Resistor codes :
4-band
(E12 & E24)
|
5-band
(E48, E96 & E192)
|
6-band (+ TempCo)
1k Resistor Color Code Common (5%) and Precision (1%) markings
Technically, a (real) pulsatance is a rate of phase 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 2p radians).
The modern convention is to express a pulsatance (preferably denoted by the symbol w) in radians per second (rad/s) and the corresponding frequency (preferably denoted by the symbol n) in hertz (Hz, or "cycle per second").
w = 2p n
In electrical engineering, the letter "i" is often used to denote a current intensity. It's thus unavailable as a name for the unit vector along the imaginary axis of the complex plane. So, the letter "j" is used instead for that purpose. The square of that imaginary number is -1. It's to the number 1 (unity) what a step sideways to the left is to a step forward. Refrain from calling this "the" square root of -1.
j 2 = -1
The value at time t of a pure sinewave signal of frequency n and/or of pulsatance w = 2pn can be conveniently represented as the real part of the following expression, where s is equal to the imaginary pulsatance jw.
|A| exp ( jq + s t ) = |A| cos ( q + wt ) + j (...)
In this, A is a positive real number and q is called the signal's phase. The number A = |A| exp(jq) is called the complex amplitude of the signal and the value of the 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 made by considering that the above remains true even if s is a complex number with a nonzero real part s.
s = s + jw = -p + jw
s is called the damping constant. A negative value of s (a positive p) does translate into a signal which is a damped sinewave, like e -pt cos(wt).
This observation may be construed as the basis for Oliver Heaviside's operational calculus, which characterizes circuits by their reaction to nonoscillatory decaying signals (p>0, w=0). This approach rests on the co-called Laplace 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 sinewave signals, involving the Fourier transform (and its own inverse). Either approach yields results applicable to any signal whatsoever.
A dipole is defined as a current-conserving two-terminal device (the total electric charge inside the device 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 the current (i) through the dipole. The difference between the tension (voltage) of the input electrode and the output tension 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 proportionality between u and i is the impedance (Z).
In this, Z is a complex number which may depend on the operating complex pulsatance (s) defined above. For example, Z may be equal to s multiplied by a (real) constant L when the voltage is proportional to the derivative of the current (such is the case for a perfect inductor of inductance L, as discussed below).
Operating at a given imaginary pulsatance s = jw, the dipole's resistance is defined as the real part of its impedance Z (the aforementioned perfect inductor has zero resistance). The imaginary part of an impedance is called reactance.
A nonzero reactance at (imaginary) pulsatance s indicates that the current and the voltage are out of phase at the corresponding operating frequency.
A linear dipole whose impedance is a real number R which does not depend on 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 which may 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 a very good practical approximation, but it's not a strict law of nature).
For completeness, we may also mention that resistance may vary greatly with temperature, 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".
Ideally, a capacitor (or electrical condenser) 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 the coefficient of proportionality is the condenser's capacity (C).
We discuss elsewhere the physical basis for that relation and how the capacity (C) can be computed from geometric parameters and/or from the characteristics of the dielectric material separating the conducting plates (armatures).
Strictly speaking, the charge on each armature is proportional to its absolute voltage (with respect to an "infinitely distant" ground) so there may be a bias in the actual charges stored on each armature. However, the only important practical quantities are variations in the charges (i.e., currents) and/or differences in voltage, so the above fiction is an adequate description.
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The quality factor Q of a system reacting to a periodic excitation is the ratio of its maximum energy to the average energy it dissipates (per radian of phase change).
Q = wL / R
I first heard about the following approach to elementary analog electronic design in the late 1970's at Ecole Polytechnique (X). It was a novelty at the time.
In an electronic circuit, a dipole is defined as a two-terminal component; whatever current enters one terminal goes out the other.
Normally, such a dipole is characterized by how the current through it varies with time as a function of the voltage across it (or vice-versa). The characteristic of an ordinary dipole thus imposes one constraint between current and voltage...
However, two types of extraordinary dipoles may be considered which greatly simplify the design of some active systems which could not otherwise be modelized by dipoles alone... One such beast is called a nullator (symbol -o-) and imposes two 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 to consist of a nullator and a norator in parallel. NPN transistor in nullator-norator terms An open circuit (zero current, any voltage) consists of a nullator and a norator in series. Less trivially, a properly polarized high-gain transistor is approximately 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 combination is the popular type of subsystem known as an operational amplifier. The gain of an operational amplifier is normally so large that some feedback must somehow occur which forces the two 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 the two extremities of a nearly perfect nullator. Conversely, the amplifier's output can be viewed as one extremity of a norator connected to the system's ground.
In practice, of course, the circuit will only be stable with the proper choice of amplifier inputs for the extremities of the nullator ("inverting" vs. "non-inverting" input). Nevertheless, the nullator-norator approach allows a quick preliminary design before final stability issues are addressed.
