domingo, 21 de marzo de 2010

Basic Circuits - Bypass Capacitors

The Function

The definition of a bypass capacitor can be found in the dictionary of electronics.

Bypass capacitor: A capacitor employed to conduct an alternating current around a component or group of components. Often the AC is removed from an AC/DC mixture, the DC being free to pass through the bypassed component.
In practice, most digital circuits such as microcontroller circuits are designed as direct current (DC) circuits. It turns out that variations in the voltages of these circuits can cause problems. If the voltages swing too much, the circuit may operate incorrectly. For most practical purposes, a voltage that fluctuates is considered an AC component. The function of the bypass capacitor is to dampen the AC, or the noise. Another term used for the bypass capacitor is a filter cap.

n the chart on the left, you can see the what happens to a noisy voltage when a by-pass capacitor is installed. Notice that the differences in voltage are pretty small (between 5 and 10 millivolts). This graph represents a small range of 4.95 volts to 5.05 volts. Random electrical noise causes the voltage to fluctuate, as you can see in graph. This is often called 'noise' or 'ripple'. The blue line, represents the voltage of a circuit that doesn't have a bypass. The pink line is a circuit that has a bypass. Ripple voltages are present in almost any DC circuit. You can see even with the bypass, the voltage does fluctuate, even though it is to a smaller degree. The key function of the bypass capacitor is to reduce the amount of ripple in a circuit. Too much ripple is bad, and can lead to failure of the circuit. Ripple is often random, but sometimes other components in the circuit can cause this noise to occur. For example, a relay or motor switching can often times cause a sudden fluctuation in the voltage. Much like disturbing the water level in a pond. The more current the other component uses, the bigger the ripple effect.

A fair question to ask is why does this small fluctuation matter? Gee, isn't the voltage close enough? The answer depends on the type of circuit you are designing. If you are just running a motor connected to a battery, or perhaps an LED, then chances are the ripple doesn't matter much to you. However, if you are using digital logic gates, things get slightly more complex, and this ripple can cause problems in the circuit.

Lets consider for just a moment what the effect of the ripple voltage is. Basic electrical theory tells us that a voltage is a difference in potential. It tells us that a current will flow across this difference in potential. We know that the larger the voltage, the larger the current. We also know the direction of the voltage determines the direction of the current.

Consider the graphs on the right. The top graph shows a pair of ripple voltages that I enlarged to make them easier to see. Just like the previous graph, the blue line represents the circuit without the bypass cap, and the other line is with the bypass cap. By looking along the bottom axis of the graph, you can see that starting at point 2 that the voltage is increasing. By looking in the Ripple Current chart, point 2 shows that the current is a relatively large magnitude in one direction. In contrast, point 5 shows the voltage and current going the other direction.
Notice the difference between the values with and without the bypass cap. By dampening the ripple voltage, the bypass cap also dampens the ripple current. I would like to point out that the Ripple Voltage chart and the Ripple Current charts clearly show an alternating current. You can see how the voltage swings, and how the current changes directions. Even though this is is a DC circuit, the ripple is causing an AC component. The bypass capacitor is helping to reduce this AC component.

The ripple current acts like an eddie or backflow in the circuit. As the fluctuating voltages and currents propogate through the circuit, differences in voltages and currents can occur that cause the circuit to fail. For example, assume that a AND gate is holding its state because the semiconductors that make up the gate are in a stable state. Transistors work by currents flowing one direction through the gate. If the current stops flowing, the transistor shuts down. If a ripple current comes through where the current momentarily flows the wrong direction, the gate will shutdown, and you will see a change it its output. This can cause a cascading failure, because one gate may be connected to many other gates.

To summarize, the bypass capacitor is used to dampen the AC component of your DC circuits. By installing bypass capacitors, your DC circuit will not be as susceptable to ripple currents and voltages.

Using Bypass capacitors

Many schematics that you find published in magazines and books leave the bypass capacitors out. They assume you know to put them in. Other times you will find a little row of capacitors (caps) stuck off in the corner of the schematic with no apparent function. These are usually the bypass (or filter) caps. If you pickup almost any digital circuit, you will find a bypass capacitor on it.

