Analog Electronics
Analog Electronics Notation
The following notation, summarized in the table below, is used throughout. A lowercase letter with an uppercase subscript, such as iB and vBE, indicates a total instantaneous value. An uppercase letter with an uppercase subscript, such as IB and VBE, indicates a dc quantity. A lowercase letter with a lowercase subscript, such as ib and vbe, indicates an instantaneous value of a time-varying signal. Finally, an uppercase letter with a lowercase subscript, such as Ib and Vbe, indicates a phasor quantity.
| Variable | Meaning |
|---|---|
| iB, vBE | Total instantaneous values |
| IB, IVB | dc values |
| ib, vbe | Total instantaneous ac values |
| Ib, Vbe | Phasor values |
Intrinsic Semiconductors
An atom is composed of a nucleus, which contains positively charged protons and neutral neutrons, and negatively charged electrons that, in the classical sense, orbit the nucleus. The electrons are distributed in various shells
at different distances from the nucleus, and electron energy increases as shell radius increases. Electrons in the outermost shell are called valence electrons, and the chemical activity of a material is determined primarily by the number of such electrons.
Elements in the periodic table can be grouped according to the number of valence electrons. Silicon (Si) and germanium (Ge) are in group IV and are elemental semiconductors. In contrast, gallium arsenide is a group III–V compound semiconductor. The elements in group III and group V are also important in semiconductors.
| Elemental Semiconductors | Compound Semiconductors | ||
|---|---|---|---|
| Si | Silicon | GaAs | Gallium arsenide |
| Ge | Germanium | GaP | Gallium phosphide |
| AlP | Aluminum phosphide | ||
| AlAs | Aluminum arsenide | ||
| InP | Indium phosphide | ||
At T = 0 K, each electron is in its lowest possible energy state, so each covalent bonding position is filled. If a small electric field is applied to this material, the electrons will not move, because they will still be bound to their individual atoms. Therefore, at T = 0 K, silicon is an insulator; that is, no charge flows through it.
When silicon atoms come together to form a crystal, the electrons occupy particular allowed energy bands. At T = 0 K, all valence electrons occupy the valence energy band. If the temperature increases, the valence electrons may gain thermal energy. Any such electron may gain enough thermal energy to break the covalent bond and move away from its original position. In order to break the covalent bond, the valence electron must gain a minimum energy, Eg, called the bandgap energy. The electrons that gain this minimum energy now exist in the conduction band and are said to be free electrons. These free electrons in the conduction band can move throughout the crystal. The net flow of electrons in the conduction band generates a current.
The energy Eν is the maximum energy of the valence energy band and the energy Ec is the minimum energy of the conduction energy band. The bandgap energy Eg is the difference between Ec and Eν, and the region between these two energies is called the forbidden bandgap. Electrons cannot exist within the forbidden bandgap. The process of an electron from the valence band gaining enough energy and moving into the conduction band is called generation.
Materials that have large bandgap energies, in the range of 3 to 6 electron–volts (eV), are insulators because, at room temperature, essentially no free electrons exist in the conduction band. In contrast, materials that contain very large numbers of free electrons at room temperature are conductors. In a semiconductor, the bandgap energy is on the order of 1 eV.
A fundamental relationship between the electron and hole concentrations in a semiconductor in thermal equilibrium is given by
no po = ni2
where no is the thermal equilibrium concentration of free electrons, po is the thermal equilibrium concentration of holes, and ni is the intrinsic carrier concentration.
In an n-type semiconductor, electrons are called the majority carrier because they far outnumber the holes, which are termed the minority carrier. In contrast, in a p-type semiconductor, the holes are the majority carrier and the electrons are the minority carrier.
Drift and Diffusion Currents
We've described the creation of negatively charged electrons and positively charged holes in the semiconductor. If these charged particles move, a current is generated. These charged electrons and holes are simply referred to as carriers.
