電子學 Microelectronics

Mathematics   Yoshio    Sep 8th, 2023 at 8:00 PM    8    0   

電子學 Microelectronics

電子學 Microelectronics

Electronics is a general term for the field of science that involves managing electric currents through circuits. Microelectronics is one of the sub-categories of electronics. Microelectronics specifically relates to manufacture of very small electronic circuits.

Impedance

Impedance is a combination of Resistance and Reactance.

Impedance, denoted Z, is an expression of the opposition that an electronic component, circuit, or system offers to alternating and/or direct electric current. Impedance is a Complex (two-dimensional) quantity consisting of two independent scalar (one-dimensional) phenomena: Resistance and Reactance.

阻抗(Impedance)又稱電阻抗,是電路中電阻、電感、電容對交流電的阻礙作用的統稱。阻抗是一個複數,實部稱為電阻(Resistance),虛部稱為電抗(Reactance)。

Reactance

Reactance is a property that opposes a change in current and is found in both inductors and capacitors. Because it only affects changing current, reactance is specific to AC power and depends on the frequency of the current. When reactance is present, it creates a 90 degree phase shift between voltage and current, with the direction of the shift depending on whether the component is an inductor or a capacitor.

Inductor

Reactance that occurs in an inductor is known as inductive reactance. When inductive reactance is present, energy is stored in the form of a changing magnetic field, and the current waveform lags the voltage waveform by 90 degrees. Inductive reactance is caused by devices in which wire is wound circularly — such as coils (including line reactors), chokes, and transformers.

\(X_L=2\mathrm{πfL}\)

XL = inductive reactance (ohms)

f = frequency (Hz)

L = inductance (henrys)

Capacitor

Reactance that occurs in a capacitor is known as capacitive reactance. Capacitive reactance stores energy in the form of a changing electrical field and causes current to lead voltage by 90 degrees. Capacitance is created when two conducting plates are placed parallel to one another with a small distance between them, filled with a dielectric material (insulator).

\(X_C=\frac1{2\mathrm{πfC}}\)

XC = capacitive reactance (ohms)

C = capacitance (farads)


電容器兩端的電壓滯後於通過電容器的電流,兩者之間的相位差為 \(\pi/2\) ;電感器兩端的電壓超前於通過電感器的電流,兩者之間的相位差為 \(\pi/2\) 。由於電壓與電流的振幅相等,阻抗的大小為 \( 1\).

Individual Voltage Vectors for a Series RLC Circuit



Admittance

Admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the reciprocal of impedance, analogous to how conductance and resistance are defined.

Admittance, denoted Y, is an expression of the ease with which alternating current ( AC ) flows through a complex circuit or system. Admittance is a two-dimensional Complex quantity comprised of two independent Scalar phenomena: Conductance(G) and Susceptance(B).

導納(Admittance)用來描述交流電通過電路或系統時的容許程度。導納是一個複數,實部稱為電導(Conductance),虛部稱為電納(Susceptance)。


Electrical
Parameter
Measuring
Unit
Symbol Description
Voltage Volt V or E Unit of Electrical Potential
V = I × R
Current Ampere I or i Unit of Electrical Current
I = V ÷ R
Resistance Ohm R or Ω Unit of DC Resistance
R = V ÷ I
Conductance Siemens G or ℧ Reciprocal of Resistance
G = 1 ÷ R
Capacitance Farad C Unit of Capacitance
C = Q ÷ V
Charge Coulomb Q Unit of Electrical Charge
Q = C × V
Inductance Henry L or H Unit of Inductance
VL = -L(di/dt)
Power Watts W Unit of Power
P = V × I  or  I2 × R
Impedance Ohm Z Unit of AC Resistance
Z2 = R2 + X2
Frequency Hertz Hz Unit of Frequency
ƒ = 1 ÷ T
Magnetic Flux Weber Wb Unit of magnetic flux
\(Φ_B\)
Electric Flux Coulomb C Unit of electric flux
\(Φ_E\)
Magnetic Flux Density Tesla T Unit of magnetic flux density
\(Φ_B\)

時間常數 \((τ )\) Time constant

The time constant – usually denoted by the Greek letter τ (tau) – is used in physics and engineering to characterize the response to a step input of a first-order, linear time-invariant (LTI) control system. The time constant is the main characteristic unit of a first-order LTI system.

Time constant, denoted as \(τ\), is a crucial concept in electrical engineering, measuring the response time of a system to a step input. In an RC circuit, \(τ = RC\), and in an RL circuit, \(τ = L/R\).