The most simple incarnation of the bypass capacitor is a cap connected directly to the power source and to ground, as shown in the diagram to the left. This simple connection will allow the AC component of VCC to pass through to ground. The cap acts like a reserve of current. The charged capacitor helps to fill in any 'dips' in the voltage VCC by releasing its charge when the voltage drops. The size of the capacitor determines how big of a 'dip' it can fill. The larger the capacitor, the larger the 'dip' it can handle. A common size to use is a .1uF capacitor. You will also see .01uF as a common value. The precise value of a bypass cap isn't very important.

So, how many bypass capacitors do you really need? A good rule of thumb I like to use is each IC on my board gets its own bypass capacitor. In fact, I try to place the bypass cap so it is directly connected to the Vcc and Gnd pins. This is probably overkill, but it has always served me well in the past, so I will recommend it to you. It turns out you can even by DIP sockets that have the bypass caps built in. I suppose once you reach more than a few capacitors per square inch, you might be able to let up a bit!

Another great place for a bypass cap is on power connectors. Anytime you have a power line heading off to another board or long wire, I would recommend putting in a bypass cap. Any long length of wire is going to act like a little antenna. It will pick up electrical noise from any magnetic field. I always put a bypass cap on both ends of such lengths of wire.

The frequency of the ripple can have a role in choosing the capacitor value. Rule of thumb is the higher the frequency, the smaller the bypass capacitor you need. If you have very high frequency components in your circuit, you might consider a pair of capacitors in parallel. One with a large value, one with a small value. If you have very complex ripple, you may need to add several bypass capacitors. Each cap is targeting a slightly different frequency. You may even need to add a larger electrolytic cap in case the amplitude of the lower frequencys is too great. For example, the circuit on the right is using three different capacitor values in parallel. Each will respond better to different frequencies. The 4.7uF cap (C4) is used to catch larger voltage dips which are at relatively low frequencies. The cap C2 should be able to handle the midrange frequencies, and C3 will handle the higher frequencies. The frequency response of the capacitors is determined by their internal resistance and inductance.


Bypass capacitors help filter the electrical noise out of your circuits. They do this by removing the alternating currents caused by ripple voltage. Most digital circuits have at least a couple of bypass capacitors. A good rule of thumb is to add one bypass capacitor for every integrated circuit on your board. A good default value for a bypass cap is 0.1uF. Higher frequencies require lower valued capacitors.

Frequency compensation

In electrical engineering, frequency compensation is a technique used in amplifiers, and especially in amplifiers employing negative feedback. It usually has two primary goals: To avoid the unintentional creation of positive feedback, which will cause the amplifier to oscillate, and to control overshoot and ringing in the amplifier's step response.


Most amplifiers use negative feedback to trade gain for other desirable properties, such as decreased distortion or improved noise reduction. Ideally, the phase characteristic of an amplifier's frequency response would be constant; however, device limitations make this goal physically unattainable. More particularly, capacitances within the amplifier's gain stages cause the output signal to lag behind the input signal by 90° for each pole they create.[1] If the sum of these phase lags reaches 360°, the output signal will be in phase with the input signal. Feeding back any portion of this output signal to the input when the gain of the amplifier is sufficient will cause the amplifier to oscillate. This is because the feedback signal will reinforce the input signal. That is, the feedback is then positive rather than negative.
Frequency compensation is implemented to avoid this result.
Another goal of frequency compensation is to control the step response of an amplifier circuit as shown in Figure 1. For example, if a step in voltage is input to a voltage amplifier, ideally a step in output voltage would occur. However, the output is not ideal because of the frequency response of the amplifier, and ringing occurs. Several figures of merit to describe the adequacy of step response are in common use. One is the rise time of the output, which ideally would be short. A second is the time for the output to lock into its final value, which again should be short. The success in reaching this lock-in at final value is described by overshoot (how far the response exceeds final value) and settling time (how long the output swings back and forth about its final value). These various measures of the step response usually conflict with one another, requiring optimization methods.
Frequency compensation is implemented to optimize step response, one method being pole splitting.

