Difference between revisions of "TF EIM Chapt3"

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=Semiconductor physics=
+
=Differentiator circuit=
  
 +
Consider the effect of a high-pass filter on a rectangular shaped input pulse with a width <math>\tau</math> and period <math>T</math>.
  
There are 5 states of matter: Solid, liquid, gas, plasma (ionized gas) , and a Bose-Einstein condensate (quantum effects on a macroscopic scale).
 
  
=Crystal Lattice=
+
[[File:TF_EIM_PulsedRCHighpass.png | 400 px]]
  
Semiconductor physics focuses on the solid state of matter in the form of crystals.
 
  
Crystals are formed when atoms are arranged in repeating structures called the crystal lattice.  The recurrent  structure which forms the crystal lattice is referred to a a cell.
+
The first thing to consider is what happens to the different parts of the square pulse as it travels to the circuit.
  
Some popular cell names are
+
You know that the circuit, as a high pass filter, will tend to attenuate low frequency (slow changing) voltages and not high frequency (quick changing) voltages.
 +
 
 +
This means no DC voltage values pass beyond the capacitor.
 +
 
 +
This means that points <math>a</math> and <math>b</math> in the pulse shown below should pass through the circuit. 
 +
 
 +
[[File:TF_EIM_SquarePulseDifferentiated.png| 200 px]]
 +
 
 +
The voltage changes in between point <math>a</math> and DC are passed according to what you select for the break frequency (<math>\omega_B = \frac{1}{RC}</math>)
 +
 
 +
In the above pulse picture it was assumed that <math>\tau >> \frac{1}{\omega_B} = RC</math>  or in other words the RC network time constant is a lot smaller than the pulse width.
 +
 
 +
If the RC time constant is larger than the pulse width then you will see the voltage stay high.  The rectangular pulse will be output as a square like pulse which is "sagging".
  
  
 
{| border="3"  cellpadding="20" cellspacing="0"
 
{| border="3"  cellpadding="20" cellspacing="0"
| [[File:TF_EIM_SimpleCubic.jpeg| 200 px]] || [[File:TF_EIM_BodyCentereCubic.jpeg| 200 px]] || [[File:TF_EIM_FaceCentereCubic.jpeg| 200 px]]
+
| [[File:TF_EIM_PulsedRCHighpassSmallRC.png | 400 px]]|| [[File:TF_EIM_PulsedRCHighpassBigRC.png | 400 px]]
|-
 
|(SC) Simple Cubic || (BCC) Body Centered Cubic|| (FCC) Face Centered Cubic
 
 
|-
 
|-
 +
|<math> RC << \tau</math> || <math>RC >> \tau</math>
 
|}
 
|}
  
The above images are suppose to represent the distribution of electrons in the crystal lattice cell.
+
== Loop Theorem==
  
 +
:<math>V_{in} = V_C + V_R = I X_C + I R</math>
  
To understand this structure more let's consider the structure of a 1-D crystal of atoms
 
  
 +
;How does <math>V_C</math> compare to <math>V_R</math> ?
  
A single free atom will have electrons bound to the atom due to the coulomb force. The bound electrons are found in various energy levels.
+
:<math>\left | \frac{V_R}{V_C} \right | = \frac{\left | IR \right |}{\left | I X_C\right |}= \frac{R}{\left | \frac{1}{i \omega C}\right |} = \omega RC</math>
  
 +
The maximum value of <math>\omega</math> is determined by the pulse width <math> \tau</math> while <math>T</math> is the period that corresponds the lowest frequency (other than DC).
  
[[File:TF_EIM_Crystal_Cell_EnergyLevels.png| 200 px]]
 
  
;Notice
+
:<math>\left | \frac{V_R}{V_C} \right |_{MAX} = \frac{RC}{\tau}</math>
:When you add the atoms to form the lattice, the potential barrier is lowered.  This allows the electrons in the higher (more valence) states to become shared among adjacent atoms.( Delocalized)
 
:Another effect is that the energy bands become wider.  As the electrons in the higher energy states are shared between atoms forming a bond, the time any one electron spends bound to any one atom becomes smaller.  SInce<math> \Delta E \Delta t \approx \hbar</math> one finds that <math>\Delta E</math> gets bigger for the higher state electrons (less bound) since their time \Delta t in a given bound state becomes less.  The electron energy states become band like structures which get wider as you move to higher energy (less bound) electron states.
 
