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School of Engineering
Honors project Switched capacitor filter design for mixed signal applications
-12V
V11Vac0Vdc
V2
TD = 0
TF = 0PW = 500nPER = 1u
V1 = 0
TR = 0
V2 = 5V3
TD = 500n
TF = 0PW = 500nPER = 1u
V1 = 0
TR = 0
V2 = 5
0
0 0
-12V
-12V
J1BC264A
J2BC264A
J3BC264A
J4BC264A
J5BC264A
J6BC264A
J7BC264A
J8BC264A
J9BC264A
J10BC264A
J11BC264A
J12BC264A
J13BC264A
J14BC264A
J15BC264A
J16BC264A
J17BC264A
J18BC264A
J19BC264A
J20BC264A
C1
10n
C2
10n
C3
50p C450p
C5
50pC650p
C7
50p C850p
C9
31.25pC1031.25p
C11
31.25pC1231.25p
C13
50pC1450p
C15
45.5pC1645.5p
C17
50pC1850p
C19
50pC2050p
C21
50pC2250p
Q1
0 0
0
Q1 0
0
0
0
0
Q1
0
0 0
0
0
Q1
Q1
0
Q1
Q1
Q1
Q2
Q1
Q1 Q1
Q2
Q2
Q2 Q2
V412Vdc
V512Vdc
0
+12V
-12VQ2
Q2
+3
-2
V+4
V-11
OUT1
U1A
TL084
Q2
+5
-6
V+4
V-11
OUT7
U1B
TL084
+10
-9
V+4
V-11
OUT8
U1C
TL084
Q2
VD
B
VD
B
VD
B
VD
B
+12
-13
V+4
V-11
OUT 14
U1D
TL084
Q2 Q2
+12V
+12V
+12V
+12V
-12V
Switch Capacitor Filter Parallel and SeriesLow-passHigh-passBand-passBand-reject-3dB-52dB
1000 10000 100000 1000000 10000000
110
100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
frequency
dB
By
Klaus Jørgensen
Napier No. 04007824
Electronic & Communication Engineering
1 May 2006
Supervisor Dr. Mohammad Y. Sharif
Page 2 of 46
Abstract
This project report presents the theory of Switch Capacitors (SC) technology and its
advantages, disadvantages and different implementations methods. The report provides a
short review, of three State Variable (SV) SC filters from Maxim, National Semiconductors
and Texas Instruments. The guidelines on how to construct a SV filter with Low-pass,
High-pass, Band-pass and Band-reject filter response are provided. Simulations have
been carried out to clarify the theory.
A SV filter has been implemented using SC method. Simulations have been carried out
using OrCAD software development tool and the results have been compared with filters
from Maxim, National Semiconductors and Texas Instruments.
Page 3 of 46
Table of Contents
1. Introduction......................................................................................................................4 2. Theory .............................................................................................................................5
2.1 Metal Oxide Semiconductor Capacitors .....................................................................5 2.2 Parasitic Capacitors ...................................................................................................6 2.3 Metal Oxide Semiconductor Field Effect Transistors Switches ..................................7 2.4 Resistor by Switch Capacitors ...................................................................................8
2.4.1 Resistor by Switch Capacitors sub conclusion ....................................................8 2.5 Switch Capacitor setup ..............................................................................................9
2.5.1 Switch Capacitor setup sub conclusion .............................................................10 3. Applications ...................................................................................................................11
3.1 Texas Instruments UAF42 .......................................................................................11 3.2 Maxim MAX7490 and MAX7491 ..............................................................................12 3.3 National Semiconductor LMF100.............................................................................13
4.2.1 Simulations sub conclusion ...............................................................................19 5. Switch Capacitor Filter...................................................................................................20
5.1 Test of different programs ........................................................................................22 5.1.1 Sub Conclusion .................................................................................................23
5.2 Switch Capacitor Filter parallel & Setup of the PSpice model ..................................24 5.3 Switch Capacitor Filter Series ..................................................................................26 5.4 Switch Capacitor Filter Parallel and Series ..............................................................27
6. State Variable Filter with Switch Capacitor ....................................................................33 6.1 State Variable Filter with Parallel or Series Switch Capacitor ..................................33 6.2 State Variable Filter with Parallel and Series Switch Capacitor ...............................37
10.1 Project Proposal.....................................................................................................41 10.2 Time Plans .............................................................................................................42 10.3 Derivation of equation ............................................................................................43 10.4 Vpulse in OrCAD....................................................................................................44 10.5 State Variable Filter with Parallel or Series Switch Capacitor ................................45 10.6 State Variable Filter with Parallel and Series Switch Capacitor .............................46
Page 4 of 46
1. Introduction
A SC filter used in an Integrated Circuit (IC) product, and usually implemented by using
Metal Oxide Semiconductor Field Effect Transistors (MOSFET) and capacitors to simulate
the behavior of a resistor, because the resistors are expensive and not easily controlled,
this means that a circuit can be built without the use of resistors, and this is very useful,
many places in the industry today, because that there is build a lot of IC today and a
resistor use a lot of space in a IC. A MOSFET or a capacitor do not use that much room,
also a resistor in a IC is very dependent of the temperature is very stable or else the value
of the resistor will change, the capacitor value in a IC is much more reliable and the value
of a capacitor can be set very accurately, typically around 0,1% accurate and it can be any
value that is required, mostly in a range from 0,1pF to 100pF, and are generally less costly
than resistors. The sampling clock frequency (fS) in a SC filter usually has to be 50 to 100
times bigger then the cut-off frequency at -3dB (fC) to minimize the effects of time-sampling
and charge-sharing, but the frequency area typically in the range from 100kHz to 2MHz,
therefore the SC filter is particularly useful in the voice and audio frequency area, because
that the clock frequency of the switches must be 50 to 100 times bigger then the signal
frequency, and therefore it is very useful in Digital Signal Processor (DSP) technology
which is often used for voice and audio applications. Figure 1.1 shows how a resistor (R) is
replaced with a capacitor (CS) and a switch MOSFET [1].
