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Chapter 4: Passive Analog Signal Processing
In this chapter we introduce filters and signal transmission theory. Filters are essential components of most analog circuits and are used to remove unwanted signals (i.e. noise) from the actual signal. Transmission lines are essential for sending signals from one device to another, such as from a detector to a data acquisition module.
I. Filters
Filters are ubiquitous in analog electronic circuitry. In fact, if you see a capacitor or an inductor in a circuit there is a good chance it is part of a filter. Filters are frequently used to clean up (i.e. remove high frequency noise) power supplies and remove spurious frequencies from a signal (frequently 60 Hz, switching power supply noise in computers, display screen noise, ground loop noise, and Radio-Frequency (RF) pick-up).
A. RC Filters:
RC filters are by far the most common filters around. They are simple to make (i.e. just a resistor and a capacitor), reliable, and involve relatively simple design calculations.
1. The Low-Pass RC Filter
The low-pass RC filter, or integrator, is used to remove high frequencies from a signal. Applications include the removal of RF pick-up noise and reducing ripple voltages on power supplies.
A generic RC low pass filter circuit is shown on the right. We have already calculated its performance in the previous chapter using Fourier analysis and complex impedances. We recall the results (equation 28):
( / 2 0 sin( )
= = φ i ω t +φ−^ π
Vout VC V e
VOUT
R
VIN
C
Where sin( φ )= 1 1 +( ω RC )^2 and. From these
quantities we can compute the Gain and Phase performance of
the filter. The gain is defined as
i t Vin Ve ω = 0
Gain = Vout V in and the phase
as ϕ = φ− π/ 2 (this is just the part after the ω t in the exponent).
The RC filter is just a voltage divider with complex impedances, so we can calculate the gain easily:
sin( ) 1 ( )
ω ω ω^2 φ
R i C i RC RC
i C V
V
in
out (^) (2)
The phase of the output voltage with respect to the input is easily computed and is given by
tan ϕ = tan( φ− π/ 2 )=cot( φ)= ω RC (3)
At ω = 1 / RC , the output voltage drops to 1 / 2 of the input voltage, and consequently the power transmission drops to 50% or -3dB^1. At this frequency, the voltage across the
resistor and the voltage across the capacitor are equal in amplitude, but ± π 2 out of
phase with the drive voltage. The average value of V^2 across either the resistor or the capacitor is down by a factor of 2 from the drive voltage. Consequently, this frequency characterizes the RC circuit completely and is called the 3dB frequency. We can rewrite the gain and phase equations in terms of the 3dB frequency:
2 1 ( / 3 )
f f dB
gain
= (4), tan ϕ = f / f 3 dB (5)
with f 3 dB = 1 / 2 π RC.
The Bode plots (log-log or semi-log plots) for the gain and phase of the low pass filter are the following:
Log(ω)
Log |V
out
/V
|in
1/RC
0
Log(Log(ωω))
Log |V
out
/V
|in
1/RC
0 Log(ω)
Phase
1/RC
0 Log(ω)
Phase
1/RC
Log(ω)
Phase
1/RC
0
(^1) A dB or decibel is a notation for quantifying a ratio of two numbers. For power, a dB is defined as
dB = 10log 10 (P/P 0 )
From this definition we can see that a ratio of 0.5 is roughly -3. Hence -3db is the same as halving a signal. For voltage or current, a dB is defined as
dB = 20log 10 (V/V 0 )
So at -3dB, the voltage (or current) has dropped to 1 / 2 of its input value.
3) RC Filter Design
When designing an RC filter, you need to think about two things:
.
- Make sure the impedance is approp
Step 1 is straightforward and depends on th
In Step 2 you pick R so as to satisfy the impe and your signal destination. Typically, th , so the
impedance of the capacitor and the impedance of the resistor will be about the same.
When filtering an input, choose the resistance to be about 10 times smaller than the input impedance of the next stage. This will prevent the next stage from loading the filter. You would also like to choose the resistance to be at least 10 times larger than the previous stage of your circuit’s output impedanc
- Choose an appropriate f 3 dB
riate for your desired output load. e frequencies you want to pass and block. dance requirements of your signal source e signals we want to pass are around f 3 dB
e.
Choose the capacitor for the appropriate f 3 dB = 1 ( 2 π RC ).
- If the signal you want to pas s is low frequency, choose f (^) 3 dB ≈ 2 fPass , and hence
C ≈ 1 ( 4 π RfPass ).
- If the signal you want to pass is high frequency, choose f (^) 3 dB ≈ fPass 2 , and hence
C ≈ 1 ( π RfPass ).