  
 +
=== <math>RC << \tau</math> ===
  
There are 14 different Bravais lattice configurations
+
When <math>RC << \tau</math>
  
{| border="3"  cellpadding="20" cellspacing="0"
 
|Crystal System|| Lattice type
 
|-
 
|Cubic || Simple, Face Centered, Body Centered
 
|-
 
|Tetragonal || Simple, Body Centered
 
|-
 
|Orthorhombic|| Simple, Face Centered, Body Centered, End Centered
 
|-
 
|Monoclinic|| Simple, End Centered
 
|-
 
|Rhombohedral|| Simple
 
|-
 
|Triclinic|| Simple
 
|-
 
|Hexagonal|| Simple
 
|-
 
|}
 
  
Most Semiconductors are made from Silicon and GaAs.
+
:<math>\left | \frac{V_R}{V_C} \right |_{MAX} = \frac{RC}{\tau} << 1 \Rightarrow V_R </math> may be ignored.
 +
 
 +
:<math>\Rightarrow V_{in} = V_C + V_R \approx V_C = \frac{Q}{C}</math>
 +
 
 +
taking the derivative with respect to time yields
 +
 
 +
: <math>\frac{d V_{in}}{dt} = \frac{I}{C}</math>
 +
 
 +
or
 +
 
 +
: <math>I = C \frac{d V_{in}}{dt}</math>
 +
 
 +
The Ohm's Law for <math>V_{out}</math> :
 +
 
 +
:<math>V_{out} = IR = \left ( C \frac{d V_{in}}{dt} \right ) R = RC \frac{d V_{in}}{dt}</math>
 +
 
 +
 
 +
The output of the above circuit when <math>RC << \tau</math> will be proportional to the derivative of the input AC source with respect to time.
 +
 
 +
=Integrator=
 +
 
 +
[[File:TF_EIM_PulsedRCLowpass.png | 400 px]]
 +
 
 +
The above circuit is a low pass filter.  When a square pulse hits it all of the low frequency components will pass through and the high frequency components will be attenuated.  This essentially makes the pulse smooth as shown below.
 +
 
 +
[[File:TF_EIM_SquarePulse.png| 200 px]]
 +
 
 +
The above is referred to as an integration circuit.
 +
 
 +
==Loop theorem==
 +
 
 +
:<math>V_{in} = IR + \frac{Q}{C}</math>
 +
 
 +
:: <math>= R \frac{dQ}{dt} + \frac{Q}{C}</math>
 +
 
 +
This differential equation may be written as
 +
 
 +
: <math>\frac{dQ}{dt} + \frac{Q}{RC} = \frac{V_{in}}{R}</math>
 +
 
 +
:<math>\Rightarrow Q = CV_{in} \left (1 - e^{-t/RC} \right)</math>
 +
 
 +
 
 +
:<math>V_{out} = \frac{Q}{C} = V_{in}\left (1 - e^{-t/RC} \right)</math>
 +
 
 +
== <math>RC >> \tau</math> ==
 +
 
 +
If <math>RC >> \tau</math>
 +
 
 +
then
 +
 
 +
::<math>t/RC << 1</math>
 +
 
 +
:<math>e^{-t/RC} \approx 1- t/RC</math>
 +
 
 +
:<math>\Rightarrow V_{out} = V_{in}\left (1 - e^{-t/RC} \right) = V_{in} \frac{t}{RC} = \frac{1}{RC} \int_0^t V_{in} dt</math>  The circuit appears to be integrating the input voltage.
 +
 
 +
If <math>RC << \tau</math> then very little integration is done because the pulse is changing at a low frequency compared to the RC constant.
 +
 
 +
=Pulse Sharpening=
 +
 
 +
The circuit below is a way to decrease the rise time of an input pulse at the expense of attenuating the pulse.  The output pulse is a "sharpened" version of the input pulse and attenuated.
 +
 