Use in operational amplifiers

Because operational amplifiers are so ubiquitous and are designed to be used with feedback, the following discussion will be limited to frequency compensation of these devices.
It should be expected that the outputs of even the simplest operational amplifiers will have at least two poles. An unfortunate consequence of this is that at some critical frequency, the phase of the amplifier's output = -180° compared to the phase of its input signal. The amplifier will oscillate if it has a gain of one or more at this critical frequency. This is because (a) the feedback is implemented through the use of an inverting input that adds an additional -180° to the output phase making the total phase shift -360° and (b) the gain is sufficient to induce oscillation.
A more precise statement of this is the following: An operational amplifier will oscillate at the frequency at which its open loop gain equals its closed loop gain if, at that frequency,
1. The open loop gain of the amplifier is ≥ 1 and
2. The difference between the phase of the open loop signal and phase response of the network creating the closed loop output = -180°. Mathematically,
ΦOL – ΦCLnet = -180°
Frequency compensation is implemented by modifying the gain and phase characteristics of the amplifier's open loop output or of its feedback network, or both, in such a way as to avoid the conditions leading to oscillation. This is usually done by the internal or external use of resistance-capacitance networks.
[edit]Dominant-pole compensation
The method most commonly used is called dominant-pole compensation, which is a form of lag compensation. A pole placed at an appropriate low frequency in the open-loop response reduces the gain of the amplifier to one (0 dB) for a frequency at or just below the location of the next highest frequency pole. The lowest frequency pole is called the dominant pole because it dominates the effect of all of the higher frequency poles. The result is that the difference between the open loop output phase and the phase response of a feedback network having no reactive elements never falls below −180° while the amplifier has a gain of one or more, ensuring stability.
Dominant-pole compensation can be implemented for general purpose operational amplifiers by adding an integrating capacitance to the stage that provides the bulk of the amplifier's gain. This capacitor creates a pole that is set at a frequency low enough to reduce the gain to one (0 dB) at or just below the frequency where the pole next highest in frequency is located. The result is a phase margin of ≈ 45°, depending on the proximity of still higher poles.[2] This margin is sufficient to prevent oscillation in the most commonly used feedback configurations. In addition, dominant-pole compensation allows control of overshoot and ringing in the amplifier step response, which can be a more demanding requirement than the simple need for stability.
Though simple and effective, this kind of conservative dominant pole compensation has two drawbacks:
1. It reduces the bandwidth of the amplifier, thereby reducing available open loop gain at higher frequencies. This, in turn, reduces the amount of feedback available for distortion correction, etc. at higher frequencies.
2. It reduces the amplifier's slew rate. This reduction results from the time it takes the finite current driving the compensated stage to charge the compensating capacitor. The result is the inability of the amplifier to reproduce high amplitude, rapidly changing signals accurately.

Often, the implementation of dominant-pole compensation results in the phenomenon of Pole splitting. This results in the lowest frequency pole of the uncompensated amplifier "moving" to an even lower frequency to become the dominant pole, and the higher-frequency pole of the uncompensated amplifier "moving" to a higher frequency.
Other methods
Some other compensation methods are: lead compensation, lead–lag compensation and feed-forward compensation.
Lead compensation. Whereas dominant pole compensation places or moves poles in the open loop response, lead compensation places a zero[3] in the open loop response to cancel one of the existing poles.
Lead–lag compensation places both a zero and a pole in the open loop response, with the pole usually being at an open loop gain of less than one.
Feed-forward compensation uses a capacitor to bypass a stage in the amplifier at high frequencies, thereby eliminating the pole that stage creates.
The purpose of these three methods is to allow greater open loop bandwidth while still maintaining amplifier closed loop stability. They are often used to compensate high gain, wide bandwidth amplifiers.

The Dominant Pole approximation

Reduction of a second order system to first order

Consider a second order system with a transfer function that is reduced to first order.

This assumes that a>>b, or that the pole at b is dominant. The coefficient "a" remains in the denominator so that the DC gain (which is also the final value of the output with a unit step input) remains unchanged. Recall that the DC gain is G(0).