Bipolar transistor biasing

Bipolar junction transistor BJT

The Nobel Prize in Physics 1956 was awarded jointly to three Bell Labs scientists John Bardeen (1908-1991), William Bradford Shockley (1910-1989), and Walter Houser Brattain (1902-1987). for their researches on semiconductors and their discovery of the transistor effect. They created the small semiconductor device call BJT in 1947.

John Bardeen, American physicist, was as co-winner of the Nobel Prize for Physics in both 1956 and 1972. he shared a second Nobel Prize in Physics in 1972, for the first successful explanation of superconductivity.


\( \small I_e = I_{b} + I_c = I_b + \beta I_b = (1+\beta) I_b \)






Transistor Configuration Summary Table
 
Transistor Configuration Common Base Common Collector
(Emitter Follower)
Common Emitter
  Voltage gain High Low Medium
  Current gain Low High Medium
  Power gain Low Medium High
  Input / output phase relationship 180°
  Input resistance Low High Medium
  Output resistance High Low Medium

Current mirror

Current feedback amplifier


For analog operation of a class-A amplifier, the Q-point is placed so the transistor stays in active mode (does not shift to operation in the saturation region or cut-off region) across the input signal's range. Often, the Q-point is established near the center of the active region of a transistor characteristic to allow similar signal swings in positive and negative directions.

High gain

Darlington Pair and Sziklai Pair

In electronics, a multi-transistor configuration called the Darlington configuration (commonly called as a Darlington pair) is a circuit consisting of two bipolar transistors with the emitter of one transistor connected to the base of the other, such that the current amplified by the first transistor is amplified further by the second one. It was invented in 1953 by Sidney Darlington.

\(\beta_{Darlington}=\beta_1\times\beta_2+\beta_1+\beta_2\)


Classification Amplifier
Type of Signal Type of
Configuration
Classification Frequency of
Operation
Small Signal Common Emitter Class A Amplifier Direct Current (DC)
Large Signal Common Base Class B Amplifier Audio Frequencies (AF)
  Common Collector Class AB Amplifier Radio Frequencies (RF)
    Class C Amplifier VHF, UHF and SHF
Frequencies



BJT - Voltage Divider Bias



Here the common emitter transistor configuration is biased using a voltage divider network to increase stability. The name of this biasing configuration comes from the fact that the two resistors RB1 and RB2 form a voltage or potential divider network across the supply with their center point junction connected the transistors base terminal as shown.

This voltage divider biasing configuration is the most widely used transistor biasing method. The emitter diode of the transistor is forward biased by the voltage value developed across resistor RB2. Also, voltage divider network biasing makes the transistor circuit independent of changes in beta as the biasing voltages set at the transistors base, emitter, and collector terminals are not dependant on external circuit values.

To calculate the voltage developed across resistor RB2 and therefore the voltage applied to the base terminal we simply use the voltage divider formula for resistors in series.

Generally the voltage drop across resistor RB2 is much less than for resistor RB1. Clearly the transistors base voltage VB with respect to ground, will be equal to the voltage across RB2.

The amount of biasing current flowing through resistor RB2 is generally set to 10 times the value of the required base current IB so that it is sufficiently high enough to have no effect on the voltage divider current or changes in Beta.

The goal of Transistor Biasing is to establish a known quiescent operating point, or Q-point for the bipolar transistor to work efficiently and produce an undistorted output signal. Correct DC biasing of the transistor also establishes its initial AC operating region with practical biasing circuits using either a two or four-resistor bias network.

In bipolar transistor circuits, the Q-point is represented by ( VCE, IC ) for the NPN transistors or (  VEC, IC ) for PNP transistors. The stability of the base bias network and therefore the Q-point is generally assessed by considering the collector current as a function of both Beta (β) and temperature.

Here we have looked briefly at five different configurations for “biasing a transistor” using resistive networks. But we can also bias a transistor using either silicon diodes, zener diodes or active networks all connected to the transistors base terminal. We could also correctly bias the transistor from a dual voltage power supply if so wished.


Straight-line amplitude plot

Amplitude decibels is usually done using \(dB=20\log_{10}\left(X\right)\) to define decibels. Given a transfer function in the form

\(H\left(S\right)=A\Pi\frac{\left(s-x_n\right)^{a_n}}{\left(s-y_n\right)^{b_n}}\)

where \(x_n\) and \(y_n\) are constants, \(s=j\omega,\;a_n,b_n>0\), and \(H\) is the transfer function:

At every value of \(s\) where \(\omega = x_n\) (a zero) increase the slope of the line by \({20a_n}\;{dB}\) per decade.

