency 100Hz is shown in Figure
2.12
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz
V(U3:OUT)
0V
5V
10VA
m
p
l
i
t
u
d
e
Cut off frequency = 100 Hz
Figure 2.12 Frequency response of Band pass filter with simulated L
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2.4.10 Realization of notch filter
Band stop filter(BSF) also called as band reject filter(BRF) or band elimination
filter(BEF) can be classified as narrow band reject filter(NBRF) and wide band reject
filter(WBRF).The notch filter is a narrow band reject filter which notches out a particular
frequency. The notch filter can be obtained in the same manner as shown in Figure 2.13 a).
The L replaced by simulated L is shown in Figure 2.13 b).The Figure 2.14 gives the complete
circuit of the notch filter with simulated inductor (LS). The PSPICE simulation circuit with
simulated inductor is given in the Appendix A.
Simulated L
Vi
RV0
+
_
C
Figure 2.13 a) Notch filter Figure 2.13 b) Notch filter with simulated L
where K is the low and high frequency gain of the filter.
The transfer function of the notch filter is given by
53164
2
66
2
53164
22
1
)]([
)(
RRRCC
R
RCss
RRRCCR
sK
sT
------------------------------------------------- (2.16)
The frequency response of notch filter for cut off frequency 100Hz is shown in Figure.2.15.
L
RCVi
31
R1 R2 R3 R5C4
A1
A2
K Vo
Vi
Simulated L
C6
R6
Figure 2.14 Complete circuit of notch filter with simulated L
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz
V(U3:OUT)
0V
5V
10VA
m
p
l
i
t
u
d
e
Notch frequency = 100 Hz
Figure 2.15 Frequency response of notch filter with simulated L
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2.5 Applications of Butterworth filter at low frequencies
The various types of Butterworth filters implemented with simulated inductor in this
chapter find many applications at low frequencies. Some of them are
1. Extracting the low frequency ECG signal from noise signal using low pass filter [58]
2. Limiting the bandwidth of a signal before Analog to Digital Conversion (digital sampling)
to obey the Sampling Theorem using low pass filter
3. Audio speech signal processing
4. Removal of hum at 50 Hz using notch filter [59]
5. Very low frequency(VLF) Receivers
6. Retrieval of low frequency sensor signal in submarines or under water applications and
many other low frequency applications depending on the requirement of filters.
2.6 Conclusion
The design of various Butterworth filters using basic LC filters is impractical to
realize at low frequencies as the size of inductors becomes hulking with more number of turns.
So the use of GIC for realizing various Butterworth filters are explained which eliminates the
use of such bulky inductor. These types of filters are used at low frequencies where it is
required to have maximally flat response. The same concept of using simulated L (LS) can be
applied to any higher order filters to get response closer to ideal response.