The graph below shows the exact response (red) and the dominant pole approximation (green) for a=8 and b=1. Following the graph is Matlab code in which you can set a with b=1 to see how accurate the dominant pole approximation is.


Higher Order

The dominant pole approximation can also be applied to higher order systems. Here we consider a third order system with one real root, and a pair of complex conjugate roots.

In this case the test for the dominant pole compare "a" against "zwn". This is because "zwn" is the real part of the complex conjugate root (we only compare the real parts of the roots when determining dominance because it is the real part that determines how fast the response decreases). Note that the DC gain of the exact system and the two approximate systems are equal.

In the examples and Matlab code below, the second order pole has zeta=0.4 and wn=1 (which yields roots with a real part of 0.4 and an imaginary part of +/-0.92j). There are three graphs. In the first graph a=0.1 (the real pole dominates), in the second graph a=4 (the complex conjugate poles dominate) and in the third graph a=0.4 (neither dominates and the response is obviously more complicated than a simple second order response). In all three graphs the exact response is in red, the approximate response in which the first order pole dominates is in green, and the approximate response in which the second order pole dominates is in blue.



The decibel (dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity) relative to a specified or implied reference level. Since it expresses a ratio of two quantities with the same unit, it is a dimensionless unit. A decibel is one tenth of a bel, a seldom-used unit.
The decibel is widely known as a measure of sound pressure level, but is also used for a wide variety of other measurements in science and engineering (particularly acoustics, electronics, and control theory) and other disciplines. It confers a number of advantages, such as the ability to conveniently represent very large or small numbers, a logarithmic scaling that roughly corresponds to the human perception of sound and light, and the ability to carry out multiplication of ratios by simple addition and subtraction.
The decibel symbol is often qualified with a suffix, which indicates which reference quantity or frequency weighting function has been used. For example, "dBm" indicates that the reference quantity is one milliwatt, while "dBu" is referenced to 0.775 volts RMS.
The definitions of the decibel and bel use base-10 logarithms. For a similar unit using natural logarithms to base e, see neper.


The decibel originates from methods used to quantify reductions in audio levels in telephone circuits. These losses were originally measured in units of Miles of Standard Cable (MSC), where 1 MSC corresponded to the loss of power over a 1 mile (approximately 1.6 km) length of standard telephone cable at a frequency of 5000 radians per second (795.8 Hz) and roughly matched the smallest attenuation detectable to an average listener. Standard telephone cable was defined as "a cable having uniformly distributed resistances of 88 ohms per loop mile and uniformly distributed shunt capacitance of .054 microfarad per mile" (approximately 19 gauge).[citation needed]
The transmission unit or TU was devised by engineers of the Bell Telephone Laboratories in the 1920s to replace the MSC. 1 TU was defined as ten times the base-10 logarithm of the ratio of measured power to reference power.[2] The definitions were conveniently chosen such that 1 TU approximately equalled 1 MSC (specifically, 1.056 TU = 1 MSC).[3] Eventually, international standards bodies adopted the base-10 logarithm of the power ratio as a standard unit, which was named the "bel" in honor of the Bell System's founder and telecommunications pioneer Alexander Graham Bell. The bel was a factor of ten larger than the TU, such that 1 TU equalled 1 decibel.[4] In many situations, the bel proved inconveniently large, so the decibel has become more common.
In April 2003, the International Committee for Weights and Measures (CIPM) considered a recommendation for the decibel's inclusion in the SI system, but decided not to adopt the decibel as an SI unit.[5] However, the decibel is recognized by other international bodies such as the International Electrotechnical Commission (IEC).[6] The IEC permits the use of the decibel with field quantities as well as power and this recommendation is followed by many national standards bodies, such as NIST, which justifies the use of the decibel for voltage ratios.