At every value of \(s\) where \(\omega = y_n\) (a pole) decrease the slope of the line by \({20a_n}\;{dB}\) per decade.

RC time constant \(\tau=RC\), \(\omega_0=\frac1\tau\)

The frequency \(\omega_0\) is called the break frequency, the corner frequency or the 3 dB frequency (more on this last name later).

These -3dB corner frequency points define the frequency at which the output gain is reduced to 70.71% of its maximum value.


Single-pole Bode plot




Prefix Symbol Multiplier Power of Ten
Pera P 1,000,000,000,000,000 1015
Tera T 1,000,000,000,000 1012
Giga G 1,000,000,000 109
Mega M 1,000,000 106
kilo k 1,000 103
none none 1 100
deci d 1/10 10-1
centi c 1/100 10-2
milli m 1/1,000 10-3
micro µ 1/1,000,000 10-6
nano n 1/1,000,000,000 10-9
angstrom Å 1/10,000,000,000 10-10
pico p 1/1,000,000,000,000 10-12

MOSFET

metal–oxide–semiconductor field-effect transistor

PMOS transistors have p-type source and drain regions, and the charge carriers are holes (positive charges), while NMOS transistors have n-type source and drain regions, and the charge carriers are electrons (negative charges).



In the 1950s the ability to grow an insulating silicon-oxide layer on semiconducting silicon was shown. The invention of the MOSFET is credited to Mohamed Atalla and Dawon Kahng when they successfully fabricated the first working sample at Bell Labs in November 1959.



The FinFET is a double-gate silicon-on-insulator device, one of a number of geometries being introduced to mitigate the effects of short channels and reduce drain-induced barrier lowering. The fin refers to the narrow channel between source and drain.


Robert Noyce

Robert Norton Noyce (December 12, 1927 – June 3, 1990) was vital to the invention of the integrated circuit. After Jack Kilby invented the first hybrid integrated circuit (hybrid IC) in 1958, Noyce in 1959 independently invented a new type of integrated circuit, the monolithic integrated circuit (monolithic IC). It was more practical than Kilby's implementation.

Why did Noyce leave Fairchild?

And he quickly rose to the rank of general manager. A decade later, Noyce and Gordon Moore left Fairchild to start a second company, Intel, which became a leader in the semiconductor industry in the 1970s and 1980s.

Lev Davidovich Landau

Wafer

P-type or N-type?

Silicon is the most common type of p-type semiconductor, while antimony is the most common type of n-type semiconductor.
Generally, Si wafers have a thickness between 0.5mm and 400 microns (0.4mm). Thin wafers between 2 and 25 microns in thickness can be needed for some scientific purposes.





MOSFET

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P-channel
N-channel
JFET MOSFET enh. MOSFET enh. (no bulk) MOSFET dep.


Quartz Crystal Circuit


The frequency of common quartz crystals lies between 32.768kHz and approx. 200MHz.


4 type Amplifiers


  •   Voltage Gain in dB:   av  =  20*log(Av)
  •   Current Gain in dB:   ai  =  20*log(Ai)
  •   Power Gain in dB:   ap  =  10*log(Ap)

Voltage amplifiers take voltage in and produce a voltage at the output.

Current amplifiers receive a current input and produce a current output.

Transconductance amplifiers convert a voltage input to a current output.

Transresistance amplifiers convert a current input and produces a voltage output.




Operational Amplifier (Op Amp)


Instrumentation amplifier

An instrumentation amplifier (sometimes shorthanded as in-amp or InAmp) is a type of differential amplifier that has been outfitted with input buffer amplifiers, which eliminate the need for input impedance matching and thus make the amplifier particularly suitable for use in measurement and test equipment.


The gain of the circuit is

\[A_v=\frac{V_{out}}{V_2-V_1}=(1+\frac{2R_1}{R_{gain}})\frac{R_3}{R_2}\]


Gain-Bandwidth Product

The gain–bandwidth product (designated as GBWP, GBW, GBP, or GB) for an amplifier is a figure of merit calculated by multiplying the amplifier's bandwidth and the gain at which the bandwidth is measured

\[\left|A_0(\omega)\right|=\frac{A_{ol}}{\sqrt{1+{\displaystyle\frac{\omega^2}{\omega_b^2}}}}\]


Slew Rate (SR)

\(\omega_{3dB}V_i\leq SR\) due to the current limitation.