The two basic processes which cause electrons and holes to move in a semiconductor are: (a) drift, which is the movement caused by electric fields, and (b) diffusion, which is the flow caused by variations in the concentration, that is, concentration gradients. Such gradients can be caused by a nonhomogeneous doping distribution, or by the injection of a quantity of electrons or holes into a region, using methods to be discussed later.
Drift Current Density
To understand drift, assume an electric field is applied to a semiconductor. The field produces a force that acts on free electrons and holes, which then experience a net drift velocity and net movement. Consider an n-type semiconductor with a large number of free electrons. An electric field E applied in one direction produces a force on the electrons in the opposite direction, because of the electrons' negative charge. The electrons acquire a drift velocity v>dn (in cm/s) which can be written as
vdn = − µnE
where µn is a constant called the electron mobility and has units of cm2/V–s. For low-doped silicon, the value of µn is typically 1350 cm2/V–s. The mobility can be thought of as a parameter indicating how well an electron can move in a semiconductor. The negative sign in the equation above indicates that the electron drift velocity is opposite to that of the applied electric field. The electron drift produces a drift current density Jn (A/cm2) given by
Jn = − e n vdn = − e n (− µn E) = +e n µn E
where n is the electron concentration (#/cm3) and e, in this context, is the magnitude of the electronic charge. The conventional drift current is in the opposite direction from the flow of negative charge, which means that the drift current in an n-type semiconductor is in the same direction as the applied electric field.
Next consider a p-type semiconductor with a large number of holes. An electric field E applied in one direction produces a force on the holes in the same direction, because of the positive charge on the holes. The holes acquire a drift velocity vdp (in cm/s), which can be written as
vdp = + µp E
where µp is a constant called the hole mobility, and again has units of cm2/V–s. For low-doped silicon, the value of µp is typically 480 cm2/V–s, which is less than half the value of the electron mobility. The positive sign in the equation above indicates that the hole drift velocity is in the same direction as the applied electric field. The hole drift produces a drift current density Jp (A/cm2) given by
Jp = + e p vdp = +e p (+µp E) = + e p µp E
where p is the hole concentration (#/cm3) and e is again the magnitude of the electronic charge. The conventional drift current is in the same direction as the flow of positive charge, which means that the drift current in a p-type material is also in the same direction as the applied electric field.
Since a semiconductor contains both electrons and holes, the total drift current density is the sum of the electron and hole components. The total drift current density is then written as
J = e n µn E + epµp E = σ E = (1 / p) E
where
σ = e n µn + e p µp
and where σ is the conductivity of the semiconductor in (ohm–cm)−1 and where ρ = 1/σ is the resistivity of the semiconductor in (ohm–cm). The conductivity is related to the concentration of electrons and holes. If the electric field is the result of applying a voltage to the semiconductor, then foregoing equation becomes a linear relationship between current and voltage and is one form of Ohm's law.
From the equation above, we see that the conductivity can be changed from strongly n-type, n >> p, by donor impurity doping to strongly p-type, p >> n, by acceptor impurity doping. Being able to control the conductivity of a semiconductor by selective doping is what enables us to fabricate the variety of electronic devices that are available.
Two factors need to be mentioned concerning drift velocity and mobility. The foregoing equations seem to imply that the carrier drift velocities are linear functions of the applied electric field. This is true for relatively small electric fields. As the electric field increases, the carrier drift velocities will reach a maximum value of approximately 107 cm/s. Any further increase in electric field will not produce an increase in drift velocity. This phenomenon is called drift velocity saturation.
The mobility values are actually functions of donor and/or acceptor impurity concentrations. As the impurity concentration increases, the mobility values will decrease. This effect then means that the conductivity is not a linear function of impurity doping.
These two factors are important in the design of semiconductor devices, but will not be considered in detail in this page.