Figure 1.1 [1]
C
R
0
Cs
0
C
R
0
Page 5 of 46
2. Theory
In this chapter the general theory of a SC resistor and the basic ways to construct and
calculate a SC resistor is described along with some of the problems there is by using a
SC resistor.
2.1 Metal Oxide Semiconductor Capacitors
Capacitance voltage measurements of Metal Oxide Semiconductor (MOS) capacitors
provide a wealth of information about the structure, which is of direct interest when one
evaluates an MOS process. Since the MOS structure is simple to fabricate, the technique
is widely used.
To understand capacitance-voltage measurements one must first be familiar with the
frequency dependence of the measurement. This frequency d ependence occurs primarily
in inversion since a certain time is needed to generate the minority carriers in the inversion
layer. Thermal equilibrium is therefore not immediately obtained.
The low frequency or quasi-static measurement maintains thermal equilibrium at all
times. This capacitance is the ratio of the change in charge to the change in gate voltage,
measured while the capacitor is in equilibrium. A typical measurement is performed with
an electrometer, which measures the charge added per unit time as one slowly varies the
applied gate voltage.
The high frequency capacitance is obtained from a small-signal capacitance
measurement at high frequency. The bias voltage on the gate is varied slowly to obtain the
capacitance versus voltage. Under such conditions, one finds that the charge in the
inversion layer does not change from the equilibrium value at the applied dc voltage. The
high frequency capacitance therefore reflects only the charge variation in the depletion
layer and the (rather small) movement of the inversion layer charge [2].
Figure 2.1 [2]
Page 6 of 46
2.2 Parasitic Capacitors
In MOS technology the capacitor CR has unfortunately some disadvantage, and these are
called parasitic capacitances, these appears because of the closeness between the
connecting terminals, and the small dimensions of the device layers. The capacitor Cp
appears from the capacitances mainly between the terminal connecting metal (a, b), and
the substrate (c). The value of this capacitor is usually around 1% of CR. The capacitor Cb
appears from the capacitances between bottom poly-silicon and the substrate (c), this has
usually a value of 10% of the CR [1].
Figure 2.2 [1]
Page 7 of 46
2.3 Metal Oxide Semiconductor Field Effect Transistors Switches
The most basic element in the design of a large scale IC is the transistor. The transistors
are made of semiconductor layers, usually a slice, or wafer, from a single crystal of silicon,
a layer of silicon dioxide and a layer of metal. These layers are patterned in a manner
which permits transistors to be formed in the semiconductor material. Silicon dioxide is a
very good insulator, so a very thin layer, typically only a few hundred molecules thick, is
required. The MOSFET transistor has three areas, labeled the source, the gate and the
drain. The area labeled as the gate region is actually a part of the original substrate
material, which is a single crystal of semiconductor material witch usually are silicon, a thin
insulating layer, usually silicon dioxide, and an upper metal layer. Electrical charge, or
current, can flow from the source to the drain depending on the charge applied to the gate
region. The semiconductor material in the source and drain region are a different type of
material than in the region under the gate, so an NPN or PNP type structure exists
between the source and drain region of a MOSFET. The source and drain regions are
quite similar, and are labeled depending on to what they are connected. The source is the
terminal, or node, which acts as the source of charge carriers; charge carriers leave the
source and travel to the drain. In the case of an N channel MOSFET, the source is the
more negative of the terminals; in the case of a P channel device, it is the more positive of
the terminals. The area under the gate oxide is called the channel. Figure 2.3 shows a
simplified diagram of a MOSFET transistor [1].
Figure 2.3 [1]
Page 8 of 46
2.4 Resistor by Switch Capacitors
In figure 2.4 there is shown a simplified version, on how a capacitor can work as a resistor.
By charting and recharges the capacitor, by using a switch, when CS is connected to
terminal 1, the capacitor is being charged by the voltage sours E1, and when the switch is
connected to terminal 2 the capacitor is recharging on to the voltage sours E2 [1].
SS Cf1
IE2-E1R
×== (2.1)
In practice the switch in figure 2.4 is replaced with two MOSFET, the gates of the
MOSFETs are controlled by two clock pulses that doesn’t overlap each outer, as shown in
figure 2.5. In order to switch the MOSFET on, the gate needs to be set “high” (a few volts),
and in the off stage the gate of the MOSFET is “low” (0 volts).
