Talents

I will have the final exam tomorrow...so I decided to type some contents that I learned in my class.
I thought it might be a good way to brush up what I know about the class.

While I was summarizing what I have learned in my sensor class, I felt that my professor really know how to draw pictures. Turnig pages of his class presentations, I was really surprised how much he put his efforts on the slides.
It might be relatively easy for him, if he really got used to doing it. That's very good blessing talent! He tolde me he drew all professors in his department in his undergrad. school.

I wish I can work as hard as my advior. :)

Contents I learned in Integrated Sensor Class (EE 382V)

This is the contents that I learned in EE 382V class in Fall semester, 2006.
The class was offered by Prof. Arjang Hassibi, my advisor.
The title of the class is "Integrated Sensor".

1. Performance metrics of sensors

1) Transfer function (f(x)): ensemble average of the output

2) Sensitivity: small signal ratio (or change) of the output for a give input

3) NDNL (Normalized differential nonlinear)
NDNL(x) = (f(x) - f`(x))/f`(x), where f`(x): the best linear fit for f(x)

4) Systematic errors and random errors
Systematic errors - e.g., gain error, offset, background interference
Random errors - e.g., thermal, quantization, shot-noise
(Note) The difference between above two errors
--> Systematic errors can be calibrated, but not for random errors

5) SNR: Signal-to-Noise Ratio
If a transfer function has flat response over input, the input noise is large.

6) Accuracy and Uncertainty
Given Yo(specific output) is observed, what is the input??
Typically we are interested in E[x|y=Y0], the representive input signal, and E[x^2|y=Y0] - (E[x|y=Y0])^2, the input noise given y=Y0.

where E[n]=0 (zero-mean) of the noise is assumed!

7) Dynamic Range (DR)
The input range where SNR is acceptable!
(DR is not defined as SNR(max) - SNR(min)!!!)
(Note)
X(MDL): Minimum detectable level --> typically limited by noise
X(HDL): Highest detectable level --> typically limited by saturation or quantization

(Note) You might require basic probability and Tylor series approximationknowledge.






2. Noises in Sensors

1) Random noises

i) Thermal noise
- e.g., CMOS, Diode, BJT, resistor
- Signal indepedent, temperature dependent
- white noise and Gaussian amplitude distribution
- originates from the random motion and random scattering of many charged species within a medium much larger than their mean free-path. In other words, Macroscopic view!!
- steady-state (stationary process) case: 4KTR [Vrms^2/Hz] (1-sided)
2KTR[Vrms^2/Hz] (2-sided)
- if a system undergoes a transient response, the noise analysis should be done in time domain

ii) Shot noise
- e.g., BJT, Diode
- Current going through junctions create shot noise
- signal dependent and temperature independent

iii) Substrate noise

(Note) It is good to know 1) Basic stochastic process, convolution, Lapalce transform, LTI system properties...etc


2) Systematic error (noise)
- all passive and active elements (due to process variation, design, and layout...etc)
- greatly reduced by proper design and layout
- Can be calibrated!

3) Noise calculation at the output and input referred-noise power in various ckts




3. Circuit Architectures for Sensors

1) Detecting sinusoidal tones
- Objective: to measure amplitude, freq, and/or phase information
- narrowband detection
- Ex) Brutal fast A/D, Superheterodyne, and Direct conversion...etc

2) Voltage Detection
- Objective: to acquire a voltage signal which has DC or low frequency components
- Useful when parasitic components are in series and the output of the transducer is large so that most of voltage source information is delivered to the detection circuit very well.
- Techniques for reducing 1/f or low frequency noise
i) Chopper stabilizer (CHS)
ii) Auto zero (AZ) - noise should be slow and additive
iii) Switching biasing - done in device level

3) Current Detection
- Objective: to acquire a current signal which has DC or low frequency components
- Useful when parasitic components are in parallel and the output of the transducer is small so that most of current source information is delivered to the detection circuit very well.
- Techniques for reducing 1/f or low frequency noise
i) CHS
ii) Switching biasing
- Transimpedance amplifier(TIA)/Capacitive transimpedance amplifier (CTIA)
4) Parametric Detection
- Objective: to acquire some characteristic of a passive electrical component
- This method typically uses "the output value change from the nominal ouput" as its measurement
- This method requires an excitation signal
- Information lies on "The difference of the outptu from the nominal output"