 +
 
 +
[[File:TF_EIM_PulseSharpnr.png| 400 px]]
 +
 
 +
==Junction Rule==
 +
 
 +
If we apply the junction rule for the currents in the above circuit
 +
 
 +
Then
 +
 
 +
:<math>I_1 +I_2 = I</math>
 +
 
 +
:<math>I_1 = \frac{V_{in}-V_{out}}{R_1}</math>
 +
:<math>I_2 = \frac{d}{dt} C\left( V_{in}-V_{out}\right) : Q =C \Delta V = C\left( V_{in}-V_{out}\right)</math>
 +
:<math>I =  \frac{V_{out}}{R}</math>
  
  
Silicon is a Face Centered Cubic cell
+
:<math>\Rightarrow</math>
 +
::<math>\frac{V_{in}-V_{out}}{R_1} + \frac{d}{dt} C\left( V_{in}-V_{out}\right) =  \frac{V_{out}}{R}</math>
  
Silicon is an insulator if in pure form with 4 weakly bound (valence) electrons. 
+
Collecting terms
  
If you replace silicon atoms in the lattice with atoms that have either 3 valance or 5 valence electrons (doping) your can create sites with either a deficient number of electrons (a missing bond) or extra electrons (complete bond with a free electron)
+
:<math>\frac{V_{in}}{R_1} + \frac{d}{dt} C\left( V_{in}\right) =  \frac{d}{dt} C\left( V_{out}\right) + \frac{V_{out}}{R} + \frac{V_{out}}{R_1} </math>
 +
:<math>\frac{d V_{in} }{dt}+ \frac{V_{in}}{CR_1}=  \frac{dV_{out}}{dt}  + \frac{(R+R_1)}{CRR_1} V_{out} </math>= constant = <math>\frac{V_0}{CR_1} =  \frac{V_0}{\tau_{in}}</math>
  
[[File:TF_EIM_SiliconLattice_np_junction.jpg| 200 px]]
 
  
By Doping silicon you can create sites with extra electrons (n-type) or sites with a deficient number of electrons (vacancies or holes ) (P-type)
+
assuming
  
 +
:<math>V_{in} = V_0 \left ( 1-e^{-t/\tau_{in}}\right )</math>
  
 +
:<math>\Rightarrow</math>
 +
::<math>\frac{dV_{out}}{dt}  + \frac{(R+R_1)}{CRR_1} V_{out} = V_0 \left ( \frac{1}{\tau_{in}} - \frac{1}{CR_1}\right )e^{-t/\tau_{in}} + \frac{V_0}{CR_1} </math>
  
=Fermi-Dirac Statistics=
+
If <math>CR_1 = \tau_{in}</math>
  
As seen above, the properties of a semi-conductor will be determined by the electron configuration in the crystal lattice.  In particular, the distribution of electrons in the conduction band of the lattice since those are the electrons which can move in the crystal in response to an applied electric field (voltage) as produce observable effect.
+
Then
  
 +
:<math>\frac{dV_{out}}{dt}  + \frac{(R+R_1)}{CRR_1} V_{out} =  \frac{V_0}{CR_1} = Constant</math>
  
==Pauli Exclusion Principle==
+
The above is a first order non-homogeneous differential equation
  
The Pauli exclusion principle states that no two electrons may occupy the same quantum state.  This principle is seen by the quantum energy levels of electrons bound to atoms.
+
To solve you first find a solution to the homogeneous equation (<math>V_{out}^H)</math> and then form a particular solution <math>(V_{out}^P)</math> based on the homogeneous solution
  
An important consequence of this principle is that, an crystal composed of atoms will filled energy bands is an insulator. You can only conduct electrons if there is an energy state available for the electron to populate.  An electric field is established by applying a potential difference which pushes an electron up to a partially filled available energy state that is shared between the atoms in the lattice, thereby allowing the electron to "travel" to adjacent atoms.
+
===Homogeneous Solution===
 +
:<math>\frac{dV_{out}^H}{dt}  + \frac{(R+R_1)}{CRR_1} V_{out}^H = 0</math>
  