Figure 2.4 [1]
Figure 2.5 [1]
2.4.1 Resistor by Switch Capacitors sub conclusion
One of reason construct a switch capacitor resistor is because it requires a very small
silicon area to create a very large resistor value, actually the silicon area decreases as the
resistor value increases, if implement a resistant of 10MΩ, a good value of the capacitor
will be in the range of 1pF to 10pF and a sample frequency around 100kHz [1].
10MΩ100k1p1
fC1R
SS
=×
⇒×
= (2.2)
A capacitor of 1pF requires a silicon area around 0.01mm2, where a resistor at 10MΩ
requires a silicone area there is at least 100 times bigger.
Cs
Cs
Page 9 of 46
2.5 Switch Capacitor setup
There are six different ways to construct a SC, to make it acts as a resistor, three which
use two switches and three which use four switches.
In the SC parallel setup (figure 2.6) the CS
is en parallel with C on the output, and the
size of the equivalent resistor is calculated
in equation 2.3 [1].
SS fC1R×
= (2.3)
Figure 2.6
In the SC series setup (figure 2.7) Cs is
placed in series between the MOSFET A
gate and source, the equation for the size
of the equivalent resistor is the same as
the parallel setup in equation 2.3.
Figure 2.7
The setup of the SC parallel and series
(figure 2.8) is there a capacitor in series
(Cs1) and one in parallel (CS2) these two
capacitors make on equivalent resistor.
( ) SSS f2C1C1R
×+= (2.4)
Figure 2.8
The setup of the SC bilinar (figure 2.9) the
CS is placed between 4 MOSFETs this
means that there always will be two
MOSFETs ON at the same time (A, D)
and (B, C).
SS fC41R××
= (2.5)
Figure 2.9
Input
Cs628p
C10n
0 0
Mos A Mos B
Q1 Q2
Mos A Mos B
Cs
628p
C10n
0
Q2Q1
Input
Mos A Mos B
Cs1
528p
Cs2100p
C10n
0
0
Q2Q1Input
Mos A Mos B
Mos C Mos D
Cs157p
C10n
0
Q2
Q2
Q1
Q1
Input
Page 10 of 46
In the negative trans-resistance
realization SC setup (figure 2.10) uses
four MOSFETs and the CS is placed in
series. When Q1 is high “1” the CS is
being charts and when Q2 is high “1” CS
is discharging, the equation for CS is the
same as in equation 2.3. Figure 2.10
In the positive trans-resistance realization
SC setup (figure 2.11) uses four
MOSFETs and the CS is placed in series.
When Q2 is high “1” CS transfer the
current from the input on to the capacitor
at the output. When Q1 is high “1” CS is
connected to ground and it is discharging.
The equation for CS is the same as in
equation 2.3. Figure 2.11
2.5.1 Switch Capacitor setup sub conclusion
The SC setups there uses two MOSFETs, has its advantage and disadvantages as well
dos the SC setups there uses four MOSFETs. The advantages of only using two
MOSFETs is that it dose not use that much space in the IC, one of the disadvantage is
that is dos not cancel out the parasitic capacitances there will be in an IC. One of the
advantages of using four MOSFETs is, that is eliminating the parasitic capacitances there
will be in an IC. The disadvantage is that this will take up more space in an IC, which can
be a problem in some cases.
C10n
0
Mos A Mos B
Mos DMos C
Cs
628p
0 0
Q2
Q2
Q1
Q1
Input
Mos A Mos B
Mos C Mos D
0 0
Cs
628p
C10n
0
Input
Q2 Q2
Q1 Q1
Page 11 of 46
3. Applications
There are several electronic companies’s who creates good SV filters in IC, and some of
them are being investigated in this chapter. Some of the company’s who manufacture SV
filters are National Semiconductor, Maxim, Analog, Texas Instruments and Burr-Brown.
3.1 Texas Instruments UAF42
The UAF42 IC is a universal active filter from Texas Instruments, which can be setup in
many different ways with Low-pass, High-pass and Band-pass filters outputs. It uses a
classical state-variable analog architecture with an inverting amplifier and two integrators.
The last opamp in figure 3.1 can be used to make the Band-reject filter with a summing of
the Low-pass and High-pass outputs. The integrators include on-chip 1nF capacitors
trimmed to 0.5%. This solves one of the most difficult problems of active filter design
obtaining tight tolerance, Low-loss capacitors, and the slew rate of the UAF42 is only
10V/ms. the open-loop voltage gain is typical 126dB at a supply voltage of ±10V and a RL
at 2kΩ, the maximum value of Q is 400, the gain bandwidth is 4MHz [3].
Figure 3.1 [3]
Figure 3.2 shows a SV filter with a Low-pass, High-pass and Band-pass filter outputs, the
external resistors can be set by the user to what the demands are for the SV filter, RF1 and
RF2 determine the centre frequency, RG determine the gain of the SV filter, RQ determine
the Q value of the SV filter.