4. Integrated Image Sensors (IS)

1) Signal path i IS
- Photonflux(ph/cm^2.sec)-->Current density(A/cm^2)-->Charge(Col)-->Voltage(V)

2) Photocurrent and quantum efficiency (QE)
- absorption coefficient(alpha) (=penetration coefficient^-1)
- Phtonflux at depth x in Si
F(x)=F0*exp(-alpha*x) [ph/Cm^2.sec]
where Fo is the photonflux incident on the surface
- Generation rate [e-h pair/Cm^3.sec]
G(x)=alpha*F(x)
- Quantum efficiency
Def: Given F0, how many electrons that contribute to the photocurrent are generated?
QE = (jph/q)/F0
where jph: photocurrent density, q = 1.6X10^19 [Col]
(Note) jph is linearly proportional to qF0
- Miscellaneous
i) shorter wavelength = high frequency
E = hv=hf, thus higher frequency, the higher energy
ii) If a photon energy is smaller than bandgap of a silicon, it will pass through it!
iii) Shorter wavelength photon generate e-h pair easily (shallow depth in the silicon)

3) Dark Current
- Photodetector current with no illumination present
- Bad:
- introduce unavoidable shot noise (qID)
- reduce signal swing
- DSNU (Vary over image sensor array)
- Sources:
- Thermal generation
- Defects - e.g., interface, material defects..etc

4) Photodiode Structure in CMOS Process
- P+/nwell
- nwell/Psub
- N+/Psub

5) Direct Integration
- Vo(t)=Vref - (Id/Cj)*Tint = Vref - ( (Iph+Idc)/Cj)*Tint
where Id = photodector current = Iph + Idc
Iph = photocurrent
Idc = dark current
Tint = Integration time
- Well capacity(Qmax): Max. # of electrons that a photodector can handle [electrons]
HDL = Iph,max = (qQmax/Tint)-Idc, where qQmax <= VDD*Cj where VDD: supply voltage 6) Integration with CTIA
- The junction capacitance of a photodiode is not changing during integration because there is no bias condition change during integration
- Output voltage is no longer a function of Cj, the juction capacitance of the photodiode, but for a feedback capacitor, which is much linear than Cj!!




5. Image Sensor Architecture and Noise

- Pixel architecture
PPS, APS, DPS
- Some common senses
After reading (i.e., word signal) the information, the integration cap. should be reset!
Thus, the integration time is always defined the time interval between two consecutive reset pulses (signals).

1) Passive Pixel Sensor (PPS)
- Direct integration, reading charges using CTIA
(Note) Here CTIA is used for reading (amplification) of charges, but not for integration
- Photodiode is reset during reading (word signal)

(Note) Reset is used for resetting CB and CF, not integration capacitor! The integration capacitance, Cj, is reset during word signal!

i) Signal transfer function

ii) Noise analysis
- Direct integration noise
- Reset noise
We are not interested in output noise, but the noise stored on the capacitors at the end of the reset. These noise will get transferred to the output during the readout!
- Readout noise
(Note) Noise fed back to the photodiode - noises are stored on Cj (junction cap. of the photodidoe) at the end of readout, which is read out during the following readout. Thus, due to previous fedback noise, the output noise will be doubled!