3) Combination Filters
If you put several RC filters in a row you can make a more sophisticated signal filter. For example, several similar low-pass filters placed on after the other will produce a steeper fall off of the gain. One can also use a low-pass followed by a high pass to
produce a bandpass filter ( f^ 3 dB , lowpass >^ f 3 dB , highpass ).
Design tip: When you are designing a bandpass filter, you should make sure that the resi r about a factor of 10. Otherwise, the second stage will load the first stage, and shift the effectiv frequency
of t f
ations of RC filters:
a a p igh voltage DC will not get through to your detection
sto in the second stage is larger than the first stage by e f 3 dB he irst stage.
4) Applications
Here are some specific applic
Blocking Capacitor
This is a high pass filter that is used to eliminate DC. Suppose that you want to measure small time-dependent sign ls that h p en to “float” on a high voltage. If you use a blocking capacitor, then the h
electronics, but the signal will get through. Choose f 3 dB low to insure that your entire
signal gets through.
Ripple Eliminator
This is a low pass filter used to build power supplies. Since most of our power is 60 Hz er supplies will convert AC to DC, but there will always be some set well below 60 Hz will work. You o not use a resistance in this case, but let bination of the loading resistance RL nd the Thevenin resistance of the previous components serve as your R. This usually acitor since R might be quite small when you use the power supply.
equently the voltage which you supply to a chip component, such as an op-amp (we the semester), may be “clean” when it comes out of the power
r with a high value of.
AC, our DC pow residual 60 Hz “ripple”. A low pass filter with f 3 dB
d a
the com
requires a large cap (^) L If the capacitor is not big enough, then f 3 dB will then shift to a higher frequency, and the
60 Hz ripple will reappear.
Chip Supply Clean Up
Fr will study these later in supply, but will pick up noise by the time it reaches the component. In this case a 10 – 100 nF capacitor placed at the supply leads of the component will remove the high frequency pick-up noise.
Noise Eliminator
Any signal line is susceptible to picking up high frequency transients; especially if there are motors or switching power supplies (or FM radio stations!) nearby. A noise eliminator is a low pass filte f 3 dB
Integrator
If you build an RC filter, but set the value of f 3 dB much higher than the highest frequency
in your signal, the filter integrates your signal. From our earlier analysis, when f << f 3 dB ,
we can see that each (low) frequency voltage component will see a π 2 same phase shift
and its amplitude will be proportional to 1 f. This is exactly the prescription for
integ in
ration. So, a high pass filter sends high frequencies out on the resistor, and the tegral of very low frequencies on the capacitor.
e ifferentiates your signal. m our earlier analysis, when , each (high) equency voltage component will see a
Differentiator
If you build an RC filter with f 3 dB lower than th lowest frequency in your signal, the
Fro f >> f 3 dB
filter d
fr same phase shift and its amplitude will be
proportional to f. This is exactly the prescription for differentiation. So, a low pass filter
A common remedy for dealing with the inherent inductance of a capacitor at high frequencies is to place a small capacitor (10 – 100 pF) in parallel with the main capacitor in RC filter. The high frequency performance of the small capacitor will generally much etter than that of the main capacitor: the small will pick-up the high frequency signal hen the main, larg ll-off f the RC filter will no longer be -20 dB/decade, but at least a low-pass filter will not art to behave like a high-pass filter (or vice-versa)!
. LC Filters:
RC filters are by far the simplest and the most common type of filter found in nalog circuits, however they suffer from a relatively slow roll off of the gain: while the ain or attenuation slope can be made steeper than -20 dB/decade, the transition region, r knee of the curve (the region where the gain changes from flat to a log-log slope), will lways have the same shape and frequency width. e complex but can be engineered to produce m
5 th^ order low-pass LC filter
5 th^ order high-pass LC filter
required filter performance.
b w er capacitor begins to have a significant inductance. The gain fa o st
B
a g o a
LC filters are mor uch sharper features and steeper fall-off regions. The standard design for LC filters is an LC ladder with an un-interrupted ground line, such as in the 5th^ order filters shown below:
The algebra required for computing the gain and phase Bode plots for these filters is generally quite cumbersome, and a computer program (i.e. Maple or Excel) is generally useful for helping with the design. A number of web-applets can also be found on the internet for determining all the inductor and capacitor values that will produce the
L
C
L
C
L
C
L
C4 C
RTH
RLOAD
L L
C C
L
C
L4 L
RTH^ C
RLOAD
Example 1: 5 th^ order Butterworth LC low-pass filter with =10 kHz for 50 Ω input
impedance and 50 Ω output impedance.
f 3 dB
This Butterworth filter uses C1 = C5 = 0. 6946 μF, C2 = C4 = 3.0642 μF, C3 = 4 μF,
L1 = L4 = 5 mH, and L2 = L3 = 9.397 mH. Butterworth filters have a very flat pass-band, and a reasonably regular phase change across the knee of the curve.
