(2007-06-08) Confusingly, complex pulsatance is often calledcomplex frequency.
Technically, a (real) pulsatance is a rate ofphase change per unit of time. It's expressed in angular units (radians or degrees) per unit of time (second). Pulsatance is also commonly called angular frequency. On the other hand, frequency is what pulsatance becomes when phases are expressed in cycles (one cycle is a phase change of 360° or 2 radians).
The modern convention is to express a pulsatance (preferably denoted by the symbol )in radians per second (rad/s) and the corresponding frequency(preferably denoted by the symbol ) in hertz(Hz, or "cycle per second").
=
In electrical engineering, the letter "i" is often used to denotea current intensity. It's thus unavailable as a name forthe unit vector along the imaginary axisof thecomplex plane. So, the letter "j" is used instead for that purpose. The square of thatimaginary number is -1. It's to the number 1 (unity) what a step sideways to the left is to a stepforward. Refrain from calling this "the" square root of -1.
j 2 = -1
The value at time t of a pure sinewave signal of frequency and/or of pulsatance = 2 can be conveniently represented as the real part of the following expression, where s is equal to the imaginary pulsatance j.
|A| exp ( j + s t ) = |A| cos ( t ) + j (...)
In this, A is a positive real number and is called the signal's phase. The number A = |A| exp(j) is called the complex amplitude of the signal and the value ofthe signal at time t is therefore simply the real part of:
A exp ( s t )
Therefore, the complex amplitude of the signal's derivative is A s. Likewise, the complex amplitude of the second derivative is A s 2, etc.
A slight generalization can be madeby considering that the above remains true even if s is a complex number with a nonzero real part .
s = + j = p + j
is called the damping constant. A negative value of (a positive p)does translateinto a signal which is a damped sinewave, like e -pt cos(t).
This observation may be construed as the basis forOliver Heaviside's operational calculus, which characterizes circuits by their reaction tononoscillatory decaying signals (p>0, =0). This approach rests on the co-calledLaplace transform(and its inverse). From a mathematical standpoint,such an analysis (which may be quite convenient)is just as sound as the more "physical" one based on sinewavesignals, involving theFourier transform(andits own inverse). Either approach yields results applicable to any signal whatsoever.
(2007-06-10) The complex number characterizing a linear dipole at a fixed frequency.
A dipole is defined as a current-conserving two-terminal device (the totalelectric charge inside thedevice doesn't change). Each terminal may also be referred to as an electrode or a pin. One of them is (somewhat arbitrarily) called "input", the other is the "output" terminal. Whatever current enters the input goes out the output; this quantity is thecurrent (i) through the dipole. The difference between the tension (voltage) of the input electrode and the outputtension is called the voltage (u) across the dipole.
A dipole for which u is proportional to i, is called a linear dipole. The coefficient of proportionalitybetween u and i is the impedance (Z).
u = Z i
In this, Z is a complex number which may dependon the operatingcomplex pulsatance (s) defined above. For example, Z may be equal to s multiplied by a (real) constant L when the voltage is proportional to thederivativeof the current (such is the case for a perfect inductor of inductance L,as discussedbelow).
Operating at a given imaginary pulsatance s = j, the dipole's resistance is defined as the real partof its impedance Z (the aforementioned perfect inductorhas zero resistance). The imaginary part of an impedanceis called reactance.
A nonzero reactance at (imaginary) pulsatance s indicates thatthe current and the voltage are out of phase at the corresponding operatingfrequency.
Resistor
A linear dipole whose impedance is a real number R which does not dependon the frequency of the signal is called a pure resistor of resistance R.
Practical resistors are never quite ideal,because any conducting element has a nonzero inductance whichmay become noticeable at very high frequencies. Also, there may be a tiny dependence of R on the amplitude of the signal (Ohm's law is avery goodpractical approximation, but it's not a strict law of nature).
For completeness, we may also mention that resistance may vary greatly withtemperature, so that a high current (which heats up the resistor)may give the apparence of a change in resistance with the amplitude of"large signals".