The ratio v/u = H(s) expressed as a function of the complex pulsatance (s) is called the transfer function. In this case, it's equal to 1 / (1+RG+RC s). Introducing the DC attenuation A = 1 / (1+RG) and the circuit's characteristic pulsatance w0 = ARC, we obtain:
H = A / (1 + j x)
The normalized variable is x = w / w0 = 2pn ( 1/RC + G/C ).
The normalized gain (in dB) of the first-order low-pass filter is obtained by plotting 20 log(|H|/A) as a function of x, using a logarithmic scale for x, as shown above. This diagram is called a Bode plot and is commonly used to chart the frequency response of any filter.
The above shape is the main reason why bandwidth is usually defined as the range of frequencies for which the signal's amplitude is attenuated by no more than a factor of Ö2 (-3 dB) from a reference gain (corresponding to low-frequency signals and/or DC in the case of a low-pass filter). As the power is the square of the amplitude, such an attenuation means that the power is divided by 2, so the above is best called "half-power bandwidth".
This definition does gives directly the "corner frequency" of any low-pass Butterworth filter, including the above first-order lowpass, which is the simplest Butterworth filter... The relation isn't so simple in other cases.
u / v = (1+RG) + jw (RC+LG) - w2 LC
We may cast this in a normalized form:
v / u = A / [ 1 + l j w/w0 - (w/w0 ) 2 ]
A = 1 / (1+RG) is the low-frequency attenuation, used as the 0 dB reference level in the above normalized Bode amplitude plot which charts the variations of the gain |v/u| in decibels, against the ratio of the pulsatance w to the nominal pulsatance (w0 ) on a logarithmic horizontal scale.
So normalized, the response of a second-order lowpass filter is characterized by the so-called damping l. For the above actual circuit, it's useful to express l by introducing the characteristic resistance R0 = Ö(L/C).
For the common case where G = 0, this means that l is simply R/R0.
In the normalized lowpass transfer function 1 / ( 1 + l s + s 2 ) different values of the damping l make the corresponding second-order filter a member of one of the general families discussed elsewhere on this page:
To clarify some of the technical literature pertaining to Chebyshev filters, it's important to distinguish the "corner" frequency (compatible with the above "nominal" frequency) from what's best called the "cutoff" frequency... The cutoff frequency of a lowpass "equiripple" Chebyshev filter is defined as the highest frequency for which the gain is equal to one of the bandpass minima (all such minima are equal in a Chebyshev filter). The cutoff frequency coincides with the corner ("nominal") frequency only in the case of a "natural" Chebyshev filter (like the 1.25 dB second-order Chebyshev filter plotted above). For high-ripple Chebyshev filters, the cutoff frequency is higher than the corner frequency. For low-ripple Chebyshev filters, it's lower (and the term "cutoff frequency" is not recommended in that case).
For future reference (and for easy comparison with the next section) we present this filter with a buffered output. If the opamp used as a voltage-follower has JFET inputs (featuring impedances measured in teraohms) this circuit can effectively be used to hold a voltage for a very long time when the input goes into a high-impedance state (as the charges in the capacitors have nowhere to go, then).
Second-order passive RC low-pass filter The impedance of a capacitor C at pulsatance w is equal to 1 / (jw). Using that, it's left as an exercise for the reader to verify the following relation (HINT: obtain w from v and u from w):
u / v = 1 + jw [ R1 C1 + R2 C2 + R1 C2 ] - w2 R1 C1 R2 C2
Critical damping is almost achieved when the first two bracketed terms are equal (R1 C1 = R2 C2 ) and the third bracketed term (R1 C2 ) 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 to work 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.
When active 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 the signal-to-noise ratio (SNR or S/N).
This second-order active lowpass filter of unity gain was among the designs introduced in 1955 by R.P. Sallen and E. L. Key (Lincoln Labs of MIT).
"A Practical Method of Designing Active Filters"
by R.P. Sallen and E.L. Key.
IRE Transactions on Circuit Theory
CT-2, 74 -85 (1955)
It can be used as a building block (along with a first-order stage) to realize all the lowpass filters described on this page, without using any inductor.
The value of l in a normalized second-order factor 1/(1+ls+s 2 ) may thus be obtained from any convenient combination of the parameters x and y.
For example, with equal resistors (x=1) we have l = 2y and a second-order Butterworth filter (l=Ö2) is obtained for y=1/Ö2 (i.e., C1 = 2 C0 ).
In practice, capacitors may only be available in a few standard values. Picking coarse values for the capacitors, we may use the following formula to compute precise matching values for the two resistors R- and R+.
We just have to choose capacitor values so that z > 1.
The voltage response does not depend on which resistor goes where, but you may want to make the input impedance larger (and/or reduce the power involved) by placing the larger resistance R+ on the input side.
Numerically, when z is large, the above expression yields a mediocre way to compute R- with ordinary floating-point arithmetic (because subtracting nearly equal quantities entails a great loss of precision). Instead, we compute R+ first (full precision is retained when quantities of like signs are added) then obtain R- from the following formula (no precision is lost in multiplications or divisions).