The use of the decibel has a number of merits:
The decibel's logarithmic nature means that a very large range of ratios can be represented by a convenient number, in a similar manner to scientific notation. This allows one to clearly visualize huge changes of some quantity. (See Bode Plot and half logarithm graph.)
The mathematical properties of logarithms mean that the overall decibel gain of a multi-component system (such as consecutive amplifiers) can be calculated simply by summing the decibel gains of the individual components, rather than needing to multiply amplification factors. Essentially this is because log(A × B × C × ...) = log(A) + log(B) + log(C) + ...
The human perception of, for example, sound or light, is, roughly speaking, such that a doubling of actual intensity causes perceived intensity to always increase by the same amount, irrespective of the original level. The decibel's logarithmic scale, in which a doubling of power or intensity always causes an increase of approximately 3 dB, corresponds to this perception.

The decibel is commonly used in acoustics to quantify sound levels relative to some 0 dB reference. The reference level is typically set at the threshold of perception of an average human and there are common comparisons used to illustrate different levels of sound pressure. As with other decibel figures, normally the ratio expressed is a power ratio (rather than a pressure ratio).
The human ear has a large dynamic range in audio perception. The ratio of the sound pressure that causes permanent damage during short exposure to the quietest sound that the ear can hear is above a trillion. Such large measurement ranges are conveniently expressed in logarithmic units: the base-10 logarithm of one trillion (1012) is 12, which is expressed as an audio level of 120 dB. Since the human ear is not equally sensitive to all sound frequencies, noise levels at maximum human sensitivity — for example, the higher harmonics of middle A (between 2 and 4 kHz) — are factored more heavily into some measurements using frequency weighting.
Further information: Examples of sound pressure and sound pressure levels

In electronics, the decibel is often used to express power or amplitude ratios (gains), in preference to arithmetic ratios or percentages. One advantage is that the total decibel gain of a series of components (such as amplifiers and attenuators) can be calculated simply by summing the decibel gains of the individual components. Similarly, in telecommunications, decibels are used to account for the gains and losses of a signal from a transmitter to a receiver through some medium (free space, wave guides, coax, fiber optics, etc.) using a link budget.
The decibel unit can also be combined with a suffix to create an absolute unit of electric power. For example, it can be combined with "m" for "milliwatt" to produce the "dBm". Zero dBm is the power level corresponding to a power of one milliwatt, and 1 dBm is one decibel greater (about 1.259 mW).
In professional audio, a popular unit is the dBu (see below for all the units). The "u" stands for "unloaded", and was probably chosen to be similar to lowercase "v", as dBv was the older name for the same thing. It was changed to avoid confusion with dBV. This unit (dBu) is an RMS measurement of voltage which uses as its reference 0.775 VRMS. Chosen for historical reasons, it is the voltage level which delivers 1 mW of power in a 600 ohm resistor, which used to be the standard reference impedance in telephone audio circuits.
The bel is used to represent noise power levels in hard drive specifications. It shares the same symbol (B) as the byte.

In an optical link, if a known amount of optical power, in dBm (referenced to 1 mW), is launched into a fiber, and the losses, in dB (decibels), of each electronic component (e.g., connectors, splices, and lengths of fiber) are known, the overall link loss may be quickly calculated by addition and subtraction of decibel quantities.
In spectrometry and optics, the blocking unit used to measure optical density is equivalent to −1 B. In astronomy, the apparent magnitude measures the brightness of a star logarithmically, since, just as the ear responds logarithmically to acoustic power, the eye responds logarithmically to brightness; however astronomical magnitudes reverse the sign with respect to the bel, so that the brightest stars have the lowest magnitudes, and the magnitude increases for fainter stars.
]Video and digital imaging
In connection with digital and video image sensors, decibels generally represent ratios of video voltages or digitized light levels, using 20 log of the ratio, even when the represented optical power is directly proportional to the voltage or level, not to its square. Thus, a camera signal-to-noise ratio of 60 dB represents a power ratio of 1000:1 between signal power and noise power, not 1,000,000:1