Figure 3.2 [3]
Page 12 of 46
3.2 Maxim MAX7490 and MAX7491
The MAX7490/MAX7491 IC’s from Maxim consist of two equal 2nd-order SC building
blocks, with Rail-to-Rail output. Each of the two filter sections, together with two to four
external resistors, can produce a standard 2nd-order filter functions such as Low-pass,
High-pass, Band-pass, and Band-reject. Three of these functions are simultaneously
available. Fourth-order filters can be obtained by cascading the two 2nd-order filter
sections. In the same way a higher order filters can easily be constructed by cascading
several MAX7490/MAX7491s. There are two ways to make the clock for the IC’s available.
One is an internal clock build in the IC with the use of an external capacitor to determine
the clock frequency, or external clocking for tighter cutoff frequency control. Sampling is
done at twice the clock frequency, further separating the cutoff frequency and Nyquist
frequency. The MAX7490/MAX7491 has an internal rail splitter that establishes an
accurate common voltage needed for single-supply operation. The MAX7490 operates
from a single +5V supply and the MAX7491 operates from a single +3V supply. Both
devices feature a low-power shutdown mode which only use 1μA supply current, and can
operate with a center frequency from 10Hz up to 40kHz, they also has a very high
accuracy for the Q value which is ±0.2% and the Clock-to-Centre Frequency Error is only
±0.2%. Figure 3.3 shows a block diagram of the MAX7490/MAX7491 IC. Figure 3.4 shows
a 2nd-Order state variable filter providing High-pass, Band-pass, and Low-pass outputs
[4].
Figure 3.3 [4]
Figure 3.4 [4]
Page 13 of 46
3.3 National Semiconductor LMF100
The LMF100 IC from National Semiconductor contains two independent SV filters with
high performance SC. With an external clock and 2 to 4 resistors, a wide range of second-
order and first-order filtering functions can be carried out by each filter block. Each block
has 3 outputs. One output can be setup to perform a High-pass, or Band-reject filter
function. The other two filter outputs makes a Band-pass and Low-pass filter functions.
The center frequency of each filter function is setup by using an external clock or a
combination of a clock and resistor ratio. A higher order of the filters can be implemented
by cascading more LMF100 IC’s, and all the classical filters, such as Butterworth, Bessel,
Elliptic, and Chebyshev, can be realized. The LMF100 IC has a very low offset, a wide
supply range from ±4V to ±15 but it can also operate with a single supply from 4V to 15V.
It can operate up to 100 kHz, the offset voltage is typically from ±5 mV to ±15 mV is hat a
very low crosstalk witch is in the range of −60dB the clock to center frequency ratio
accuracy is typically in the range of ±0.2%. Figure 3.5 shows a block diagram of the
LMF100 IC. Figure 3.6 shows LMF100 IC setup for a SV filter with a High-pass, Band-
pass, Low-pass and Band-reject filter with the use of an external operational amplifier [5].
Figure 3.5 [5]
Figure 3.6 [5]
Page 14 of 46
4. State Variable Filter
A SV filter is a filter there contains all filter types implanted in to one circuit, i.e. Low-pass,
High-pass, Band-pass and Band-reject filter in one circuit (figure 4.1), the outputs of the
filters is accessible at the same time. To create a SV filter circuit there are required four
operational amplifiers, two of the amplifiers performs as summing function, the two outer
performs as integrators. The passive component values is chosen by the designer, but
must be consistent with the amplifier operating range and input signal [6, 7].
4.1 Calculations
R3R2R1 == (4.1)
R5R4 = (4.2)
R9R8 = (4.3)
C2C1= (4.4)
R1R2Gain Lowpass = (4.5)
R1R3 Gain Highpass = (4.6)
C2C1R5R4R2R3
π21fC ××××
××
= (4.7)
⎟⎠⎞
⎜⎝⎛ ++××
+=
31
21
R11R7R1
R7R6 Gain Bandpass
RR
(4.8)
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
××××
×⎟⎠⎞
⎜⎝⎛ ++
×+
=C2R5R3R2
C1R4
R31
R21
R11
1R7
R7R6Q (4.9)
To calculate the value of R4 and R5 the program MathCAD where used to drive the
formula for R4 and R5, R5 is typed as R4 in equation 4.10, so MathCAD can drive a
formula for R4 shown in equation 4.11.
fc12π
R3R2 R4⋅ R4⋅ C1⋅ C2⋅
⋅
R4
12 π R2 C1 C2⋅⋅⋅⋅
R2 C1 C2 R3⋅⋅⋅( )1
2
fc⋅
1−2 π R2 C1 C2⋅⋅⋅⋅
R2 C1 C2 R3⋅⋅⋅( )1
2
fc⋅
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
:=
(4.10)
(4.11)
Page 15 of 46
To be able to calculate the values for the resistors in the SV filter some of them have to be
chosen before it is possible [6, 7].