==Fermi Function (F[E]) ==
 
  
The energy levels in a crystal lattice may be grouped into a conduction band and a valence band that are separated by a forbidden energy region called the band gap.
+
:<math>\Rightarrow \frac{dV_{out}^H}{dt}  = - \frac{(R+R_1)}{CRR_1} V_{out}^H \equiv \frac{1}{CR_{\parallel}} V_{out}^H</math>
 +
:<math> V_{out}^H = D e^{-t/CR_{\parallel}}</math>
  
<math>F[E] \equiv</math> the probability that a given energy level <math>E</math>  is occupied by an electron.
+
=== Particular Solution===
 +
:<math>\frac{dV_{out}}{dt}  + \frac{(R+R_1)}{CRR_1} V_{out} =  \frac{V_0}{CR_1} = Constant</math>
  
using Statistical Thermodynamic one can calculate the above Fermi function as
+
:<math>V_{out} = V_{out}^P + V_{out}^H</math>
  
:<math>F[E] = \frac{1}{e^{\left( E-E_f\right)/kT}+1}</math>
+
::<math> V_{out}^H = D e^{-t/CR_{\parallel}}</math>
  
where
+
:<math>\Rightarrow</math>
 +
::<math>V_{out}^P = \frac{CR_{\parallel}}{CR_1}V_0</math>
  
:<math>E_f =</math> Fermi energy of the given crystal lattice
+
=== Apply Boundary Conditions===
:<math>k = 1.38 \times 10^{-23} J/deg K</math>
 
:<math>T =</math> temperature of lattice in Kelvin
 
  
;Note: At room temperature: kT = 1/40 eV
+
:<math>V_{out} =\frac{R_{\parallel}}{R_1}V_0 + D e^{-t/CR_{\parallel}}</math>
  
The number of electrons per unit volume within the energy band<math> E + dE</math> is represented as
+
:<math>V_{out}(t=0) = 0 \Rightarrow  D = - \frac{R_{\parallel}}{R_1}V_0</math>
  
<math>N(E) dE = \rho(E) dE F(E)</math>
+
:<math>V_{out} =\frac{R_{\parallel}}{R_1}V_0 (1- e^{-t/CR_{\parallel}})</math>
 +
::<math>= \frac{R}{R+R_1}V_0 (1- e^{-t/\tau_{out}})</math>
  
 
where  
 
where  
  
:\<math>rho(E)  = \frac{2^{7/2}m^{3/2} \pi}{h^3}E^{1/2}=</math> density of states
+
:<math>\tau_{out} \equiv CR_{\parallel} = C \frac{RR_1}{R+R_1} =\frac{R}{R+R_1} \tau_{in} < \tau_{in} </math>
 +
 
 +
;Note
 +
:<math>\tau_{out}</math> is always less than <math>\tau_{in}</math>
 +
 
 +
or in other words the rise time of the output pulse is short than the rise time of the input pulse
  
 +
or in other words the output pulse is sharper (rising faster) than the input pulse
  
 +
: <math>\frac{\tau_{out}}{\tau_{in}} = \frac{R}{R+R_1}</math>
  
 +
;So the output pulse is faster by a factor of <math>\frac{R}{R+R_1}</math> and attenuated by the same factor!
  
 
[[Forest_Electronic_Instrumentation_and_Measurement]]
 
[[Forest_Electronic_Instrumentation_and_Measurement]]

Latest revision as of 16:52, 12 February 2015

Differentiator circuit

Consider the effect of a high-pass filter on a rectangular shaped input pulse with a width [math]\tau[/math] and period [math]T[/math].


TF EIM PulsedRCHighpass.png


The first thing to consider is what happens to the different parts of the square pulse as it travels to the circuit.

You know that the circuit, as a high pass filter, will tend to attenuate low frequency (slow changing) voltages and not high frequency (quick changing) voltages.

This means no DC voltage values pass beyond the capacitor.

This means that points [math]a[/math] and [math]b[/math] in the pulse shown below should pass through the circuit.