\(\tau=RC\)

\(\omega_{3dB}=\frac1\tau\)

Unity Gain Frequency \(\omega_t\)


For RC circuit,

\(C=\frac QV\)

\(i=C\frac{dV_c}{dt}\)

\({(i_o)}_{MAX}=C{(\frac{dV_o}{dt})}_{MAX}=SR\)

\(SR=\frac{I_{sat}}CA\)


Input Bias Current

Op-amp input bias currents are the small currents required by the input terminals of the op-amp in order to function properly. If the input stage is comprised of bipolar junction transistors (BJTs), then the bias currents are the actual base currents required by these transistors.

Input Bias Cancellation \(R3=R1\parallel R2\) parallel combination of DC resistance.


Miller integrator

\(C=\frac QV\)

\(i=C\frac{dV_c}{dt}\)

\(v=-\frac1C\int i\;dt\)

\(v_o=-\frac1{RC}\int_0^tv_i(t)\;dt\)

\(\int\sin(\omega t)\rightarrow\frac1\omega\cos(\omega t)\)


the high frequecy part of Low pass filter works as a integrator.


\(C=\frac QV\)


Differentiator

\(I=C\frac{dV_{in}}{dt},\;V_{out}=-IR\)

\(V_{out}=-RC\frac{dV_{in}}{dt}\)

\(\frac{d\;\sin(\omega t)}{dt}\rightarrow\omega\cdot\cos(\omega t)\)

The circuit becomes sensitive to high frequency noise that, when amplified, dominates the input signal.



ODE equations applied to RC Op-Amp circuits


\(6v(t)+\frac{8d}{dt}v(t)+\frac{10d^2}{dt^2}v(t)+\frac{4d^3}{dt^3}v(t)=12v_{in}\)

\(\begin{align*} \left[4D^3+10D^2+8D+6\right]v_{out}&=12v_{in} \\\\ \left[D^3+\frac52D^2+2D+\frac32\right]v_{out}&=3v_{in} \\\\ \therefore \frac{v_{out}}{v_{in}}&=\frac3{D^3+\frac52D^2+2D+\frac32} \\\\ \end{align*}\)

\(\dddot{v_{out}}=3\left[v_{in}-\frac12v_{out}\right]-\frac52\ddot{v_{out}}-2\dot{v_{out}}\tag{4}\label{ac}\)

\(\frac{V_{out}}{V_{in}}=2\cdot\frac{\frac32}{s^3+\frac52s^2+2s+\frac32}\)





Diode

P-N junction diode

The invention of the p–n junction is usually attributed to American physicist Russell Ohl of Bell Laboratories in 1939. Two years later (1941), Soviet experimental physicist Vadim Lashkaryov reported discovery of p–n junctions in Cu2O and silver sulphide photocells and selenium rectifiers. The modern theory of p-n junctions was elucidated by William Shockley in his classic work Electrons and Holes in Semiconductors (1950).

DNA diameter: 2.5 nm
Silicon atom: 0.21 nm (2.1 Å)

The heighth of a n-layer silicon atom square pyramid \[\;(n-1)\times\frac{\sqrt2}2\times0.21nm\]

3nm thick silicon wafer stocks 21 atoms

1.8nm thick silicon wafer stocks 13 atoms



Intel’s CFET technology 4x 3 nm


TSMC 2D transistor





Diode


Diode law

\[i(v)=I_s(e^\frac v{\eta V_T}-1)\]

\[I_s: Saturation\;current\]


Rapid Analysis

Battery-plus-resistance Model

\(V_{DO}, r_D\)


The Constant-Voltage-Drop Model

\(V_D\)


Small Signal Model

\(i_D=I_S(e^\frac{V_D}{nV_t}-1)\)

\(\cong I_Se^\frac{(V_D+v_d)}{nV_t}\) Assume \(V_D>0.5V\)

\(=I_Se^\frac{V_D}{nV_t}e^\frac{v_d}{nV_t}\)

\(=I_De^\frac{v_d}{nV_t}\)

\(\cong I_D(1+e^\frac{v_d}{nV_t})\)

\(v_d\ll nV_t\) (must less than 10mv)

\(ii_D=I_D+\frac{v_d}{(\frac{nV_t}{I_D})}\) .... (dc)+(ac)

\(i_d=\frac{v_d}{({\frac{nV_t}{I_D}})}\)

\(r_d=\frac{nV_t}{I_D}={\left.{(\frac{\partial i}{\partial v})}^{-1}\right|}_{V_D}\)

\(i_d=\frac{v_d}{r_d}\)

\(r_d\propto\frac1{I_D}\)

Voltage Regulator, as the diode voltage variation diminished

line regulation \(\frac{\triangle V_o}{\triangle V_s}\)