Common reference levels and corresponding units

Absolute and relative decibel measurements
Although decibel measurements are always relative to a reference level, if the numerical value of that reference is explicitly and exactly stated, then the decibel measurement is called an "absolute" measurement, in the sense that the exact value of the measured quantity can be recovered using the formula given earlier. For example, since dBm indicates power measurement relative to 1 milliwatt,
0 dBm means no change from 1 mW. Thus, 0 dBm is the power level corresponding to a power of exactly 1 mW.
3 dBm means 3 dB greater than 0 dBm. Thus, 3 dBm is the power level corresponding to 103/10 × 1 mW, or approximately 2 mW.
−6 dBm means 6 dB less than 0 dBm. Thus, −6 dBm is the power level corresponding to 10−6/10 × 1 mW, or approximately 250 μW (0.25 mW).
If the numerical value of the reference is not explicitly stated, as in the dB gain of an amplifier, then the decibel measurement is purely relative. The practice of attaching a suffix to the basic dB unit, forming compound units such as dBm, dBu, dBA, etc, is not permitted by SI However, outside of documents adhering to SI units, the practice is very common as illustrated by the following examples.

Absolute measurements
Electric power
dBm or dBmW
dB(1 mW) — power measurement relative to 1 milliwatt. XdBm = XdBW + 30.
dB(1 W) — similar to dBm, except the reference level is 1 watt. 0 dBW = +30 dBm; −30 dBW = 0 dBm; XdBW = XdBm − 30.
Since the decibel is defined with respect to power, not amplitude, conversions of voltage ratios to decibels must square the amplitude, as discussed above.

dB(1 mVRMS) — voltage relative to 1 millivolt, regardless of impedance. Widely used in cable television networks, where the nominal strength of a single TV signal at the receiver terminals is about 0 dBmV. Cable TV uses 75 Ω coaxial cable, so 0 dBmV corresponds to −78.75 dBW (-48.75 dBm) or ~13 nW.
dBμV or dBuV
dB(1 μVRMS) — voltage relative to 1 microvolt. Widely used in television and aerial amplifier specifications. 60 dBμV = 0 dBmV.
Probably the most common usage of "decibels" in reference to sound loudness is dB SPL, referenced to the nominal threshold of human hearing:[11]
dB (sound pressure level) — for sound in air and other gases, relative to 20 micropascals (μPa) = 2×10−5 Pa, the quietest sound a human can hear. This is roughly the sound of a mosquito flying 3 metres away. This is often abbreviated to just "dB", which gives some the erroneous notion that "dB" is an absolute unit by itself. For sound in water and other liquids, a reference pressure of 1 μPa is used.[12]
dB — relative to 1 Pa, often used in telecommunications.
dB sound intensity level — relative to 10−12 W/m2, which is roughly the threshold of human hearing in air.
dB sound power level — relative to 10−12 W.
dB(A), dB(B), and dB(C)
These symbols are often used to denote the use of different weighting filters, used to approximate the human ear's response to sound, although the measurement is still in dB (SPL). These measurements usually refer to noise and noisome effects on humans and animals, and are in widespread use in the industry with regard to noise control issues, regulations and environmental standards. Other variations that may be seen are dBA or dBA. According to ANSI standards, the preferred usage is to write LA = x dB. Nevertheless, the units dBA and dB(A) are still commonly used as a shorthand for A-weighted measurements. Compare dBc, used in telecommunications.
dB HL or dB hearing level is used in audiograms as a measure of hearing loss. The reference level varies with frequency according to a minimum audibility curve as defined in ANSI and other standards, such that the resulting audiogram shows deviation from what is regarded as 'normal' hearing.[citation needed]
dB Q is sometimes used to denote weighted noise level, commonly using the ITU-R 468 noise weighting[citation needed]
dB(Z) - energy of reflectivity (weather radar), or the amount of transmitted power returned to the radar receiver. Values above 15-20 dBZ usually indicate falling precipitation.
dBsm - decibel (referenced to one) square meter, measure of reflected energy from a target compared to the RCS of a smooth perfectly conducting sphere at least several wavelengths in size with a cross-sectional area of 1 square meter. "Stealth" aircraft and insects have negative values of dBsm, large flat plates or non-stealthy aircraft have positive values.[14]
Radio power, energy, and field strength
dBc — relative to carrier — in telecommunications, this indicates the relative levels of noise or sideband peak power, compared with the carrier power. Compare dBC, used in acoustics.
dB(J) — energy relative to 1 joule. 1 joule = 1 watt per hertz, so power spectral density can be expressed in dBJ.
dB(mW) — power relative to 1 milliwatt. When used in audio work the milliwatt is referenced to a 600 ohm load, with the resultant voltage being 0.775 volts. When used in the 2-way radio field, the dB is referenced to a 50 ohm load, with the resultant voltage being 0.224 volts. There are times when spec sheets may show the voltage & power level e.g. -120 dBm = 0.224 microvolts.
dBμV/m or dBuV/m
dB(μV/m) — electric field strength relative to 1 microvolt per meter.
dB(fW) — power relative to 1 femtowatt.
dB(W) — power relative to 1 watt.
dB(kW) — power relative to 1 kilowatt.
Antenna measurements
dB(isotropic) — the forward gain of an antenna compared with the hypothetical isotropic antenna, which uniformly distributes energy in all directions. Linear polarization of the EM field is assumed unless noted otherwise.
dB(dipole) — the forward gain of an antenna compared with a half-wave dipole antenna. 0 dBd = 2.15 dBi
dB(isotropic circular) — the forward gain of an antenna compared to a circularly polarized isotropic antenna. There is no fixed conversion rule between dBiC and dBi, as it depends on the receiving antenna and the field polarization.
dB(quarterwave) — the forward gain of an antenna compared to a quarter wavelength whip. Rarely used, except in some marketing material. 0 dBq = -0.85 dBi
Other measurements
dBFS or dBfs
dB(full scale) — the amplitude of a signal (usually audio) compared with the maximum which a device can handle before clipping occurs. In digital systems, 0 dBFS (peak) would equal the highest level (number) the processor is capable of representing. Measured values are always negative or zero, since they are less than the maximum or full-scale. Full-scale is typically defined as the power level of a full-scale sinusoid, though some systems will have extra headroom for peaks above the nominal full scale.
dB(hertz) — bandwidth relative to 1 Hz. E.g., 20 dB-Hz corresponds to a bandwidth of 100 Hz. Commonly used in link budget calculations. Also used in carrier-to-noise-density ratio (not to be confused with carrier-to-noise ratio, in dB).
dBov or dBO
dB(overload) — the amplitude of a signal (usually audio) compared with the maximum which a device can handle before clipping occurs. Similar to dBFS, but also applicable to analog systems.
dB(relative) — simply a relative difference from something else, which is made apparent in context. The difference of a filter's response to nominal levels, for instance.
dB above reference noise. See also dBrnC.