TF EIM SquarePulseDifferentiated.png

The voltage changes in between point [math]a[/math] and DC are passed according to what you select for the break frequency ([math]\omega_B = \frac{1}{RC}[/math])

In the above pulse picture it was assumed that [math]\tau \gt \gt \frac{1}{\omega_B} = RC[/math] or in other words the RC network time constant is a lot smaller than the pulse width.

If the RC time constant is larger than the pulse width then you will see the voltage stay high. The rectangular pulse will be output as a square like pulse which is "sagging".


TF EIM PulsedRCHighpassSmallRC.png TF EIM PulsedRCHighpassBigRC.png
[math] RC \lt \lt \tau[/math] [math]RC \gt \gt \tau[/math]

Loop Theorem

[math]V_{in} = V_C + V_R = I X_C + I R[/math]


How does [math]V_C[/math] compare to [math]V_R[/math] ?
[math]\left | \frac{V_R}{V_C} \right | = \frac{\left | IR \right |}{\left | I X_C\right |}= \frac{R}{\left | \frac{1}{i \omega C}\right |} = \omega RC[/math]

The maximum value of [math]\omega[/math] is determined by the pulse width [math] \tau[/math] while [math]T[/math] is the period that corresponds the lowest frequency (other than DC).


[math]\left | \frac{V_R}{V_C} \right |_{MAX} = \frac{RC}{\tau}[/math]

[math]RC \lt \lt \tau[/math]

When [math]RC \lt \lt \tau[/math]


[math]\left | \frac{V_R}{V_C} \right |_{MAX} = \frac{RC}{\tau} \lt \lt 1 \Rightarrow V_R [/math] may be ignored.
[math]\Rightarrow V_{in} = V_C + V_R \approx V_C = \frac{Q}{C}[/math]

taking the derivative with respect to time yields

[math]\frac{d V_{in}}{dt} = \frac{I}{C}[/math]

or

[math]I = C \frac{d V_{in}}{dt}[/math]

The Ohm's Law for [math]V_{out}[/math] :

[math]V_{out} = IR = \left ( C \frac{d V_{in}}{dt} \right ) R = RC \frac{d V_{in}}{dt}[/math]


The output of the above circuit when [math]RC \lt \lt \tau[/math] will be proportional to the derivative of the input AC source with respect to time.

Integrator

TF EIM PulsedRCLowpass.png

The above circuit is a low pass filter. When a square pulse hits it all of the low frequency components will pass through and the high frequency components will be attenuated. This essentially makes the pulse smooth as shown below.

TF EIM SquarePulse.png

The above is referred to as an integration circuit.

Loop theorem

[math]V_{in} = IR + \frac{Q}{C}[/math]
[math]= R \frac{dQ}{dt} + \frac{Q}{C}[/math]

This differential equation may be written as

[math]\frac{dQ}{dt} + \frac{Q}{RC} = \frac{V_{in}}{R}[/math]
[math]\Rightarrow Q = CV_{in} \left (1 - e^{-t/RC} \right)[/math]


[math]V_{out} = \frac{Q}{C} = V_{in}\left (1 - e^{-t/RC} \right)[/math]

[math]RC \gt \gt \tau[/math]

If [math]RC \gt \gt \tau[/math]

then

[math]t/RC \lt \lt 1[/math]
[math]e^{-t/RC} \approx 1- t/RC[/math]
[math]\Rightarrow V_{out} = V_{in}\left (1 - e^{-t/RC} \right) = V_{in} \frac{t}{RC} = \frac{1}{RC} \int_0^t V_{in} dt[/math] The circuit appears to be integrating the input voltage.

If [math]RC \lt \lt \tau[/math] then very little integration is done because the pulse is changing at a low frequency compared to the RC constant.

Pulse Sharpening

The circuit below is a way to decrease the rise time of an input pulse at the expense of attenuating the pulse. The output pulse is a "sharpened" version of the input pulse and attenuated.