Diffusion Current Density
In the diffusion process, particles flow from a region of high concentration to a region of lower concentration. This is a statistical phenomenon related to kinetic theory. To explain, the electrons and holes in a semiconductor are in continuous motion, with an average speed determined by the temperature, and with the directions randomized by interactions with the lattice atoms. Statistically, we can assume that, at any particular instant, approximately half of the particles in the high-concentration region are moving away from that region toward the lower-concentration region. We can also assume that, at the same time, approximately half of the particles in the lower-concentration region are moving toward the high-concentration region. However, by definition, there are fewer particles in the lower-concentration region than there are in the high-concentration region. Therefore, the net result is a flow of particles away from the high-concentration region and toward the lower-concentration region. This is the basic diffusion process.
For example, consider an electron concentration that varies as a function of distance x. The diffusion of electrons from a high-concentration region to a low-concentration region produces a flow of electrons in the negative x direction. Since electrons are negatively charged, the conventional current direction is in the positive x direction.
The diffusion current density due to the diffusion of electrons can be written as (for one dimension)
Jn = e Dn(dn/dx)
where e, in this context, is the magnitude of the electronic charge, dn/dx is the gradient of the electron concentration, and Dn is the electron diffusion coefficient.
[T]he hole concentration is [understood to be] a function of distance. The diffusion of holes from a high-concentration region to a low-concentration region produces a flow of holes in the negative x direction. (Conventional current is in the direction of the flow of positive charge.)
The diffusion current density due to the diffusion of holes can be written as (for one dimension)
Jp = − e Dp (dp/dx)
where e is still the magnitude of the electronic charge, dp/dx is the gradient of the hole concentration, and Dp is the hole diffusion coefficient. Note the change in sign between the two diffusion current equations. This change in sign is due to the difference in sign of the electronic charge between the negatively charged electron and the positively charged hole.
The mobility values in the drift current equations and the diffusion coefficient values in the diffusion current equations are not independent quantities. They are related by the Einstein relation, which is
Dn / µn = Dp / µp = k T / e ~ 0.026 V (approximately)
at room temperature.
The total current density is the sum of the drift and diffusion components. Fortunately, in most cases only one component dominates the current at any one time in a given region of a semiconductor.
Excess Carriers
Up to this point, we have assumed that the semiconductor is in thermal equilibrium. In the discussion of drift and diffusion currents, we implicitly assumed that equilibrium was not significantly disturbed. Yet, when a voltage is applied to, or a current exists in, a semiconductor device, the semiconductor is really not in equilibrium. In this section, we will discuss the behavior of nonequilibrium electron and hole concentrations.
Valence electrons may acquire sufficient energy to break the covalent bond and become free electrons if they interact with high-energy photons incident on the semiconductor. When this occurs, both an electron and a hole are produced, thus generating an electron–hole pair. These additional electrons and holes are called excess electrons and excess holes.
When these excess electrons and holes are created, the concentrations of free electrons and holes increase above their thermal equilibrium values. This may be represented by
n = no + δn
and
p = po + δp
where no and po are the thermal equilibrium concentrations of electrons and holes, and δn and δp are the excess electron and hole concentrations.
If the semiconductor is in a steady-state condition, the creation of excess electrons and holes will not cause the carrier concentration to increase indefinitely, because a free electron may recombine with a hole, in a process called electron–hole recombination. Both the free electron and the hole disappear causing the excess concentration to reach a steady-state value. The mean time over which an excess electron and hole exist before recombination is called the excess carrier lifetime.
Excess carriers are involved in the current mechanisms of, for example, solar cells and photodiodes.
The pn Junctions
In the preceding sections, we looked at characteristics of semiconductor materials. The real power of semiconductor electronics occurs when p- and n-regions are directly adjacent to each other, forming a pn junction. One important concept to remember is that in most integrated circuit applications, the entire semiconductor material is a single crystal, with one region doped to be p-type and the adjacent region doped to be n-type.
The Equilibrium pn Junction
The interface at x = 0 is called the metallurgical junction. A large density gradient in both the hole and electron concentrations occurs across this junction. Initially, then, there is a diffusion of holes from the p-region into the n-region, and a diffusion of electrons from the n-region into the p-region. The flow of holes from the p-region uncovers negatively charged acceptor ions, and the flow of electrons from the n-region uncovers positively charged donor ions. This action creates a charge separation, which sets up an electric field oriented in the direction from the positive charge to the negative charge.