2) Active Pixel Sensor (APS)
- General advantages of APS over PPS
--> reading is nondestructive, faster, and multiple reading is possible

i) Types of APS
(1) 3T APS: photodiode, direct integration
(2) 4T APS: photogate, direct integration
(3) 1.75T/1.5T APS: Pinned diode, direct integration
(4) CTIA APS: photodiode, integration with CTIA (integration over feedback cap.)

ii) 3T APS
(Note) Reset signal here is used for resetting the integrating capacitor, Cj!
- Anti-blooming scheme is possible during integration
- The very bottom current source (transistor with Vbias gate biasing) is used as a current source for all source followers.
(Note) Why is not the reset switch Pmos? Due to the size of Pmos implementation, in order to have large fill factor, we perfer NMOS to PMOS!

iii) 1.75T/1.5T APS
A floating diffusion capacitor in these architecture is the part of the reading action


3) Charge-Coupled Devices

i) Photogate - collect and generate e-h pair
- NMOS(VG>0) and PMOS(VG<0)>Eph))
(2) Low QE due to interface states
- Thich gate oxide is desired
(1) In order to avoid gate leakage due to large gate voltage
(2) In order to make a bump small in a potential well
Smaller a bump, better the transfer efficiency!!

ii) CCD Basic
(1) CCD is a "dynamic" analog (charge) shift register
(2) CCD is clocked and all operations are in "transient mode"
(3) Charge is coupled from one gate to the next gate by fringing electric field, potential and carrier gradient

- Surface potential control in CCD
(1) Doping method
(2) Oxide step
(Note) Higher doping, higher threshold voltage for the same doping material. If the doping materials are different, higher doping region makes threshold voltage lower!

ii) CCD Architectures
(1) FT-CCD (Frame transfer CCD) - most popular
(2) Full FT-CCD - need mechanical shutter
(3) IT - CCD (Interline transfer CCD) - faster, small fill factor

iv) Charge transfer efficiency (CTE)
- Why is CTE 1000%?
--> main causes (1) lack of time to complete tranfer
(2) charge trapping - captured by interface states
(3) due to a bump in a potential well
- transfer mechanisms
: a combination of carrier diffusion and carrier drift


4) Technology
- What are the considerations for image sensor fabrications?

5) Stuff not covered!
- Fixed Pattern Noise (FPN) - nonuniformity
: spatial variation in pixel output values under uniform illumination due to device and interconnnect parameter variations (mismatches) across the sensor
: gain FPN and offset FPN
: Calibrations





6. Integrated Magnetic Field Sensors

1) Detecting alternative magnetic field

2) Detecting DC magnetic field
- Hall sensor (uses Hall effect)




7. Backbone Ckt for MEMS and Packaging




8. Biological and Chemical Sensors

1) General Biosensor Platforms
- Chanllenges
(Biology)
High detection sensitivity
Detection in presence of interference
Parallel detection
Amplification and lable-free
(Electronics)
Low cost
Battery operated
Portable or hand-held
Biocompatible/Implantable

2) Affinity based biosensor
- Many biological molecules, when brought together, can go to a lower energy state by binding
- Molecular affinity
(1) The bindings are inherently probablitistic
(2) Thermal energy results in a postivie probability of detachment
(3) Non-specific binding is also possible, although less probable
- Process procedures
(1) Sample exposure
(2) Incubation (i.e., hybridization)
(3) Detection
- Noise in biosensors
: Noise is dominant by biochemical moise, not by electronic noise

3) Electrochemical Detection
- Molecular binding changes the characteristics of the electrode-electrolyte interface
- Biosensor can detect the changes (parametric detection)
- Study of Prof. Hassibi's circuit (Elecrtrochemical sensor microarray)

------------------ END of Class -----------------------------------------------

Impedance of a Degenerated Circuit

I had to review the series and parallel RLC circuits for my RFIC class. In typical analog IC regime, I rarely deal with L so that I didn't have a chance to study it, but in RFIC I have one more component, L. Is this bless or curse? Well, of course, it depends on how to use it. Since I have one more degree of freedom to tweek a knob so that there is more freedom to design circuits. However, I have one more degree of freedom that make me consider many other consequences to consider.

Today, I would like to discuss about a source load degenerated circuit. This is a particularly important in a LNA design because inductively degenerated LNA gives possibly good noise figure while this provides good input matching as well. The side effect of this, a good one, is that it also boost Gm (transconductance) so that it helps the LNA gain. Let's see why it give me that!

If you derive the input impedance equation, it will look like above. One interesting term is (gm*Z)/(jwCgs). Depending on Z, it can be resistance, capacitance, and negative resistance.