Ω=== 10kR3R2R1 (4.12)
10kΩR7 = (4.13)
Ω== 10kR9R8 (4.14)
1nFC2C1 == (4.15)
vv1Gain = (4.16)
0.707Q = (4.17)
10kHzfC = (4.18)
When this values is been chosen it is possible to calculate the value of R4 and R5
( )
( )15.9kΩ
10k1n1n10k10k
1n1n10kπ21R4
fC2C1R3R2
C2C1R2π21R4
C
=⎟⎟⎠
⎞⎜⎜⎝
⎛ ××××⎟⎠⎞
⎜⎝⎛
××××=
⇒⎟⎟⎠
⎞⎜⎜⎝
⎛ ××××⎟⎠⎞
⎜⎝⎛
××××=
(4.19)
The relations between R6 and R7 defines the value of Q, the bigger R6 is then R7 the
bigger the value of Q will be, the value of R7 and Q has been chosen so now the value of
R6 can be calculated (equation 4.20). In appendix 10.3 the whole derivation of equation
4.20 is shown.
( )
( ) 11.21kΩ10k
1n15.9k10k10k1n15.9k
10k1
10k1
10k1
1
10k0.707R6
R7
C2R5R3R2C1R4
R31
R21
R11
1R7QR6
C2R5R3R2C1R4
R31
R21
R11
1R7
R7R6Q
=−
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
××××
×⎟⎠⎞
⎜⎝⎛ ++
×=
−
⎟⎠⎞
⎜⎝⎛
××××
×⎟⎠⎞
⎜⎝⎛ ++
×=⇒
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
××××
×⎟⎠⎞
⎜⎝⎛ ++
×+
=
(4.20)
R10 is chosen to 10kΩ like R8 and R9, because the gain is chosen to be 1v/v. The signals
from the High-pass and Low-pass filter is added together to produce the Band-reject
output, and the gain of the Band-reject filter is 1v/v. Therefore all the resistors of the Band-
reject filter have to bee the same value (R8, R9 and R10).
Page 16 of 46
4.2 Simulations
Figure 4.1 shows the SV filter circuit with all four types of filters, High-pass (U1A), Low-
pass (U1C), Band-pass (U1B) and Band-reject (U1D), and the output signals from the
filters is shown in figure 4.2 U1A and U1D performs as summing function, and U1B and
U1C performs as integrators. One of the advantages of a SV filter is that by changing the
value of only a few components the parameters for the whole circuit is changed. If the
value of R1 is changed the gain is changed, the ratio between R4 and R5 determined
where the fC should bee, and the ratio between R6 and R7 specifics the value of Q.
+3
-2
V+4
V-11
OUT1
U1A
TL084
+5
-6
V+4
V-
11
OUT7
U1B
TL084
+10
-9
V+4
V-
11
OUT 8
U1C
TL084
C1
1n C2
1nR1
10k
R2
10k
R3
10k
R4
16k R5
16k
R6
11k
R710k
VD
B
VD
B
VD
B
VDB
0
00
V112V
V212V0
+12V
-12V
+12V
+12V
+12V
-12V
-12V
-12V
V31Vac0Vdc
0
+12
-13
V+4
V-
11
OUT 14
U1D
TL084
R810k
R9
10k
R10
10k
0 +12V
-12V
Figure 4.1 [7]
State Variable Filter
Low-passHigh-passBand-passBand-reject-3db-40dB
1000 10000 100000 1000000 10000000
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
frequency
dB
Figure 4.2
Page 17 of 46
R1 controls the gain for all four filters but the
filters losses some of there sharpness. In figure
4.3 R1 has been change to 1kΩ, which give a
gain of 20dB, which is equal to a gain 10 times.
102020Log
20dBLogAv 1010 =⇒⎟
⎠⎞
⎜⎝⎛= (4.21)
Figure 4.3
R2 controls the gain of the Low-pass filter, in
figure 4.4 R2 is changes to 100kΩ which gives
a gain of 20dB which is equal to 10 times
(equation 4.21)
Figure 4.4
R3 controls the gain of the High-pass filter, in
figure 4.5 R3 is changes to 100kΩ which gives
a gain of 20dB which is equal to 10 times
(equation 4.21)
Figure 4.5
R10 controls the gain of the Band-reject filter,
in figure 4.6 R10 is changes to 100kΩ which
gives a gain of 20dB which is equal to 10 times
(equation 4.21)
Figure 4.6
State Variable Filter (Low-pass gain of 20dB)Low-passHigh-passBand-passBand-reject
1000 10000 100000 1000000 10000000
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
frequency
dB
State Variable Filter (All gain of 20dB)Low-passHigh-passBand-passBand-reject
1000 10000 100000 1000000 10000000
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
frequency
dB
State Variable Filter (High-pass gain of 20dB)Low-passHigh-passBand-passBand-reject
1000 10000 100000 1000000 10000000
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
frequency
dB
State Variable Filter (Band-pass gain of 20dB)Low-passHigh-passBand-passBand-reject
1000 10000 100000 1000000 10000000
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
frequency
dB
Page 18 of 46
The relationship between R6 and R7
determinant the Q value for all four filters, in
figure 4.7 R7 has been changes to 1kΩ, this
gives a Q of 4, see equation 4.22 [7].