Capacitor
Ideally, a capacitor (or electricalcondenser)is a two-terminal device which stores opposite charges (q and -q) on two opposing armatures, connected to each terminal. That charge (q) is proportional to the voltage (U) across the terminals and thecoefficient of proportionality is the condenser'scapacity (C).
q = C u
We discusselsewhere the physical basis for thatrelation and how the capacity (C) can be computed from geometric parametersand/or from the characteristics of the dielectric material separatingthe conducting plates (armatures).
Inductor
(2007-06-07) The ratio of maximal energy stored to power dissipated.
The quality factor Q of a system reacting to a periodic excitationis the ratio of its maximum energy to the average energy itdissipates (per radian of phase change).
Q = L / R
(2007-06-09) Strange dipoles embodied by active electronic components.
I first heard about the following approach to elementary analog electronic designin the late 1970's at Ecole Polytechnique (X). It was a novelty at the time.
In an electronic circuit, adipole is defined as a two-terminal component;whatever current enters one terminal goes out the other.
Normally, such adipole is characterized by howthe current through it varies with timeas a function of the voltage across it (or vice-versa). The characteristic of anordinary dipole thus imposesoneconstraint between current and voltage...
However, two types ofextraordinary dipoles may be considered whichgreatly simplify the design of some active systems which could not otherwisebe modelized by dipoles alone... One such beast is called a nullator (symbol ) and imposestwo constraints: Zero current, zero voltage. On the other hand, a so-called norator dipole (symbol ) imposes no constraints at all: Any current, any voltage. Neither of those can be realized by itself but they can appear in complementary pairs which make the total number of constraints just right(i.e., one constraint per dipole connecting two nodes).
For example, a short-circuit (zero voltage, any current) can be considered toconsist of a nullator and a noratorin parallel. An open circuit (zero current, any voltage) consists of a nullator and anoratorin series. Less trivially, a properly polarized high-gaintransistor isapproximately equivalent to a norator from collector (C)to emitter (E) and a nullator from base (B) to emitter (E).
A nearly perfect embodiment of a useful nullator-norator combinationis the popular type of subsystem known as an operational amplifier. Thegain of an operational amplifier is normally so large thatsome feedback must somehow occur which forces thetwo high-impedance inputs of the amplifier to be at nearly the same voltage (or else the output "saturates" at either the lowest or the highest value allowed). The amplifier's inputs may thus be construed as thetwo extremities of a nearly perfect nullator. Conversely, the amplifier's output can be viewedas one extremity of a norator connected to the system's ground.
In practice, of course, the circuit will only be stable with the proper choiceof amplifier inputs for the extremities of the nullator ("inverting" vs. "non-inverting" input). Nevertheless, the nullator-norator approach allows a quick preliminary designbefore final stability issues are addressed.
(2007-06-07) First-order low-pass RC filter and its half-power bandwidth.
At left is the standard first-order passive RC low-pass attenuator (usually, = 0). At zero output current, the input voltageu is to R+Z what the output v is to Z. In other words:
u / v = 1 + R/Z = 1 + R (+jC)
The ratio v/u = H(s)expressed as a function of thecomplex pulsatance (s) is called the transfer function. In this case, it's equal to 1 / (1+R+RC s). Introducing the DC attenuation A = 1 / (1+R) and the circuit's characteristic pulsatance 0 = ARC,we obtain:
H = A / (1 + j x)
The normalized variable is x = 0 = 2 ( 1/RC + /C ).
Thenormalized gain (in dB) of the first-order low-pass filteris obtained by plotting 20 log(|H|/A) as a functionof x, using a logarithmic scale for x, as shown above. This diagram is called a Bode plot and is commonlyused to chart the frequency response of any filter.
The above shape is the main reason whybandwidth is usually defined as the range of frequencies for whichthe signal's amplitude is attenuated by no more than a factor of 2 (-3 dB) from a reference gain (corresponding to low-frequency signals and/or DCin the case of a low-pass filter). As the power is the square of the amplitude, such an attenuationmeans that the power is divided by 2, so the above isbest called "half-power bandwidth".
This definition does gives directly the "corner frequency" of any low-passButterworth filter,including the above first-order lowpass, which isthe simplest Butterworth filter... The relation isn't so simple in other cases.