The reason is that if you consider the term's magnitude, (gm*Z)/(wCgs). However, it has 1/j multiplied. What does 1/j term do? This term rotates (gm*Z)/(wCgs) by-90 degree shift in phase diagram (rotate by 90 degree in counterclockwise) - Re and Im for x-axis and y-axis, respectively.

Keep in mind what 1/j term does.

Let Z = jwL (inductor),

The last term will be (gm*L)/Cgs - resistance!, but it is NOT a real physical resistor so that it does not generate any thermal noise! This is a great advantage! If you make L such that it gives you the right resistor for good input matching at resonant frequency, it is good for noise and matching - you threw a stone, and caught two birds!! Is that all? Don't be so greedy~ Well, there is one more advantage....since this is series RLC circuit, voltage across L and C will be boosted by Q times - will be explained in detail later.

In the same manner, you can substitute Z = 1/(jwC) and Z = R.

The last term, then, will give you capactance and negative resistance, respectivley.

Why? Since 1/jw term give you -90 degree rotation ( couterclockwise 90 degree rotation),

Z=jwL (postive part of the y-axis) --> resistance

Z=R (postive part of the x-axis) --> capacitance

Z=1/(jwC) (negative part of the y-axis) --> negative resistance

Hope this is not very confusing any more...

Now, let's review how this also boost gm by Q times --> Gm = Q*gm! (gm: transistor transconductance and Gm: system transconductance)

From series RLC circuit analysis, we know that at resonant frequency the voltage across the capacitor and the inductor is Q times larger than what is applied. Thus, the voltage across Cgs is boosted by Q time at resonant frequency.

Remember Q boosting and input matching work at resonant frequency (or its neighbor frequencies)

That's all for today!

Lunch with Prof. Powers

Today, I had lunch with Prof. Powers, who is my master advisor, and my old labmates in Peiwei restaurant in Hankok. I ordered a scallop orange peel dish (it seemed the most expensive they had accidentally :) ). The lunch was for congrating Hyeonsu's qual and Changyong's defense. I really need to get ready for qual..., but I know I still need to learn many stuffs :)

Port vs Terminal

Today, I would like to discuss about some confusing terms - port and terminal.

In a sense, there is not much difficulty in understanding both terms. Many people know that port or terminal is a gate that signals go into/out of a circuit. Yes, they mean that!

Then, a question might arise if there is any subtle difference in them because typically rf or high frequency circuits' in and out part we call a port, otherwise, typically in low frequency or typical analog circuit designers use a terminal for the same meaning. For example, if you refer to the famous CMOS IC design (Razavi's) book, in chapter 8 - feedback, the author mention about impedance transformation due to feedback effect calls terminal modification, not port modification.

Why designer in high frequency circuits call a port circuit's in and out and why low frequency circuit designers call it a terminal?

Of course, there is a subtle difference between two terms.

Let's find out what is the difference.

In RF signal measurements, we cannot simply put a probe to a circuit to measure voltage or current. This is because the wavelength of the signal to be measured is small so that its phase/amplitude changes depending on the length of the probe and its wire. That's why it is difficult to measure rf signal with a conventional oscilloscope.

The reason that a port is used in high frequencies is that we need to define signal and ground condition accurately. Due to phase shift sensitivity of a short wavelength, we need to define signal and ground condition all way such a way that they need to be defined exactly. In typical low frequency analog circuit, wavelengths are long engough it does not matter you use relatively long probe wire or not. For example, if a wavelength of 300m and your probe is 30Cm long, the phase change due to probe length etc is not an issue. However, if a wavelength is 13mm, and your probe has 30Cm long, depending on its actual length, ie. 29.3Cm or 30.21Cm, will affect the oscilloscope measure amplitude (due to phase shift). Importantly, at high frequencies, due to short wavelengths, it is critical to define signal and ground condition accurately.