Zener diode

breakdown diode


Full-Wave Rectification

bridge rectifiers

\[V_o=V_s-2V_{Do}\]

peak inverse voltage





Peak rectifier

\(V_r\) ripple

\(V_c(t)=V_pe^{-t/RC}=V_p-V_{r\_p}\)

\(t\cong T,\;RC\gg T\)

\(V_pe^\frac{-T}{RC}=V_p(1-\frac T{RC})\)

\(\because e^x=1+x+\frac{x^2}{2!}+....\)

\(V_p(1-\frac T{RC})\cong V_p-V_{r\_p}\)

\(V_{r\_p}=V_p\frac T{RC}\)

Another view \(\triangle Q(discharge)=\triangle Q(charge)\)

\(\triangle Q(discharge)=\triangle i\times t=\frac{V_p}RT\)

\(\triangle Q(charge)=C\triangle V=CV_{r\_p}\)

\(\because C=\frac{\triangle Q}{\triangle V}\)

\(\frac{V_p}RT=CV_{r\_p}\)

\(V_{r\_p}=\frac{V_p}{RC}T\)


\(V_p\cos\omega(-\triangle t)=V_p-V_r\)

\(V_p(1-\frac12{(\omega\triangle t)}^2)=V_p-V_r\)

conduction angle \(\theta_c=\omega\triangle t=\sqrt{\frac{2V_r}{V_p}}\)

conduction interval \(\triangle t\)

\(i_{Dmax}=\frac{V_p}R(1+2\mathrm\pi\sqrt{\frac{2{\mathrm V}_{\mathrm p}}{{\mathrm V}_{\mathrm r}}})\)


Root mean square

Statistically, the root mean square (RMS) is the square root of the mean square, which is the arithmetic mean of the squares of a group of values. RMS is also called a quadratic mean and is a special case of the generalized mean whose exponent is 2. Root mean square is also defined as a varying function based on an integral of the squares of the values which are instantaneous in a cycle.


\(V_{rms}=\sqrt{\frac1T\int_{t_0}^{t_0+T}v^2(t)dt}\)

\(f_{rms}=\sqrt{\frac1{T_2-T_1}\int_{T_1}^{T_2}\lbrack f{(t)}^2dt\rbrack}\)

\(x_{RMS}=\sqrt{\frac1n\overset n{\underset{i=1}{\sum x_i^2}}}\)

A sinusoidal curve
1. Peak amplitude \(\hat {u}\)
2. Peak-to-peak amplitude \(2\hat {u}\)
3. Root mean square amplitude \(\hat u/\sqrt2\)
4. Wave period (not an amplitude)


Sinusoidal wave or fully rectified sinusoidal wave: \(X_{pp}\;=\;2\sqrt2X_{RMS}\)
Triangular wave or sawtooth wave: \(X_{pp}\;=\;2\sqrt3X_{RMS}\)
Square wave: \(X_{pp}\;=\;2X_{RMS}\)
Half-rectified sinusoidal wave: \(X_{pp}\;=\;4X_{RMS}\)


\[Ripple\;factor\;=\;\frac{RMS\;value\;of\;A.C\;component\;of\;output}{DC\;component\;of\;output}\]



super diode


\(\mathrm A({\mathrm V}_{\mathrm i}-{\mathrm V}_{\mathrm o})={\mathrm V}_{\mathrm o}+0.7\)

\(V_o=\frac A{1+A}V_i-\frac{0.7}{1+A}\)


Dual diode Limiter

Diode Clipper

Diode Clamper


Peak to Peak detector



Physical operation of diodes

Intrinsic Semiconductors

It shows a low electrical conductivity under room temperature and its conductivity depends on its temperature. Therefore, intrinsic semiconductors are generally not used in electronic devices due to their low electrical conductivity.

Extrinsic Semiconductors

valence electrons \(n\cong10^{10}/cm^3\)

free electrons ~ The electrons are not bound within the atom and free to move and also called conduction electrons as they conduct electricity.

free holes ~ A hole is the absence of a negative-mass electron.

Current \(j\) is caused by free electrons and movement of valence electrons due to the existance of silicon holes.

Mass action law

In electronics and semiconductor physics, the law of mass action relates the concentrations of free electrons and electron holes under thermal equilibrium. It states that, under thermal equilibrium, the product of the free electron concentration \(n\) and the free hole concentration \(p\) is equal to a constant square of intrinsic carrier concentration \(n_i\)

\(np=n_i^2\)



\(E_0=V_T\ln\left(\frac{N_D\cdot N_A}{n_i^2}\right)\)

This electric field created by the diffusion process has created a “built-in potential difference” across the junction with an open-circuit (zero bias) potential


Forward Bias



Reverse Bias



Current

Diffusion current = the movement caused by variation in the carrier concentration.