Example 2: 5 th^ order Chebyshev LC low-pass filter with =10 kHz for 50 Ω input
impedance and 50 Ω output impedance.
f 3 dB
This Chebyshev filter uses C1 = C5 = 1.125 μF, C2 = C4 = 1.486 μF, C3 = 1.505 μF,
If the LC ladder is infinite and has an impedance Z 0 , then if we add an extra LC ladder “rung”, the total impedance should not change, and we obtain the following relation
i ω L + Z 0
0 (^0 00) i C
i L Z i C Z L Z C Z
From th is relation we can extract an expression for Z 0 , and we obtain
2 2 0
L
C
i L L Z
If we consider that L and C are the inductance and capacitance, respectively, of a short
section Δl of transmission line, then as we take the limit Δl→0, we have L→0 and
C→0, but L/C→ constant. In this limit the equation for Z 0 becomes
C
L
Z 0 = (13)
here L and C are the inductance and capacitance, respectively, of the transmission line,
real impedance, and consequently consume power. The explanation for this apparent paradox that since the network is infinite, power is flowing fro ad infinitum, so that power is constantly moving down the transmission line, though it is wer is consumed at the end of the transmission line when we attach a load resistor.
The LC ladder model is include the wire resistance which can contribute to signal attenuation.
hould be terminated with an load impedance of Z if the transmission line is longer than 1/10 of the
w and Z 0 is called the characteristic impedance of the transmission line.
It may seem surprising that a network of inductors and capacitors can have a
is m one LC ladder rung to the next
not dissipated in either the inductors or capacitors. Of course, the po
a high frequency model of transmission line and does not
Transmission Line Impedance Matching
A transmission line with a characteristic impedance of Z 0 s 0 wavelength of the signal (recall wavelength^ =^ c / f , where c is the speed of light in the
transmission line and f is the signal frequency). Ifthe transmission line is not properly
terminated, then the signal will be partially reflected back towards the source upon arrival at the load.
There are three main types of transmission lines: wires, twisted pairs, and coaxial cables. In this section we go over their performance characteristics.
1. Wires
Plain wires are the simplest and the cheapest transmission lines available: they include the wires you use to connect components on your breadboard and electrical grid ower lines. If the wires are kept parallel, then the transmission line will provide some rotection to external fields and noise at low frequencies. While wire transmission lines ver possible since they are susceptible to
isted pair of wires provides good protection from outside fields and noise and ls without difficulty. The electro-magnetic eld of a signal kept close to the wire pair. A twisted pair can transmit analog signals at
cable, such as RJ thernet cables which has a characteristic impedance of Z 0 = 100 Ω. Twisted pair ansmission lines are easy to make: just take two wires of equal length and twist them
is in the electric and agnetic fields that carry the signal though the cable, instead of in the potential energy of e current carrying electrons. The speed of light in a coaxial cable is usually 60-70% of n vacuum. The characteristic impedance of most coaxial cables used in
ercise 4-2: Design a band-pass filter which will only pass frequencies near 0 kHz. Do this by combining 2 different RC filters (one high pass and one low pass). : Design a notch filter which will attenuate 1 kHz by at least 60 dB,
p p are simple, they should be avoided whene interference and high frequency pick-up. At high frequencies a wire transmission line actually becomes an antenna (for both transmission and reception).
2. Twisted Pairs
A tw can transmit relatively high frequency signa fi up to 250 kHz (sometimes even 1 MHz) and digital signals up to 100 MHz. Twisted pairs are quite common and are used in many computer communications E tr together!
3. Coaxial Cables
Coaxial cable is the best form of transmission line, short of a waveguide. In the limit of perfect conductors, signals on coaxial cables are impervious to external fields and do not radiate either. Coaxial cables can be used for frequencies up to 1 GHz. At high frequencies, a significant fraction of the transmitted energy/power m th the speed of light i industry and labs is Z 0 = 50 Ω -- coaxial cable for cable TV is an exception, it has a characteristic impedance of 75 Ω.
Design Exercises:
Design Exercise 4-1: Design a high-pass filter, an RC circuit that can filter out 60 Hz and 120 Hz but yet still pass signals in the kHz region.
Design Ex 1
Design Exercise 4- but will pass 1.2 kHz and 0.8 kHz with less than 6 dB of attenuation, and can drive a 100 kΩ load. Assume that the signal source has 50 Ω Thevenin impedance (i.e. function generator).