From the arguements above, I can now say that a port is one that has well-defined signal and ground condition and a terminal is relatively not well defined on them. For example, a coaxial cable is two conductive material running through across the cable and one is defined for a signal and the other is for ground. Both signal and ground lines go together always so that they are well defined anywhere we use them. However, a wire can be used for a terminal connection. If we just use a single wire to communicate a signal between external and internal part of a circuit, we are automatically assumed that the ground potential is well-defined so that we can just pass a signal without much attention to where the ground potential is defined. Therefore, just a wire to transfer a signal is enough to deliver energy to a circuit.

It maybe a subtle and quite often they are being used interchangeably. But, it is good to know!

weekly facts

1. Parametric sensing is possible by detecting the change at the output from the nominal output. Thus, most importantly, the information lies in how much deviate from the nominal vlaue. This deviation, or change, comes from the parameter component (sensitive to a particular meausre) value change from its nominal value. This deviation eventually changes the output value so that we can sense input measure!

2. delta function
The unit of delta function is delta($) = [1/$]. This is becuase a delta function is a general function and it is defined only under the integration, which is 1. Since we integrate in term of d$ [$] and the integration result is 1, the delta function itself should have a unit of [1/$]

3. Differental mode noise analysis
In noise analysis, if noise sources are uncorrelated and additive, the half-circuit concept used in signal analysis cannot be used in noise analysis. This is because due to lack of correlation, the middle cut in the origianl circuit will not provide the half-circuit with a good virtual ground. The best way, not necessarily the easist way, is to derive the noise effect of each source individually!!

Noise analysis in differential mode circuits

The use of differential mode including a differential pair is probably the most important circuit topology in IC. However, people, particulary me, often are confused differential mode circuit noise analysis with its signal analysis. For example, I typically use Half-circuit concept to analyze a differential pair signal analysis (i.e., signal transfer function) for convenience. This half-circuit concept greatly reduce amount of work that I have to undergo to analyze the differential circuit otherwise.

The prime reason that this half-circuit concept is possible is due to the circuit's symmetricity. In order for the circuit to be perfect symmetric, left- and right- side of the circuit components should have the same components and values - ideally no random variation. Another requirement is the input should be 180 out of phase each other so that the drain voltage of the tail current source should be "virtual ground"! (Vin1 and Vin2 are 180 out of phase - same magnitude, but opposite sign!) The input relationship shows that both Vin1 and Vin2 are 100% correlated! This fact should be paid special attention because this fact is important discrimination in noise analysis.

Anyway, in signal analysis, the half-circuit concept is very useful and it is mere double the differential output voltage with differential input voltage. (or, Vout,dm/2 = (......)Vin,dm/2 ---> Thus, Vout,dm = 2(......)Vin,dm).

The noise analysis of the differential pair should be treated differently from the signal analysis case. Since noise sources are typically uncorrelated in the differential pair - all the noise sources in the differential pair (i.e., drain current noise, load resistance noise...etc), the virtual gound concept cannot be used in noise analysis. This fact, the drain voltage of the tail current source is no longer virtual ground, makes it impossible to use half-circuit concept in noise analysis in the differential pair. Thus, we simply derive the noise effect of each source individually! If you still want to use half-circuit concept, which is inappropriate in exact sense, then you should use it with special attentions!!!!


The following is a fine example that I had for midterm exam - yr 2006, fall.

Let's fine the output noise power!
Can we use half circuit concept with special care? Um...yes....
To be exact, I should not use half-circuit concept, which Vin is now shorted and circuit gets divided a half exactly. Then, the noise sources in the half-circuit concept are Vn, (VR1)/2 from R1 thermal noise and VR2.
Notice Vn and R2 are noise sources, which do not have any correlation with the other half-circuit. Thus, the noise power will be added! Due to symmetricity, the noise power will be doubled! (Assumption: noise is additive) However, (VR1)/2 is used for half-circuit concept. This is the part we should give special attention to. Since one noise voltage source has been divided into two for half-circuit concept, the noise voltage, (VR1)/2, is correlated to the other half-circuit. Thus, now in (VR1)/2 perspective, the half-circuit can be treated just like signal analysis case - virtual ground!. Thus, at the output, the noise voltage is doubled in differential mode consideration! This mean in noise power calculation, the noise power due to this term will be quadrapled!!!!
Therefore, the answer is ....