Drift current = the movement caused by electric fields.

Drift (under E) \(j\;=\;\rho v,\;j_n=q_n\mu_nE,\;j_p=q_p\mu_pE\)

\(v=\mu E\)

Diffusion

\(j_n=q_nD_n\frac{dn}{dx},\;j_p=-q_pD_p\frac{dp}{dx}\)

\(j_n=q_nn\mu_nE+q_nD_n\frac{dn}{dx}=0\)

\(\frac{D_n}{\mu_n}\frac1ndn=-Edx\)

\(\frac{D_n}{\mu_n}\int_{n(x_1)}^{n(x_2)}\frac1ndn=-\int_{x_1}^{x_2}Edx\)

\(v_t\ln\frac{n(x_2)}{n(x_1)}=v(x_2)-v(x_1)\)

Built-in voltage

\(n_{p_0}e^\frac v{v_t},p_{n_0}e^\frac v{v_t}\)



diffusion length \(L_n=\sqrt{D_n\tau_n},\;L_p=\sqrt{D_p\tau_p}\)

life time \(\tau_n\) \(\tau_p\), minority carrier \(e^{-\frac x{L_n}},e^{-\frac x{L_p}}\)

\(j_p=qD_p\frac{dp}{dx}=q\frac{D_p}{L_p}p_{no}(e^\frac v{v_t}-1)\)

\(j_n=qD_n\frac{dn}{dx}=q\frac{D_n}{L_n}n_{po}(e^\frac v{v_t}-1)\)

\(j=j_p+j_n=(q\frac{D_p}{L_p}p_{no}+q\frac{D_n}{L_n}n_{po})(e^\frac v{v_t}-1)\)

\(j_s=(q\frac{D_p}{L_p}p_{no}+q\frac{D_n}{L_n}n_{po}),\;Saturate\;current\)


reverse bias current \(j=-j_s\)


\(\)

\(\)






\(\oint\overset\rightharpoonup E\cdot d\overset\rightharpoonup A=\frac Q{\varepsilon_s}\)

\(N_Ax_1=N_Dx_2\)

\(E_{max}=\frac{qN_Ax_1}{\varepsilon_s}\)

\(\frac{qN_Ax_1^2}{2\varepsilon_s}+\frac{qN_Dx_2^2}{2\varepsilon_s}\)

\(V_0+V_R=\frac{qN_Ax_1^2}{2\varepsilon_s}+\frac{qN_Dx_2^2}{2\varepsilon_s}\)

\(w=x_1+x_2\)

\(x_1=\sqrt{\frac{2\varepsilon_s}q\frac{N_D}{N_A(N_A+N_D)}(V_0+V_R)}\)

\(x_2=\sqrt{\frac{2\varepsilon_s}q\frac{N_A}{N_D(N_A+N_D)}(V_0+V_R)}\)

\(W=x_1+x_2=\sqrt{\frac{2\varepsilon_s}q(\frac1{N_A}+\frac1{N_D})(V_0+V_R)}\)

Junction capacitance (reverse bias)

\(C_j={\left.\frac{dQ}{dV_R}\right|}_{V_R}=\frac{qN_Ddx_2}{dV_R}=\frac{qN_Adx_1}{dV_R}\)

\(C_j=\frac{\varepsilon_s}{\sqrt{2{\frac{\varepsilon_s}q}({\frac1{N_A}}+\frac1{N_D})(V_0+V_R)}}=\frac{\varepsilon_s}W\)

\(dQ=idt=CdV,\;dt=\frac{CdV}i\)

Diffusion Capacitance (forward bias)

\(Q_p\cong qp_{no}e^\frac v{v_t}L_p\)

\(j_p=qD_p\frac{dp}{dx}=q\frac{D_p}{L_p}p_{n0}(e^\frac v{v_t}-1)\)

\(j_p=\frac{Q_p}{\tau_p}\)

\(C_D=\frac{dQ_p+dQ_n}{dv}=\tau_p\frac{dj_p}{dv}+\tau_n\frac{dj_n}{dv}\)

\(C_D=\frac{\tau_p}{v_t}j_p+\frac{\tau_n}{v_t}j_n\)

\(C_D=\frac1{v_t}(\frac{\tau_p\cdot j_p}j+\frac{\tau_n\cdot j_n}j)j\)

\(C_D=\frac1{v_t}(\tau_t)j\)

\(C_D=\frac{\tau_t}{v_t}j_s(e^\frac v{v_t}-1)\)

\(r_d=\frac{nV_t}I\)


The resistance is very large at reversed bias and we can assume it is open.