(2007-06-07) Second-order rolloff is 40 dB per decade (roughly 12 dB per octave).
The second-order passive RLC low-pass filter at left is like itsfirst order counterpart, except thatthe resistor R becomes the impedance R+jL. Therefore, u/v is 1+(R+jL)(+jC)
u / v = (1+R)+ j (RC+L)2 LC
We may cast this in a normalized form:
v / u = A / [ 1 + j 2]
A =
1 / (1+R)
2 = =
1+R
LC
A = 1 / (1+R) is the low-frequency attenuation, used as the 0 dB reference level inthe above normalized Bode amplitude plot which charts the variationsof the gain |v/u| in decibels, against the ratio of the pulsatance to the nominal pulsatance (0 ) on a logarithmic horizontal scale.
So normalized, the response of a second-order lowpass filter ischaracterized by the so-called damping . For the above actual circuit,it's useful to express by introducing the characteristic resistance R0 = (L/C).
=
RC+L
=
R/R0 + R0
1+R
( 1+R )½
For the common case where = 0, this means that is simply R/R0.
In the normalized lowpass transfer function 1 / ( 1 + s + s 2) differentvalues of the damping make the correspondingsecond-order filter a member of one of the general families discussedelsewhere on this page:
Damping
0
Perfect (ideal) resonator, no damping. R = 0 and = 0.
Linkwitz-Riley filter: Two cascaded identical first-order filters.
> 2
Two first-order filters with distinctcorner frequencies (whosegeometric mean is 1 andwhose sum is ).
above). For high-ripple Chebyshev filters, the cutoff frequency is higher than the cornerfrequency. For low-ripple Chebyshev filters, it's lower (and the term "cutoff frequency" is not recommended in that case).
(2014-05-20) The resulting second-order filter can almost achieve critical damping.
The impedance of a capacitor C at pulsatance is equal to 1 / (j). Using that, it's left as an exercise for the reader to verify the followingrelation ( obtain w from v and u from w):
u / v = 1 + j [ R1C1 + R2C2 + R1C2] 2 R1C1R2C2
Critical damping is almost achieved when the first two bracketed terms are equal (R1 C1 = R2 C2 ) and the third bracketed term (R1C2) is very small compared to that common value (i.e, the impedance of the first stage ought to be much lower than the impedance of the second one).
This illustrates a fairly general principle: To cascade several RC filters, we usually want the early stages towork in a low impedance regime (fairly high current) so that they can feed small currents to later stages without being significantly affected.
This is a key advantage of active filters; they can always have a low impedance output. So does the above filter when endowed with its voltage-follower. Without some active section like this, the output of the filter would be of limited use.
(2007-06-16) Active second-order filters and/or resonators without inductors.
Whenactive components are used for signal processing,the DC gain of a lowpass filter should be kept close to unity. A larger gain would impose limitations on the input amplitudes (in order to prevent saturation of the output signal) whereas a much smaller gain would worsen thesignal-to-noise ratio(SNR or S/N).
This second-order active lowpass filter of unity gainwas among the designs introduced in 1955 by R.P. Sallen and E. L. Key (Lincoln Labs of MIT).
It can be used as a building block (along with a first-order stage)to realizeall the lowpass filters described on this page, without using anyinductor.
2 = = 1 / RC = ( x + 1/x ) y v = u / [ 1 + j 2]
The value of in a normalized second-orderfactor 1/(1+s+s 2) may thus be obtained from any convenient combination of theparameters x and y.
For example, with equal resistors (x=1) we have = 2y and a second-order Butterworth filter (2) is obtained for y=1/2 (i.e., C1 = 2 C0 ).
In practice, capacitors may only be available in a few standard values. Picking coarse values for the capacitors, we mayuse the following formula to compute precise matching valuesfor the two resistors R and R+.
R = R ( z
z 2-1
)
where R = 1/0
C0C1
and z = (½ )
C1 / C0
We just have to choose capacitor values so that z > 1.