E[Vn,out^2] = A^2{2E[VR2^2] + 2(1+2(R2/R1)^2)E[Vn^2] + 4((R2/R1)^2)E[VR1^2]}
the orange colored number "2" is due to uncorrelated noise sources, in psd, they are added.
the red colored numer "4" indicates that there was correlation in noise source in half-circuit, thus it was treated as signal half-circuit case. Thus, its noise power are quadrapled!

That's all, folks!!!

Qual/defense party

Many of my friends had passed their defenses and qualifying exams successfully this semester. Congrat! Particularly, Hyeonsu, one of my best friends, passed his qual this semester. Those who passed the defense/qual exams this semester are having a party with all other Korean friends. However, I am sitting on my desk and reading class materials to review what I have learned this semester.

Well....Hope people continue to success in their lives! Good luck!

Input referred noise

Today, I would like to discuss about "input-referred noise". Most of discussions are from Prof. Razavi's famous CMOS analog design book .

Actual (output) noise calculation can be done by killing all the input sources (i.e., voltage sources --> short and current sources --> open). This is how we can measure the noise in a laboratory. Killing all the input sources at the input terminals and putting probes at the output terminals to measure the noise. Note that the noise we can measure is "output" noise! It can never be "input".

Then, a question arises "what the heck do we want to create "input referred noise" term?".
The simple final answer is that we want to compare the noise of one circuit to the other circuit under fair comparison condition. What make input referred noise be the fair comparison measure compared to output noise? Consider the following case. Let us assume that we have two amplifiers with same noise source at the input. However, one with the gain of 1 (circuit 1) and the other with the gain of 100 (circuit 2). Now, let's measure the output noise at the output terminal. The circuit 1 has the same noise as the source noise. On the other hands, the output noise of circuit 2 becomes 100 times larger (in power quantity, 10,000 times larger) than the same noise source at the input. Can you say the 2nd circuit has larger noise than the 1st circuit? Is this fair comparison? Remember the signal also gets 100 times larger! This is one of many reasons that we would like to consider Signal-to-noise ratio (SNR), rather than only signal! One very important observation here is SNR does not depend on the gain!

In order to overcome this unfair comparsion of noise people developed a new concept called "input referred noise". The idea is to represent the effect of all noise sources in the circuit by noise sources at the input such that the output noise is equal to the output noise by simply mutliplying with gain of an amplifier.

Some important observations...

1) The input-referred noise and the input signal are both multiplied by the gain as they are processed by the circuit. Thus, the input-referred noise indicates how much the input singal is corrupted by the circuit's noise, i.e., how small an input the circuit can detect with acceptable SNR! (this input is called minimum detectable level). --> fair comparison is possible. Observing the same output SNR, what is the input SNR! Then, we can compare it with other circuit's SNR so that we can determine the noise contribution from a circuit.

2) If the circuit has a finite input impedance and is driven by a finite source impedance we have to use both voltage input noise source and current input noise source for completeness. Votlage input noise source is in series with a source and a circuit and current input noise source is in paralle with a source terminal. Let's first find when we can model sole voltage input referred noise source is sufficient. If Zs (source impedance) is zero (0), then even if input current noise source exist at the input, its current all flow through the source side. None of current from the input current noise source goes to the circuit side. Only survivor is the voltage noise source! Since the source terminal shows zero impedance and the circuit input terminal has some finite impedance, all the voltage will effectively show on the input impedance of the circuit. Conversely, if the source impedance is open (infinate impedance), all the current from the input noise source goes to the circuit! (Since a source terminal has infinite impedance, the some finite input terminal impedance of the circuit is relatively said that it has almost zero impedance in relative sense!). However, the voltage noise source all will be delivered to the source side due to its infinite impedance. THUS, to make a long story short, if a source impedance is in somewhere between 0 and infinite impedance, we have to have both voltage noise source and current noise source at the input terminal! - voltage source is in series and current source is in paralle!


Now....let's see how we can calculate the input noise !
There are two typical ways to find them. The first one is comparison method. (less often used). The other is gain divison methods.



............ INCOMPLETE!