If no voltage is applied to the pn junction, the diffusion of holes and electrons must eventually cease. The direction of the induced electric field will cause the resulting force to repel the diffusion of holes from the p-region and the diffusion of electrons from the n-region. Thermal equilibrium occurs when the force produced by the electric field and the force
produced by the density gradient exactly balance.
The positively charged region and the negatively charged region comprise the space-charge region, or depletion region, of the pn junction, in which there are essentially no mobile electrons or holes. Because of the electric field in the space-charge region, there is a potential difference across that region. This potential difference is called the built-in potential barrier, or built-in voltage, and is given by
Vbi = (k T / e) ln (Na Nd / ni2) = VT ln (Na Nd / ni2)
where VT ≡ kT /e, k = Boltzmann's constant, T = absolute temperature, e = the magnitude of the electronic charge, and Na and Nd are the net acceptor and donor concentrations in the p- and n-regions, respectively. The parameter VT is called the thermal voltage and is approximately VT = 0.026 V at room temperature, T = 300 K.
The potential difference, or built-in potential barrier, across the space-charge region cannot be measured by a voltmeter because new potential barriers form between the probes of the voltmeter and the semiconductor, canceling the effects of Vbi. In essence, Vbi maintains equilibrium, so no current is produced by this voltage. However, the magnitude of Vbi becomes important when we apply a forward-bias voltage, as discussed later on.
Reverse-Biased pn Junction
Assume a positive voltage is applied to the n-region of a pn junction. The applied voltage VR induces an applied electric field, EA, in the semiconductor. The direction of this applied field is the same as that of the E-field in the space-charge region. The magnitude of the electric field in the space-charge region increases above the thermal equilibrium value. This increased electric field holds back the holes in the p-region and the electrons in the n-region, so there is essentially no current across the pn junction. By definition, this applied voltage polarity is called reverse bias.
When the electric field in the space-charge region increases, the number of positive and negative charges must increase. If the doping concentrations are not changed, the increase in the fixed charge can only occur if the width W of the space-charge region increases. Therefore, with an increasing reverse-bias voltage VR, space-charge width W also increases.
Because of the additional positive and negative charges induced in the space-charge region with an increase in reverse-bias voltage, a capacitance is associated with the pn junction when a reverse-bias voltage is applied. This junction capacitance, or depletion layer capacitance, can be written in the form
Cj = Cj0 ( 1 + VR / Vbi ) - 1/2
where Cjo is the junction capacitance at zero applied voltage. The junction capacitance will affect the switching characteristics of the pn junction. The voltage across a capacitance cannot change instantaneously, so changes in voltages in circuits containing pn junctions will not occur instantaneously.
The capacitance–voltage characteristics can make the pn junction useful for electrically tunable resonant circuits. Junctions fabricated specifically for this purpose are called varactor diodes. Varactor diodes can be used in electrically tunable oscillators, such as a Hartley oscillator.
As implied in the previous section, the magnitude of the electric field in the space-charge region increases as the reverse-bias voltage increases, and the maximum electric field occurs at the metallurgical junction. However, neither the electric field in the space-charge region nor the applied reverse-bias voltage can increase indefinitely because at some point, breakdown will occur and a large reverse bias current will be generated. This concept will be described further on.
Forward-Biased pn Junction
We have seen that the n-region contains many more free electrons than the p-region; similarly, the p-region contains many more holes than the n-region. With zero applied voltage, the built-in potential barrier prevents these majority carriers from diffusing across the space-charge region; thus, the barrier maintains equilibrium between the carrier distributions on either side of the pn junction.