R- = R 2 / R+
Such filters are named after the British radio engineer Stephen Butterworth (1885-1958) who first described them in 1930.
"On the Theory of Filter Amplifiers" (1930) by
Stephen Butterworth
Experimental Wireless and the Radio Engineer,
vol. 7, pp. 536-541.
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 in 1958.
The normalized transfer function of an order-n lowpass Butterworth filter is of the form 1/Bn(s) where Bn is a Butterworth polynomial of order n.
For any n, | Bn (x) | is Ö2, so the attenuation of a Butterworth filter at its 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 lowpass filter of order 2n with a 6 dB attenuation at the corner frequency.
This is particularly useful in combination with a similar highpass filter tuned 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 the design of audio systems, where low frequencies are directed to one loudspeaker and high frequencies to another. Modern professional active audio crossovers are often based on a fourth-order Linkwitz-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).
Linkwitz-Riley active crossovers were first made commercially available by Sundholm and Rane in 1983. Nowadays, this may well be the most popular design for professional audio crossovers.
The basic properties of Chebyshev polynomials can be put to good use in filter design, by explicitly allowing ripples of amplitude e in the frequency response.
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The parametrization of Cauer filters is general enough to include Butterworth filters and both types of Chebyshev filters.
Those filters are named after the German scientist Wilhelm Cauer (1900-1945). They're also called elliptic filters, complete Chebyshev filters or Zolotarev filters to honor the work of Egor Zolotarev (1847-1878) whose results were applied to filter theory by Wilhelm Cauer in 1933.
The Optimum "L" filter, or Legendre filter, was introduced in 1958 by Athanasios Papoulis (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 a Butterworth filter (which is monotonic with unimodal derivatives) and the faster roll-off of a (non-monotonous) Chebyshev filter.
The Gegenbauer polynomials are a generalization of the Legendre polynomials (which correspond to the special case l = ½). They are named after Leopold Gegenbauer (1849-1903).
For a given value of l, the Gegenbauer polynomials are recursively defined:
The generating function of those Gegenbauer polynomials is:
( 1 - 2xt + t2 ) -l = ån Cn(x) t n
If G = |G| exp(jj) is the complex gain of a discrete low-pass filter, the following approximative relation holds, far from its corner frequencies, because it holds far from the corner frequency of every elementary such filter (the transfer function of higher-order filters is the product of transfer functions of order 1 or 2).
j » p/2 d ( Log |G| ) / d ( Log w )
The Bayard-Bode relations where developed in 1936 by Marcel Bayard (1895-1956, X1919-S).
The class of orthogonal polynomials named after the German mathematician and astronomer (Friedrich) Wilhelm Bessel (1784-1846) was only introduced in 1948 by H.L. Krall and O. Fink. The filters themselves were first presented by W.E. Thomson in 1949 and are best called Bessel-Thomson filters (BT for short).
"Delay Networks Having Maximally Flat Frequency Characteristics"
W.E. Thomson. Proc. IEEE, part 3, vol. 96, pp. 487-490 (Nov. 1949).
The group delay of a filter whose gain is G = |G| exp(jj) is defined to be:
tg = - dj / dw
The transfer function is qn(0)/qn(s) where qn is the n-th reverse Bessel polynomial, as tabulated below:
These filters are to Bessel filters with respect to group delay what Chebyshev filters are to Butterworth filters with respect to amplitude gain. In either case, better pass-band flatness of the frequency response for the desired property is achieved by allowing some ripples, foregoing the strict monotonicity featured in Butterworth filters (for amplitude gain) or Bessel filters (for group delay).
"Plain Old Telephone Service" (POTS) requires only the voiceband (300 Hz to 3400 Hz) corresponding to the spoken human voice. PCM digitalized voice corresponds to the 0-4 kHz range (8 kHz sampling rate).
This is strictly for standard telephony (voice). By contrast, "CD quality" digital audio involves a 44.1 kHz sampling rate, corresponding to an upper audio limit of 22.05 kHz. The "audio range" is most often quoted as going from 20 Hz to 20 kHz, although you've certainly not heard a 20 kHz tone since you were an infant (and never will again, if you ever did)... The highest vocal note in classical repertoire is G7 (3136 Hz). The last key on an 88-key grand piano is at 4186 Hz.
The final "twisted pair" which goes to the telephone subscriber is able to carry a much broader signal, up to 1.1 MHz or more. ADSL service makes use of that entire 0-1104 kHz band by 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 much more 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 increased to 276 kHz per "Annex M" of the ADSL2 standard). This explains the "A" for "asymmetric" in the ADSL 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 the cube 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 W.
Order-4 DSL Filter
Order-4 DSL filter
by Ben Kamen.
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 signals which 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 n.C (u-v). This is precisely the current that would flow if the switched capacitor assembly was replaced by a resistor of value 1 / n.C
More generally, a single capacitor can be connected via switches to any number of points.
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Take the Mystery Out of the
Switched-Capacitor Filter by Richard Markell.
Wikipedia :
Switched capacitor