Thermal Noise

Acknowledgement:
Most of materials are excerpted from Jim hellums's short note and van der Ziel's book.

Brownian motion is created by the impact of the molecules of the fluid surrounding the object due to the fluctuations in the ambient thermal energy(i.e., temperature)

Let's first find it out about equipartition theorem...because I have heard many times from other physics folks including my professor.

The equipartition theorem is a principle of calssical (non-quantum) statistical mechanics which states that the internal energy of a system composed of a large number of particles at thermal equilibrium will distribute itself evenly among each of the quadratic degree of freedom allowed to the particles of the system. (from Wikipedia). The theorem says that the mean internal energy associated with each degree of freedom of a monatomic (single atom) ideal gas is the same.
The equation is ...
(1/2)M*vx^2 = (1/2)KT
where M: the mass of the particle
vx: instantaneous veolcity component in the x-direction
K: Boltzmann's constant
T: absolute temperature

Einstein showed that
xd^2 = 2Dt
where xd: displacement of x in the x-direction
D: diffusion constant of the particle

The termal noise in circuits is nothing more than Brownian motion of electrons due to the ambient temperature.

***** test *****

\int_{-\infty}^{\infty}e^{-x^{2}}\;dx=\sqrt{\pi}

Later Johnson (experimental physicist) and Nyquist (theorectician) in Bell labs found the termal agitation of electricity in conductors produces a random variation in the potential between the ends of the conductors. The electromotive force developed across the end of the conductor due to Thermal noise is unaffected by the presence or absence of direct current. This can be explained by the fact that electron thermal velocities in a conductor are much greater (~10^3 times) than electron drift veolocities. The amplitude distribution of Thermal noise is Gaussian in three dimensions (Central limit theorem), which can be illustrated by a random walk process.

1) van der Ziel's Derivation of thermal noise
Consider a resistor R in parallel with a capacitor C. As a result of the random thermal agitation of the electrons in the resistor, the capacitor will be charged and discharged at random. The average energy stored in the capacitor will be
(1/2)CV^2 = (1/2)KT
or
V^2 = KT/C
where V^2: the mean square value of the voltage fluctuation empressed across the capacitor.



.....INCOMPLETE!

Dear all visitors

Wazup~?

I would like to build this blog for my study and possibly to share my humble insights with others.
I think it is always good to share ideas and to discuss about stuffs of the same interest. I have seen many friends and other folks who don't want to share their ideas and even disgrace others' ideas. That's bad. At least that is far away from what I have learned in my life lessons.
Anyway, so I would like to keep many insights I have developed (if I am permitted to say...-.-;) and to share with others.
That's the motivation of creating this blog.

Today, I went to Buffet place for dinner with my budies (Hyeonsu and Wonjin). The food was okay...One dish I like was crab legs.
Now I have to back to my office and reading some materials.

Let me go back to my desk....

About Natural/Forced & Transient/Steady-state Responses

(Note) Discussion of transient response and steady-state error is moot if the system does not have stability.

In order to explain stability,

[Total response] = [Natural response] + [Forced response]
= [Homogeneous solution] + [Particular solution]
~= [Transient response] + [Steady-state response]

where ~= "approximately equals to"

As we see above "Natural response" = "Homogeneous solution" ~= "Transient response" and "Forced response" = "Particular solution" ~= "Steady-state response".
(Note) Above relationships are only value for linear systems!!

Are you still confused? Think about a step response of the first order system, if the 1st order system is RC low pass filter and input is 1V step, the output will initially undergo exponential response, but eventually it will converge to 1V in time domain. The output eventually converges 1V because its input (force signal) was 1V. Thus, the steady-state response is the response that eventually converges to a particular value by responding to a forced input!

Let's find it out why Nautal response is not exactly identical to transient response...etc.
The transient response is the sum of the natural and forced response while natural response is large. The steady-state response is also the sum of the natural and forced response, but while the natural response is small. Thus, the transient and steady-state responses are what you actually see on the plot!
The natural and forced responses are the underlying mathematical components of those responses. Since transient and steady-state responses include both natural and forced responses in the responses, they should be defined by associating with "acceptable error"!

