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>
 
|}
 
|}
There are 14 different Bravais lattice configurations
 
  
{| border="3"  cellpadding="20" cellspacing="0"
+
== Loop Theorem==
|Crystal System|| Lattice type
+
 
|-
+
:<math>V_{in} = V_C + V_R = I X_C + I R</math>
|Cubic || Simple, Face Centered, Body Centered
+
 
|-
+
 
|Tetragonal || Simple, Body Centered
+
;How does <math>V_C</math> compare to <math>V_R</math> ?
|-
+
 
|Orthorhombic|| Simple, Face Centered, Body Centered, End Centered
+
:<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>
|-
+
 
|Monoclinic|| Simple, End Centered
+
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).
|-
+
 
|Rhombohedral|| Simple
+
 
|-
+
:<math>\left | \frac{V_R}{V_C} \right |_{MAX} = \frac{RC}{\tau}</math>
|Triclinic|| Simple
+
 
|-
+
=== <math>RC << \tau</math> ===
|Hexagonal|| Simple
+
 
|-
+
When <math>RC << \tau</math>
|}
+
 
 +
 
 +
:<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>
 +
 
 +
 
 +
:<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>
  
Most Semiconductors are made from Silicon and GaAs.
+
=== Apply Boundary Conditions===
  
 +
:<math>V_{out} =\frac{R_{\parallel}}{R_1}V_0 + D e^{-t/CR_{\parallel}}</math>
  
Silicon is a Face Centered Cubic cell
+
:<math>V_{out}(t=0) = 0 \Rightarrow  D = - \frac{R_{\parallel}}{R_1}V_0</math>
  
Silicon is an insulator if in pure form with 4 weakly bound (valence) electrons.  if you replace silicon atoms in the lattice with atoms that have either 3 valance or 5 valence electrons your can create sites with either a deficient number of electrons (a missing bond) or extra electrons (complete bond with a free electron)
+
:<math>V_{out} =\frac{R_{\parallel}}{R_1}V_0 (1- e^{-t/CR_{\parallel}})</math>
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)
+
::<math>= \frac{R}{R+R_1}V_0 (1- e^{-t/\tau_{out}})</math>
  
 +
where
  
[[File:TF_EIM_SiliconLattice_np_junction.jpg| 200 px]]
+
:<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