41n16k10k10k
1n16k
10k1
10k1
10k1
11k
1k11kQ =⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
××××
×⎟⎠⎞
⎜⎝⎛ ++
×+
=
(4.22)
Figure 4.7
R4 and R5 have an equal influence on the fC
frequency, in figure 4.8 R4 has been changes
to 6kΩ which gives an fC of 16,24kHz if
equation 4.7 on page 14 is used.
16.24kHz1n1n6k16k10k
10kπ2
1fC =××××
××
=
(4.23)
Figure 4.8
R4 and R5 have an equal influence on the fC
frequency, in figure 9 R4 has been changes to
26kΩ which gives an fC of 7,8kHz if equation
4.7 on page 14 is used.
7,8kHz1n1n16k6k210k
10kπ2
1fC =××××
××
=
(4.24)
Figure 4.9
R4 and R5 have an equal influence on the fC
frequency, in figure 10 R5 has been changes
to 6kΩ which gives an fC of 16,24kHz if
equation 4.7 on page 14 is used.
16.24kHz1n1n6k6k110k
10kπ2
1fC =××××
××
=
(4.25)
Figure 4.10
State Variable Filter (All Q of 4)Low-passHigh-passBand-passBand-reject
1000 10000 100000 1000000 10000000
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
frequency
dB
State Variable Filter (R4 at 6kohm)Low-passHigh-passBand-passBand-reject
1000 10000 100000 1000000 10000000
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
frequency
dB
State Variable Filter (R4 at 26kohm)Low-passHigh-passBand-passBand-reject
1000 10000 100000 1000000 10000000
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
frequency
dB
State Variable Filter (R6 at 6kohm)Low-passHigh-passBand-passBand-reject
1000 10000 100000 1000000 10000000
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
frequency
dB
Page 19 of 46
State Variable Filter (R6 at 26kohm)Low-passHigh-passBand-passBand-reject
1000 10000 100000 1000000 10000000
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
frequency
dBR4 and R5 have an equal influence on the fC
frequency, in figure 11 R5 has been changes to
26kΩ which gives an fC of 7,8kHz if equation
4.7 on page 14 is used.
7,8kHz1n1n6k26k110k
10kπ2
1fC =××××
××
=
(4.26)
Figure 4.11
To find out how good performance the SV filter have, the value of Q is going to be tester
for the Low-pass filter section, the test values of Q is chosen to 0.5, 5, 10, 50 and 200 to
se how the filter will handle these values. To calculate R6 equation 4.20 on page 15 is
going to be used. Table 4.1 shows the values of R6. Figure 4.12 shows the result of
simulated values of Q.
Value of Q. Value of R6
0.5 5kΩ
5 140kΩ
10 290kΩ
50 1.49MΩ
200 5.99MΩ
Table 4.1
Figure 4.12
4.2.1 Simulations sub conclusion
The reason that SV filter response for the High-pass and Band-reject, begins to go down
at roughly 2MHz is that there is used a TL084 [8] opamp from Texas Instrument in the
simulations which have a bandwidth of 3MHZ, this is don because nothing is ideal in this
world, so it was chosen to use the TL084 opamp in steed off a ideal opamp, that will give a
impossible result to achieve in reel life.
State Variable Filter (Q from 0.5 to 200)Q = 0.5Q = 5Q = 10Q = 50Q = 200
1000 10000 100000 1000000 10000000
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
30
40
50
frequency
dB
Page 20 of 46
5. Switch Capacitor Filter
There is created a passive Low-pass filter, there is used as a reference to how the filter
response for the SC filter should lock like. The center frequency for the Low-pass filter is
chosen to be 10kHz, this frequency is going to be used for all the simulations in this
chapter.
Chosen values, fC = 10kHz, C = 10nF
CRπ21fC ×××
= (5.1)
1,592kΩ10n10kπ2
1Cfπ2
1RC
=×××
⇒×××
= (5.2)
On figure 5.1 and 5.2 the results of equation 5.2 are shown, the circuit on figure 5.1 is
converted into a SC filter (figure 5.3).
R
1.592kC10n
00
V11Vac0Vdc
VDB
Figure 5.1
Switch CapacitorLow-pass
-3dB
1000 10000 100000 1000000 10000000
-50
-40
-30
-20
-10
0
frequency
dB gain
Figure 5.2
Page 21 of 46
To calculate the value of CS in the SC filter, equation 5.3 has to be rewritten, so that the
value of CS can be found, fS >> fC, fs has to be 50 to 100 times bigger then fC to avoid anti-
aliasing in the SC filter.
Cπ2Cff SS
C ×××
= (5.3)
1MHz10010k100ff CS =×⇒×= (5.4)
628,3pF1M
10nπ210kC
Cπ2Cff
S
SSC
=×××
=
⇒××
×=
(5.5)
The whole divagation of equation 5.5 is shown in appendix 10.3
1,592kΩ1M628,3p
1fC
1RSS
=×
⇒×
= (5.6)
Figure 5.3 shows the SC filter circuit and figure 5.4 shows the frequency response, as it is
shows the fC is at 45.8kHz and not at 10kHz as it should be, the reason for this is mainly
that the capacitors inside the MOSFET has not been taken in to considering for this circuit.