HOWEVER, from my experience, people are often using those terms interchangeably. For examle, "natural response" = "transient response" and "forced response" = "steady-state"!

Is it still confusing???

Let's see an example!

Given: C(s) = (2/5)/s + (3/5)/(s+5)
If I take inverse Laplace transform, c(t) = 2/5 + (3/5)exp(-5t).
To be exact, (2/5) in c(t) is called "forced response" and (3/5)exp(-5t) in c(t) is called "natural response" of the system. If you plot c(t) with Matlab, you will see the response at the beginning is the sum of two terms in c(t), but soon the second term (3/5...term) dies out. Thus, transient response is typically associated with "acceptable error". If the output is not within the acceptable error, the response until it is within the error is called "transient". Otherwise, "steady-state response". For example, if I define "acceptable error" as 1mV, then the response before 1mV error is called "transient", otherwise (so within 1mV error) the response is called "steady-state response"!

Laplace and Fourier transforms

It was a quite long ago when I first met Laplace transform. Later, I also learned Fourier (most famous transform in EE in general ^^) and z-transform. However, it took me quite while to understand them and actually use them. I am sure I don't understand them all of their properties and usefulness.

I basically understood that Fourier transform was a subset of Laplace transform. That's all I understood in my undergrad. Now? Not much...actually.
However, last year I had to solve one of equations using Laplace tranform and had a chance to re-think about its property.
The followings are what I gave thought about Lapalce and Fourier transform.

1. Laplace is a linear tranform (so is Fourier transform). Thus, it should be applied to linear systems only.

2. Frequency reponse of a system is basically the response seen from jw axis out of 3D figure in s-domain.

3. jw-axis in S-domain is defined as when sigma = 0, which means all the transient responses die out so that only response we can get out of the frequency response is "steady-state" response. Therefore, you don't expect any transient response while you solve a question in frequency response. This is simple, but often confusing. For example, you want to calculate noise response of a system. Let's assume that only noise source is a resistor - thermal noise. From typical text book says, a resistor's power spectral density (PSD) is equal to 4KTR [Vrms^2]. If you analyze a circuit with this equation, you are intersted in noise analysis in steady-state!!! Noise power you calculate in the end only counts "steady-state" noise power. You never see how noise power changes over time because we didn't consider transient response! In conclusion, the in frequency domain (i.e. 4KTR in this example) is valid only when Linear time-invariant steady-state case! (when sigma = 0 in Laplace domain)

* I think this is a good insight I have overlooked for long time!

4. Many other properties are stated in many books....so...this is it!!! This is all I want to mention!

About DC path

A capacitor is one of most important components in analog/digital circuit design in general. For example, a switched capacitor circuit is one of prevalent circuit topology in analog designs.
While we face many capacitor components in circuits, I am often heard that "DC path does not exist". This DC path implies that charges can go onto a capacitor. The heart insight of understanding DC path term is to recognize that there is no DC voltage drop if there is no DC path exists.

One of frequent uses of "DC path" term is when we talk about "ac coupling". Since a capacitor effectively blocks a DC signal, DC cannot pass through a capacitor. For example, there is typically a big fat C (capacitor) precedes a low-noise amplifier (LNA) input. This C effectivly block a DC signal and let an ac signal pass through without affecting a bias condition of the LNA. In other words, ac signal can go into the big fat C because the impedance of this big fat C at high frequencies is small so that ac signal has no problem of commuting through it. Since there is no DC path exist, this big fat C won't drop a bias (DC) voltage across it.

Remember if there is no DC path, there is NO DC voltage drop!

Another example - there is the rule called "Charge conservation rule", which is a good tool when we analyze a switched capacitor circuit transfer function. This rule can be applied when there is no DC path existence. If there is a DC path, the charges on the capacitor can go in and out depending on the voltage and capacitor value givne at that node. Also, it is important to recognize that this rule is applied to a node.

First edit

I am typicall not a good website keeper! I am not sure how long I can update this blog and keep it useful to provide good insights for circuits/modelings.
but, I will try...