If a positive voltage vD is applied to the p-region, the potential barrier decreases. The electric fields in the space-charge region are very large compared to those in the remainder of the p- and n-regions, so essentially all of the applied voltage exists across the pn junction region. The applied electric field, EA, induced by the applied voltage is in the opposite direction from that of the thermal equilibrium space-charge E-field. However, the net electric field is always from the n- to the p-region. The net result is that the electric field in the space-charge region is lower than the equilibrium value. This upsets the delicate balance between diffusion and the E-field force. Majority carrier electrons from the n-region diffuse into the p-region, and majority carrier holes from the p-region diffuse into the n-region. The process continues as long as the voltage vD is applied, thus creating a current in the pn junction. This process would be analogous to lowering a dam wall slightly. A slight drop in the wall height can send a large amount of water (current) over the barrier.
This applied voltage polarity (i.e., bias) is known as forward bias. The forward-bias voltage vD must always be less than the built-in potential barrier Vbi.
As the majority carriers cross into the opposite regions, they become minority carriers in those regions, causing the minority carrier concentrations to increase. . These excess minority carriers diffuse into the neutral n- and p-regions, where they recombine with majority carriers, thus establishing a steady-state condition.
Ideal Current–Voltage Relationship
an applied voltage results in a gradient in the minority carrier concentrations, which in turn causes diffusion currents. The theoretical relationship between the voltage and the current in the pn junction is given by
The parameter IS is the reverse-bias saturation current. For silicon pn junctions, typical values of IS are in the range of 10−18 to 10−12 A. The actual value depends on the doping concentrations and is also proportional to the cross-sectional area of the junction. The parameter VT is the thermal voltage, and is approximately VT = 0.026 V at room temperature. The parameter n is usually called the emission coefficient or ideality factor, and its value is in the range 1 ≤ n ≤ 2.
The emission coefficient n takes into account any recombination of electrons and holes in the space-charge region. At very low current levels, recombination may be a significant factor and the value of n may be close to 2. At higher current levels, recombination is less a factor, and the value of n will be 1. Unless otherwise stated, we will assume the emission coefficient is n = 1.
This pn junction, with nonlinear rectifying current characteristics, is called a pn junction diode.
PN Junction Diode
For a forward-bias voltage, the current is an exponential function of voltage. With only a small change in the forward-bias voltage, the corresponding forward-bias current increases by orders of magnitude. For a forward-bias voltage vD > +0.1 V, the (−1) term in the equation above can be neglected. In the reverse-bias direction, the current is almost zero.
The diode can be thought of and used as a voltage controlled switch that is off
for a reverse-bias voltage and on
for a forward-bias voltage. In the forward-bias or on
state, a relatively large current is produced by a fairly small applied voltage; in the reverse-bias, or off
state, only a very small current is created.
When a diode is reverse-biased by at least 0.1 V, the diode current is iD = −IS. The current is in the reverse direction and is a constant, hence the name reverse-bias saturation current. Real diodes, however, exhibit reverse-bias currents that are considerably larger than IS. This additional current is called a generation current and is due to electrons and holes being generated within the space-charge region. Whereas a typical value of IS may be 10−14 A, a typical value of reverse-bias current may be 10−9 A or 1 nA. Even though this current is much larger than IS, it is still small and negligible in most cases.
Resistors
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Resistor Values
In 1952, the IEC (International Electrotechnical Commission) decided to define the resistance and tolerance values into a norm, to ease the mass manufacturing of resistors. These are referred to as "preferred values" or "E-series", and they are published in standard IEC 60063:1963. These standard values are also valid for other components like capacitors, inductors and Zener diodes.
The standardization of resistor values serves several important purposes. When manufacturers produce resistors with different resistance values, these end up approximately equally spaced on a logarithmic scale. This helps the supplier to limit the number of different values that have to be produced or kept in stock. By using standard values, resistors from different manufacturers are compatible for the same design, which is favorable for the electrical engineer.
Aside from the preferred values, many other standards related to resistors exist. Example include the standard sizes for resistors and the marking of resistors with color codes or numerical codes. Power ratings of resistors are not defined in a norm, and, therefore, often deviate from the above described series.