Visualization of (a) Drift transport and (b) Diffusion transport in a semiconductor with corresponding current components with electron and hole



P: Majority carrier \(p=N_A=10^{15}cm^{-3}\), Minority carrier \(n=10^5cm^{-3}\)

N: Majority carrier \(n=N_D=10^{16}cm^{-3}\), Minority carrier \(p=10^4cm^{-3}\)

MOSFET

Metal Oxide Semiconductor Field Effect Transistor


Enhancement mode MOSFETS are easier to control and can switch states faster than Depletion mode MOSFETS.


Cross-section of N-channel enhancement (a) and depletion (b) type MOSFET


Cross-section of P-channel enhancement and depletion type MOSFET


N-channel Enhancement MOSFET


\(V_{gs}>V_t\) threshold voltage the N channel exists.

There are Inversion layer \(n^+\) and Depletion layer in N channel

\(j_n=qn^+\mu_nE,\;E=\frac{V_{DS}}L\)

\(n^+\;of\;channel\;\propto(V_{GS}-V_t)\)

\(V_{OV}=V_{GS}-V_t\) over drive voltage




N-channel Depletion MOSFET

BJT vs MOSFET


Here \(I_D\) is the drain current, \(V_{DS}\) is the drain-source voltage, \(V_{GS}\) is the gate-source voltage, \(V_T\) is the threshold voltage, \(L\) is the length of the transistor (in the direction that the current flows), \(W\) is the width of the transistor, \(C\) is the specific capacitance of the gate in F/m², and \(\mu\) is the mobility.

Off mode

Off region \(V_{GS}\leq V_T\;\)

\(I_{DS}=0\)

Triode mode (ohmic mode)

Linear region \({V_{GS}>V_T\;,\;}V_{DS}\leq V_{GS}-V_T\)

\(I_{DS}=\mu_nC_{ox}\frac WL\left[(V_{GS}-V_T)V_{DS}-\frac{V_{DS}^2}2\right](1+\lambda V_{DS})\)

Saturation (active mode)

Saturation region \({V_{GS}>V_T\;,\;}V_{DS}>V_{GS}-V_T\)

\(I_{DS}=\frac12\mu_nC_{ox}\frac WL{(V_{GS}-V_T)}^2(1+\lambda V_{DS})\)

threshold voltage

\(V_T = \frac{2t_{ox}}{\epsilon_{ox}}\sqrt{\epsilon_{\text{semi}}N_Ak_BT \ln \left (\frac{N_A}{n_i} \right )} +\frac{2k_BT}{e} \ln \left (\frac{N_A}{n_i} \right ) +V_{fb}\)

Bode plot

Poles and Zeros

In electrical engineering and control theory, a Bode plot /ˈboʊdi/ is a graph of the frequency response of a system. It is usually a combination of a Bode magnitude plot, expressing the magnitude (usually in decibels) of the frequency response, and a Bode phase plot, expressing the phase shift.

As originally conceived by Hendrik Wade Bode in the 1930s, the plot is an asymptotic approximation of the frequency response, using straight line segments.



These -3dB corner frequency points define the frequency at which the output gain is reduced to 70.71% of its maximum value.


Term Magnitude Phase
Constant: K 20log10(|K|) K>0:  0°       
K<0:   ±180°
Pole at Origin

(Integrator) \(\frac{1}{s}\)

-20 dB/decade passing through 0 dB at ω=1 -90°
Zero at Origin

(Differentiator) \(s\)

+20 dB/decade passing through 0 dB at ω=1
(Mirror image, around x axis,of Integrator)
+90°
(Mirror image, around x axis, of Integrator about )
Real Pole \[\frac{1}{\frac{s}{\omega_0}+1}\]
  1. Draw low frequency asymptote at 0 dB.
  2. Draw high frequency asymptote at -20 dB/decade.
  3. Connect lines at ω0.
  1. Draw low frequency asymptote at 0°
  2. Draw high frequency asymptote at -90°
  3. Connect with a straight line from 0.1·ω0 to 10·ω0
Real Zero \[\frac{s}{\omega_0}+1\]
  1. Draw low frequency asymptote at 0 dB.
  2. Draw high frequency asymptote at +20 dB/decade.
  3. Connect lines at ω0.
(Mirror image, around x-axis, of Real Pole)
  1. Draw low frequency asymptote at 0°
  2. Draw high frequency asymptote at +90°
  3. Connect with a straight line from 0.1·ω0 to 10·ω0
(Mirror image, around x-axis, of Real Pole about 0°)
Underdamped Poles