The setup of switching pulses V2 and V3 are shown in appendix 10.4
Cs628p
C10n
V11Vac0Vdc
V2
TD = 0
TF = 0PW = 500nPER = 1u
V1 = 0
TR = 0
V2 = 5V3
TD = 500n
TF = 0PW = 500nPER = 1u
V1 = 0
TR = 0
V2 = 5
0
0 0
00
J1BC264A
J2BC264A
VD
B
Figure 5.3
Switch Capacitor Filter responce
filter responce-3dB
1000 10000 100000 1000000
-55
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
frequency
dB
Figure 5.4
Page 22 of 46
5.1 Test of different programs
In steed of using MOSFET transistors, where there is small capacitance inside, there has
to be taken into considerations. And since there is no ideal MOSFET transistors in OrCAD
10.0 there was tried to use ideal switches in steed off (figure 5.5), but it is was not possible
to generate a correct frequency response with ideal switches as shown on figure 5.6.
VD
B
Cs1628p
C110n
V41Vac0Vdc
0 00
1 2U1
01 2U2
0
Figure 5.5
Switch Capacitor Filter response with switches
filter responce
1000 10000 100000 1000000
110
100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
frequency
dB
Figure 5.6
The program Tina Pro was tried, except there is no ideal MOSFET in Tina Pro either. The
output from the circuit in figure 5.7 is shown in figure 5.8 with a frequency at 10kHz at the
input. It is only possible to do this simulation at one frequency and not over a wide range to
create a frequency response. Figure 5.8 showns which affect the switches has on the
output signal. The input has the value of 1Volt peek to peek and the frequency is set to
10kHz which means that the output signal has to have a value of 0.707 Volt peek to peek,
because 10kHz is at the -3dB point (equation 5.7).
707.0203-Log anti
20dBLog anti3dB =⎟
⎠⎞
⎜⎝⎛⇒⎟
⎠⎞
⎜⎝⎛=− (5.7)
C2
628p
-+
Input
tSW1
tSW2 Output
C1
10n
Figure 5.7
Page 23 of 46 T
Input
Output
0.749
-0.693
Time (s)0.00 50.00u 100.00u 150.00u 200.00u
Vol
tage
(V)
-1.00
-500.00m
0.00
500.00m
1.00
-0.693
0.749
Input Output Input
Output
Figure 5.8
Also the program Multisim 7 from Electronics Workbench was used to simulate a switch.
However that did not give a very good result, figure 5.9 shows two of the diagrams there
was tried to simulade in Multisim 7.
V2
1 V 1 V 0.5usec 1usec
J11 V 0mV
V112 V
R11kOhm
R2
1kOhm
R3
1kOhm
R41kOhm
V312 V
J21mV 0mV
V4
5 V 5 V 0.5usec 1usec
Figure 5.9
5.1.1 Sub Conclusion
None of the three programs gave a relay good result, the one there presented the best
result was OrCAD 10.0. Therefore is was decided to try to fix the problem in that program,
and because that was the program there is used most until now.
Page 24 of 46
5.2 Switch Capacitor Filter parallel & Setup of the PSpice model
In the SC parallel Low-pass filter circuit in figure 5.10 the C1 is en parallel with C on the
output and the size of the C1 to make the equivalent resistor is calculated in equation 5.8.
628,3pF1M
10n10kπ2f
Cfπ2CS
C1 =
×××⇒
×××= (5.8)
The filter response is shown on figure 5.11 with the cutoff frequency at -3dB, at 45.8kHz
and not at 10kHz as it should be the main reason for this is all the values there are in the
PSpice model of the BC246A MOSFET, these values are shown in figure 5.12. These
values can be adjusted to make a more ideal MOSFET, and to obtain a better frequency
response.
J1BC264A
J2BC264A
C1628p
C210n
V11Vac0Vdc
V2
TD = 0
TF = 0PW = 500nPER = 1u
V1 = 0
TR = 0
V2 = 5V3
TD = 500n
TF = 0PW = 500nPER = 1u
V1 = 0
TR = 0
V2 = 5
00 0
0 0
VD
B
Figure 5.10
Switch Capacitor Filter responce
filter responce-3dB
1000 10000 100000 1000000
-55
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
frequency
dB
Figure 5.11
Page 25 of 46
Figure 5.12
After the PSpise model figure 5.12, hade been adjusted to the best possible (figure 5.14) a
new frequency response is created as shown in figure 5.13, this frequency response is not
ideal, but it was not possible to obtain a better frequency response. Therefore a decision
was made, to use this PSpice model (figure 5.14) for the rest of the simulations in this
paper. Switch Capacitor Filter response with ideal MOSFET
filter responce-3dB
1000 10000 100000 1000000
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
frequency
dB
Figure 5.13
Figure 5.14
BC264A PSpice model
.model BC264A NJF(Beta=2m Betatce=-.5 Rd=1 Rs=1
+ Lambda=2.667m Vto=-1.249 Vtotc=-2.5m Is=33.57f
+ Isr =322.4f N=1 Nr=2 Xti=3Alpha=311.7u Vk=243.6
+ Cgd=3.35p M=.3622 Pb=1 Fc=.5 Cgs=3.598p
+ Kf=14.38E-18 Af=1)
New BC264 PSpice model
.model BC264A NJF(Beta=0.285m Betatce=-1.5 Rd=1
+ Rs=1 Lambda=1p Vto=-2.249 Vtotc=0 Is=0.1f
+ Isr=0.1f N=1 Nr=2 Xti=0 Alpha=0p Vk=0 Cgd=0 M=0
+ Pb=1 Fc=0 Cgs=0 Kf=0 Af=0)
Page 26 of 46
5.3 Switch Capacitor Filter Series
In the SC series circuit figure 5.15, the C1 is placed in series between the MOSFET J1’s
gate and source, the equation for the size of the equivalent resistor is the same as for the
parallel setup in equation 5.2 (1.592kΩ) the size of C1 is calculated in equation 5.9 and
equation 5.10 [6] is a control equation. The filter response in figure 5.16 is not ideal
because of the MOSFET is not ideal, because of the PSpice model in figure 5.14 has been
modified.