TF EIM PulseSharpnr.png

Junction Rule

If we apply the junction rule for the currents in the above circuit

Then

[math]I_1 +I_2 = I[/math]
[math]I_1 = \frac{V_{in}-V_{out}}{R_1}[/math]
[math]I_2 = \frac{d}{dt} C\left( V_{in}-V_{out}\right) : Q =C \Delta V = C\left( V_{in}-V_{out}\right)[/math]
[math]I = \frac{V_{out}}{R}[/math]


[math]\Rightarrow[/math]
[math]\frac{V_{in}-V_{out}}{R_1} + \frac{d}{dt} C\left( V_{in}-V_{out}\right) = \frac{V_{out}}{R}[/math]

Collecting terms

[math]\frac{V_{in}}{R_1} + \frac{d}{dt} C\left( V_{in}\right) = \frac{d}{dt} C\left( V_{out}\right) + \frac{V_{out}}{R} + \frac{V_{out}}{R_1} [/math]
[math]\frac{d V_{in} }{dt}+ \frac{V_{in}}{CR_1}= \frac{dV_{out}}{dt} + \frac{(R+R_1)}{CRR_1} V_{out} [/math]= constant = [math]\frac{V_0}{CR_1} = \frac{V_0}{\tau_{in}}[/math]


assuming

[math]V_{in} = V_0 \left ( 1-e^{-t/\tau_{in}}\right )[/math]
[math]\Rightarrow[/math]
[math]\frac{dV_{out}}{dt} + \frac{(R+R_1)}{CRR_1} V_{out} = V_0 \left ( \frac{1}{\tau_{in}} - \frac{1}{CR_1}\right )e^{-t/\tau_{in}} + \frac{V_0}{CR_1} [/math]

If [math]CR_1 = \tau_{in}[/math]

Then

[math]\frac{dV_{out}}{dt} + \frac{(R+R_1)}{CRR_1} V_{out} = \frac{V_0}{CR_1} = Constant[/math]

The above is a first order non-homogeneous differential equation

To solve you first find a solution to the homogeneous equation ([math]V_{out}^H)[/math] and then form a particular solution [math](V_{out}^P)[/math] based on the homogeneous solution

Homogeneous Solution

[math]\frac{dV_{out}^H}{dt} + \frac{(R+R_1)}{CRR_1} V_{out}^H = 0[/math]


[math]\Rightarrow \frac{dV_{out}^H}{dt} = - \frac{(R+R_1)}{CRR_1} V_{out}^H \equiv \frac{1}{CR_{\parallel}} V_{out}^H[/math]
[math] V_{out}^H = D e^{-t/CR_{\parallel}}[/math]

Particular Solution

[math]\frac{dV_{out}}{dt} + \frac{(R+R_1)}{CRR_1} V_{out} = \frac{V_0}{CR_1} = Constant[/math]
[math]V_{out} = V_{out}^P + V_{out}^H[/math]
[math] V_{out}^H = D e^{-t/CR_{\parallel}}[/math]
[math]\Rightarrow[/math]
[math]V_{out}^P = \frac{CR_{\parallel}}{CR_1}V_0[/math]

Apply Boundary Conditions

[math]V_{out} =\frac{R_{\parallel}}{R_1}V_0 + D e^{-t/CR_{\parallel}}[/math]
[math]V_{out}(t=0) = 0 \Rightarrow D = - \frac{R_{\parallel}}{R_1}V_0[/math]
[math]V_{out} =\frac{R_{\parallel}}{R_1}V_0 (1- e^{-t/CR_{\parallel}})[/math]
[math]= \frac{R}{R+R_1}V_0 (1- e^{-t/\tau_{out}})[/math]

where

[math]\tau_{out} \equiv CR_{\parallel} = C \frac{RR_1}{R+R_1} =\frac{R}{R+R_1} \tau_{in} \lt \tau_{in} [/math]
Note
[math]\tau_{out}[/math] is always less than [math]\tau_{in}[/math]

or in other words the rise time of the output pulse is short than the rise time of the input pulse

or in other words the output pulse is sharper (rising faster) than the input pulse

[math]\frac{\tau_{out}}{\tau_{in}} = \frac{R}{R+R_1}[/math]
So the output pulse is faster by a factor of [math]\frac{R}{R+R_1}[/math] and attenuated by the same factor!

Forest_Electronic_Instrumentation_and_Measurement