(Complex conjugate poles)

\[\begin{gathered}
\frac{1}{{{{\left( {\frac{s}{{{\omega _0}}}} \right)}^2} + 2\zeta \left( {\frac{s}{{{\omega _0}}}} \right) + 1}} \\
0 < \zeta < 1 \\
\end{gathered} \]
  1. Draw low frequency asymptote at 0 dB.
  2. Draw high frequency asymptote at -40 dB/decade.
  3. Connect lines at ω0.
  4. If ζ<0.5, then draw peak at ω0 with amplitude
        |H(jω0)|=-20·log10(2ζ), else don't draw peak (it is very small).
  1. Draw low frequency asymptote at 0°
  2. Draw high frequency asymptote at -180°
  3. Connect with straight line from \[\omega = \frac{{{\omega _0}}}{{{{10}^\zeta }}}{\text{ to }}{\omega _0} \cdot {10^\zeta }\]
Underdamped Zeros

(Complex conjugate zeros)

\[\begin{gathered}
{\left( {\frac{s}{{{\omega _0}}}} \right)^2} + 2\zeta \left( {\frac{s}{{{\omega _0}}}} \right) + 1 \\
0 < \zeta < 1 \\
\end{gathered} \]

  1. Draw low frequency asymptote at 0 dB.
  2. Draw high frequency asymptote at +40 dB/decade.
  3. Connect lines at ω0.
  4. If ζ<0.5, then draw peak at ω0 with amplitude
         |H(jω0)|=+20·log10(2ζ), else don't draw peak (it is very small).
(Mirror image, around x-axis, of Underdamped Pole)
  1. Draw low frequency asymptote at 0°
  2. Draw high frequency asymptote at +180°
  3. Connect with straight line from \[\omega = \frac{{{\omega _0}}}{{{{10}^\zeta }}}{\text{ to }}{\omega _0} \cdot {10^\zeta }\]
(Mirror image, around x-axis, of Underdamped Pole)

The table assumes \(\omega_0>0\). If \(\omega_0<0\), magnitude is unchanged, but phase is reversed.

For multiple order poles and zeros, simply multiply the slope of the magnitude plot by the order of the pole (or zero) and multiply the high and low frequency asymptotes of the phase by the order of the system.

band width

3dB frequency means where the power becomes half of the maximum. So, if Power gain is 1/2, then gain in dB=10log(1/2)=-3dB, that's why it is known as 3dB frequency. As 'lower' is word used here, it is related to Band Pass Filters. This is gain (in dB) vs frequency response of an ideal low pass filter.


Nyquist plot

Nichols plot


Operational Amplifier

OpAmp

Vout = AOL (V+ – V–)


Operational Amplifier General Conditions
  1. The basic Op-amp construction is of a 3-terminal device, with 2-inputs and 1-output, (excluding power connections).
  2. The two main laws associated with the operational amplifier are that it has an infinite input impedance, ( Z = ∞ ) resulting in “No current flowing into either of its two inputs” and zero input offset voltage V-= V+.
  3. Op-amps sense the difference between the voltage signals applied to their two input terminals and then multiply it by some pre-determined Gain, ( A ).
  4. This Gain, ( A ) is often referred to as the amplifiers “Open-loop Gain”.
  5. Op-amps can be connected into two basic configurations, Inverting and Non-inverting.
  6. Open-loop gain is the gain without positive or negative feedback. Ideally, the gain should be infinite, but typical real values range from about 20,000 to 200,000 ohms.
  7. Input impedance is the ratio of input voltage to input current. It is assumed to be infinite to prevent any current flowing from the source to the amplifiers.
  8. The output impedance of an ideal operational amplifier is assumed to be zero. This impedance is in series with the load, thereby increasing the output available for the load.
  9. The bandwidth of an ideal operational amplifier is infinite and can amplify any frequency signal from DC to the highest AC frequencies. However, typical bandwidth is limited by the Gain-Bandwidth product, which is equal to the frequency where the amplifier’s gain becomes unity.
  10. The ideal output of an amplifier is zero when the voltage difference between the inverting and the non-inverting inputs is zero. Real world amplifiers do exhibit a small output offset voltage.

A component-level diagram of the common 741 op amp. Dotted lines outline;:

 current mirrors;  differential amplifier;  class A gain stage;  voltage level shifter;  output stage.