The voltage response does not depend on which resistor goes where, butyou may want to make the input impedance larger (and/or reduce thepower involved) by placing thelarger resistance R+ on theinput side.
with ordinaryfloating-point arithmetic (because subtracting nearlyequal quantities entails a great loss of precision).Instead, we compute R+ first(full precisionis retained when quantities of like signs areadded)then obtain R from thefollowing formula (no precision is lost in multiplications or divisions).
R = R 2 / R+
(2007-06-09) The lowpass filters with the flattest low-frequency responses.
Such filters are named after the British radio engineer Stephen Butterworth(1885-1958) who first described them in 1930.
Little is known [ 1 | 2 ] about the life of Stephen Butterworth (MSc,OBE). He served in the British National Physical Laboratory (NPL) and joined the Admiralty scientific staff in 1921. He retired from the Admiralty Research Laboratory in 1945 and passed away in1958.
The normalized transfer function of an order-n lowpass Butterworth filter is of the form 1/Bn(s) where Bnis a Butterworth polynomial of order n.
n
Normalized Butterworth Polynomial Bn(s)
0
1
1
1 + s
2
1 + s 2 + s 2
3
( 1 + s ) ( 1 + s + s 2)
4
( 1 + s + s 2) ( 1 + s + s 2)
5
( 1 + s ) ( 1 + s ()/2 + s 2) ( 1 + s ()/2 + s 2)
6
( 1 + s (2)/2 + s 2) ( 1 + s + s 2) ( 1 + s (2)/2 + s 2)
2m
m
[ 1 + 2 s sin(2k1)/2n + s 2 ]
k=1
2m+1
(1+s)
m
[ 1 + 2 s sin(2k1)/2n + s 2 ]
k=1
For any n, | Bn(x) | is 2, so the attenuationof a Butterworth filter atits corner frequency is always -3 dB (well, -3.0103 dB, to be more precise).
Cascading two identical lowpass Butterworth filters of order n gives a lowpassfilter of order 2n with a 6 dB attenuation at the corner frequency.
This is particularly useful in combination with a similar highpass filtertuned to the same frequency... Since both output amplitudes are halved at that crossover frequency,their sum remains at the 0 dB level.
Such a feature is desirable in thedesign of audio systems, where low frequencies are directed to one loudspeakerand high frequencies to another. Modern professional active audio crossovers are often based on a fourth-orderLinkwitz-Riley design (LR-4). With digital signal processing (DSP)Linkwitz-Riley crossovers of order 8 are available (LR-8).
The basic idea was credited to Russ Riley in a paper published by Siegfried Linkwitz in 1976 (both Linkwitz and Riley were HP R&D engineers).
"Active Crossover Networks for Non-coincident Drivers" Siegfried H. Linkwitz, J. Audio Eng. Soc.,vol. 24, pp. 2-8(1976).
Linkwitz-Riley active crossovers were first made commercially available bySundholm and Rane in 1983. Nowadays, this may well be the most popular design forprofessional audio crossovers.
(2007-06-11)
The basic properties of Chebyshev polynomials can be put to good usein filter design, by explicitly allowing ripples of amplitude in the frequency response.
Those filters are named after the German scientistWilhelm Cauer(1900-1945). They're also called elliptic filters,complete Chebyshev filters or Zolotarev filters to honor the work ofEgorZolotarev (1847-1878) whoseresultswere applied to filter theory by Wilhelm Cauer in 1933.
(2007-06-12) "Optimum L filters".
The Optimum "L" filter, or Legendre filter, was introduced in 1958byAthanasiosPapoulis (1921-2002). Among all filters with a monotonic frequency response,the Legendre filter has the maximal roll-off rate. Its features are thus intermediate between the slow roll-off of aButterworth filter (which ismonotonic with unimodal derivatives)and the faster roll-off of a (non-monotonous)Chebyshev filter.
(2007-06-13)
TheGegenbauerpolynomials are a generalization of the Legendre polynomials (which correspondto the special case = ½). They are named afterLeopoldGegenbauer (1849-1903).