(From https://eepower.com/resistor-guide/resistor-standards-and-codes/resistor-values/)
Preferred Values or E-series
To understand the E-series sets of preferred values, we will examine the E12 in detail. E12 means that every decade (0.1-1.0, 1-10, 10-100, etc.) is divided in 12 steps on a logarithmic scale. The size of every step is equal to:
10(1/12) = 1.21
Thus, every value is 21% or 1.21 times higher than the previous value in the series, rounded to whole numbers. Because of this, all resistors with a tolerance of 10% overlap. The series looks as follows: 1– 1.2 – 1.5 – 1.8 – 2.2 – 2.7 – 3.3 – 3.9 – 4.7 – 5.6 – 6.8 – 8.2 – 10 etc.
All of these values can be powers of ten (1.2 – 12 – 120, etc.).
While the E12 series is the most common, other series are also available. It is a good practice to specify resistors from a low series when tolerance requirements are not high.
The most common series are:
| 6 | 20% tolerance |
| 12 | 10% |
| 24 | 5% (also available with 1%) |
| 48 | 2% |
| 96 | 1% |
| 192 | 0.5% (also used for resistors with 0.25% and 0.1%). |
Some other common series:
| E6 series (tolerance 20%) | |||||
|---|---|---|---|---|---|
| 10 | 15 | 22 | 33 | 47 | 68 |
| E12 series (tolerance 10%) | |||||
|---|---|---|---|---|---|
| 10 | 12 | 15 | 18 | 22 | 27 |
| 33 | 39 | 47 | 56 | 68 | 82 |
| E24 series (tolerance 5% and 1%) | |||||
|---|---|---|---|---|---|
| 10 | 11 | 12 | 13 | 15 | 16 |
| 18 | 20 | 22 | 24 | 27 | 30 |
| 33 | 36 | 39 | 43 | 47 | 51 |
| 56 | 62 | 68 | 75 | 82 | 91 |
Resistor Colour Codes
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Capacitors
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The World is Analog
The world is fundamentally analog, and so are the ways we interact with it. Even in the digital age, we still need analog ICs.
Everything is going digital: cell phones, televisions, video disks, hearing aids, motor controls, audio amplifiers, toys, printers, and what have you. Analog design is obsolete or will be shortly, or so most people think.
Imminent death has been predicted for analog since the advent of the PC. But analog is still here, and analog integrated circuits (ICs) have, in fact, been growing at almost exactly the same rate as digital ones. A digital video disk (DVD) player has more analog content than the VCR—which is analog—ever did.
The explanation is rather simple: the world is fundamentally analog. Hearing is analog. Vision, taste, touch, and smell—all analog. So are lifting and walking. Generators, motors, loudspeakers, microphones, solenoids, batteries, antennas, lamps, LEDs, laser diodes, and sensors are fundamentally analog components.
The Digital Revolution is Also Analog
The digital revolution is constructed on top of an analog reality. This fact simply won't go away. Somewhere, somehow, you have to get into and out of the digital system and connect to the real world.
Unfortunately, the predominance and glamor of digital has done harm to analog. Too few analog designers are being educated, creating a void. This leaves decisions affecting analog performance to engineers with a primarily digital background.
In integrated circuits, the relentless pressure toward faster digital speed has resulted in ever-decreasing supply voltages, which are anathema to high-performance analog design.
In a 350 nm process operating at 3.3 V, there's still enough headroom for a high-performance analog design, though 5 V would be better. At 180 nm (1.8 V), the job becomes elaborate and time-consuming, and performance starts to suffer. At 120 nm (1.2 V), analog design becomes very difficult, even with reduced performance. At 90 nm, analog design is all but impossible.
There are "mixed signal" processes that purportedly allow digital and analog circuitry on the same chip. A 180 nm process, for example, will have some devices that can work with a higher supply voltage (e.g. 3 V). Such additions are welcome, if marginal. However, the design models are often inadequate and oriented toward digital design.