Lenny Z Perez M

The Miller’s theorem

The Miller’s theorem establishes that in a linear circuit, if there exists a branch with impedance Z, connecting two nodes with nodal voltages V1and V2, we can replace this branch by two branches connecting the corresponding nodes to ground by impedances respectively Z / (1-K) and KZ / (K-1), where K = V2 / V1.

In fact, if we use the equivalent two-port network technique to replace the two-port represented on the right to its equivalent, it results successively:

And, according to the source absorption theorem, we get the following:

As all the linear circuit theorems, the Miller’s theorem also has a dual form:

Miller's dual theorem

If there is a branch in a circuit with impedance Z connecting a node, where two currents I1 and I2 converge, to ground, we can replace this branch by two conducting the referred currents, with impedances respectively equal to (1+ a) Z and (1+ a) Z / a, where a = I2 / I1.

In fact, replacing the two-port networkby its equivalent, as in the figure,

it results the circuit on the left in the next figure and then, applying the source absorption theorem, the circuit on the right.

Miller's theorem applies to the process of creating equivalent circuits. This general circuit
theorem is particularly useful in the high-frequency analysis of certain transistor amplifiers at
high frequencies.

The Miller Theorem (and "Effect")

Suppose that we have two networks separated by a bridging element Y. The equivalent circuits shown above represent particular important examples of such a situation

Further, suppose that we can establish the following "gain relationship" by independent means:

and, thus, we may write

If everything else remains unchanged, this bridged configuration can be replaced by a configuration of "decoupled" networks as follows:

where by equivalence we must have

The Classic Solution to the "Miller Effect"
The Cascode Amplifier

Lenny Z Perez M