For a given value of , the Gegenbauer polynomials are recursively defined:
C0(x) = 1
C1(x) = 2 x
Cn(x) = 1/n[ (2n+22) x Cn-1(x) (n+22) Cn-2(x) ]
The generating function of those Gegenbauer polynomials is:
1/24()41/3()5 x 2 + 1/3()6 x 44/45()7 x 6 + 2/315()8 x 8
(2007-06-10) The correlation between phase delay and attenuation slope
If G = |G| exp(j) is the complex gain of a discrete low-pass filter, the following approximative relation holds,far from its corner frequencies, because itholds far from the corner frequency of every elementary such filter (the transfer function of higher-order filters is the product oftransfer functions of order 1 or 2).
/2 d ( Log |G| ) / d ( Log )
The Bayard-Bode relations where developed in 1936 byMarcel Bayard(1895-1956, X1919-S).
(2007-06-10) Optimizing phase linearity andgroup delay to preserve signalshape.
Theclass oforthogonal polynomialsnamed after the German mathematician and astronomer(Friedrich)WilhelmBessel (1784-1846) was only introduced in 1948 byH.L. Krall and O. Fink. The filters themselves were first presented by W.E. Thomson in 1949 and arebest called Bessel-Thomson filters (BT for short).
The group delay of a filter whosegain is G = |G| exp(j) is defined to be:
(2007-06-11) Preserving digital pulses in the "time domain".
(2007-06-13) Ripples allow bettergroup delay flatness than with Bessel filters.
These filters are toBessel filters with respect to group delay whatChebyshev filters are toButterworth filters with respect to amplitude gain. In either case,better pass-band flatness of the frequency response for the desired propertyis achieved by allowing some ripples,foregoing the strict monotonicity featuredin Butterworth filters (for amplitude gain) orBessel filters (for group delay).
(2007-06-14) Allowing POTS bellow 3400 Hz and blocking digital data above 25 kHz.
"Plain Old Telephone Service" (POTS) requires only the voiceband (300 Hz to 3400 Hz) corresponding to the spoken human voice. PCM digitalized voice corresponds tothe 0-4 kHz range (8 kHz sampling rate).
The final "twisted pair" which goes to the telephone subscriber is able to carrya much broader signal, up to 1.1 MHz or more. ADSL service makes use of that entire 0-1104 kHz bandby dividing it into 256 channels, each 4.3125 kHz wide.
Those channels are numbered from 0 to 255. The lowest one is the voiceband reserved for POTS. Next are 5 silent channels which provide a wide gap (from 4 kHz to 25 kHz)so a simple so-called "DSL filter" can safely block the digital frequencies(above 25.875 kHz) for POTS devices (telephone and/or FAX).
The remaining 250 channels, from 25.875 kHz to 1104 kHz,are used specifically for digital service. With ADSL, there's typically muchmore traffic downstream (downloading) than upstream (uploading). Only a small portion of the bandwidth is allocated to upstream traffic (normally, the 26 channels from 25.875 kHz to 138 kHz, but this can be increasedto 276 kHz per "Annex M" of the ADSL2 standard). This explains the "A" for "asymmetric" in theADSL acronym; Such an Asymmetric Digital Subscriber Line is nominally 8.92 times faster one way (223+1 download channels) than the other (25+1 upload channels). In practice, a 4 to 1 ratio seems more common nowadays
A third-order lowpass filter with a nominal corner frequency of 3243.375 Hz will produce an attenuation at 25.875 kHz roughly equal to thecube of the frequency ratio (1/8). This means an amplitude ratio of less than 0.002 (-54 dB). A typical fourth order filter will provide -72 dB.
The characteristic impedance of a telephone line is 600 .
(2014-05-14) A capacitor C switched between A and B at frequency is like a resistor R = 1 / .C between A and B.
It's not necessary to have an SPDT switch (single pole, double throw) as in the conceptual sketch show above. Instead, we can drive two ordinary switches (SPST) by two non-overlapping signalswhich never turn on both switches at the same time.
This allows a charge C(u-v) to be transferred at each switching cycle,which yields an average current equal to .C (u-v). This is precisely the current that would flow if the switched capacitor assemblywas replaced by a resistor of value 1 / .C
More generally, a single capacitor can be connected via switches to any number of points.
(2014-05-14)