628,3pF1M
10n10kπ2f
Cfπ2CS
2CS =
×××⇒
×××= (5.9)
1.592kΩ1M628.3p
1fC
1RSS
=×
⇒×
= (5.10)
J1BC264A
J2BC264A
C1
628p
C210n
V11Vac0Vdc
V2
TD = 0
TF = 0PW = 500nPER = 1u
V1 = 0
TR = 0
V2 = 5V3
TD = 500n
TF = 0PW = 500nPER = 1u
V1 = 0
TR = 0
V2 = 5
0 0
0 0
VD
B
Figure 5.15
Switch Capacitor Filter Series
filter response-3dB
1000 10000 100000 1000000
-55
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
frequency
dB
Figure 5.16
Page 27 of 46
5.4 Switch Capacitor Filter Parallel and Series
The circuit of the SC parallel and series 5.17, is one capacitor placed in series (C1) and
one in parallel (C2) these two capacitors creates one equivalent resistor. The equation for
the size of the equivalent resistor is the equal to equation 5.2 (1.592kΩ) used in the
parallel circuit. The size of C1 and C2 together, is calculated in equation 5.11 [6]. The value
of C1 and C2 can be any size, as long as the total value produce a capacitor size of
628.3pF (equation 5.11) and equation 5.12 [6] is a control equation. In figure 5.18 C1 =
528pF and C2 = 100pF. In figure 5.19 C1 = 100pF and C2 = 528pF. In figure 5.20 C1 =
314pF and C2 = 314pF.
628,3pF1M
10n10kπ2f
Cfπ2CS
2CS =
×××⇒
×××= (5.11)
( ) ( ) Ω=×+
⇒×+
= kMpp
592.11100528
1fCC
1RS21
(5.12)
J1BC264A
J2BC264A
C1
528p C310n
V11Vac0Vdc
V2
TD = 0
TF = 0PW = 500nPER = 1u
V1 = 0
TR = 0
V2 = 5V3
TD = 500n
TF = 0PW = 500nPER = 1u
V1 = 0
TR = 0
V2 = 5
0
0
0
0
C2100p
0
VD
B
Figure 5.17
Switch Capacitor Filter Parallel and Seriesfilter response-3dB
1000 10000 100000 1000000
-65
-60
-55
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
frequency
dB
Figure 5.18
Page 28 of 46 Switch Capacitor Filter Parallel and Series
filter response-3dB
1000 10000 100000 1000000
-65
-60
-55
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
frequency
dB
Figure 5.19
Switch Capacitor Filter Parallel and Series
filter response-3dB
1000 10000 100000 1000000
-65
-60
-55
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
frequency
dB
Figure 5.20
5.4.1 Switch Capacitor Filter Parallel and Series sub conclusion
Figure 5.18 and 5.19 don’t show an ideal filter response, the two filter responses in figure
5.18 and 5.19 is opposite each other. That is because C1 and C2 have different values in
each case, if C1 and C2 have the same value as in figure 5.20 the two capacitors cancel
each other out and generate and perfect filter response as shown in figure 5.20. But
another reason that the filter response in figure 5.18 and 5.19 is not ideal could be
because of the MOSFET is not ideal, because of the PSpice model in figure 5.14 has been
modified.
Page 29 of 46
5.5 Switch Capacitor Filter Bilinear
In the SC bilinear circuit the capacitor C1 is placed between four MOSFETs, in figure 5.21,
this means that there always will be two MOSFETs ON at the same time (J1, J4) and (J2,
J3). This will eliminate the effect from parasite capacitors in the circuit. The equation for
the size of the equivalent resistor is the same as for the parallel circuit in equation 5.2
(1.592kΩ) and with C2 at 10nF the -3dB point should be at 10kHz but it is at 20kHz in
figure 5.22. If the sample frequency fS is changes to 2MHz the value of C1 becomes
78.5pF and the result of this gives a -3dB point at 20kHz. If the same test is don with a
sample frequency fS of 500kHz the value of C1 becomes 314pF and the result of this gives
a -3dB point at 20kHz. If the original PSpice file (figure 5.12) is used the -3dB point moves