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Introduction
Review of Fundamentals
General Theme
Components at the
Macro Level - Ideal World
Micro Level - Real World
Basic Quantities
Current
Voltage
Resistance
Capacitance
Inductance
Time  Frequency
Measurement in each case can be
Relative
Absolute
With respect to a reference
Usually common which we call ground
Basic Laws
Ohms
Kirchoff
Voltage
Current
Basic Models
Thevenin
Most frequently used
Norton
Basic Components
Resistor
Capacitor
Inductor
Wire
Special case of a resistor
Components
Resistor
R
A
At the physical level we have
l
R
A
l
As l increases / decreases
R increases / decreases
A increases / decreases
R decreases / increases
Discrete Component Model
We model the resistor as follows
L  10 nH
C  5 pf
Intuitively
At DC - We speak of resistance
L is short
C is open
At AC - we now speak of impedance
L has finite non-zero impedance
C is finite impedance
1
CS
R
 LS 
RCS  1
Z    LS  R||
Z   

R 2 1  LC 2
  L
2
1   RC 
Checking the boundaries
For
 = 0,
|z()| = R
2
2

z   L RC
Observe magnitude of z
Begins to increase again
Because of the inductive and capacitive elements
We get a phase shift
The value is given by
  1  2

L

2 
 R  1  LC  
 1  tan 1  
 2  tan 1  RC 
If we now plot Z(R) vs frequency for various values of R we get
R = 10k, 1k, 0.1k
10000 1 10
4
1000
1 10
1 10
3
3
z( w )
z( w )
100
100
10
10
10
6
1 10
6
10
7
1 10
1 10
8
w
9
1 10
1000
z( w )
10
1 10
1 10
10
6
1 10
6
10
11
1 10
11
10
7
1 10
1 10
8
w
9
1 10
3
100
Typical
Values
10
1, 2, 3,
10, 11,
10k, 11k, 12k...
10
6
1 10
6
10
Tolerances
Off the shelf values
1%, 5%, 10%
7
1 10
1 10
8
w
9
1 10
10
1 10
11
1 10
11
10
4...
12...
10
1 10
11
1 10
11
10
Wattages
1/8, 1/4, 1/2, 1, 2, 5, 10...
Color Code
Black
Brown
Red
Orange
Yellow
Green
Blue
Violet
Gray
White
0
1
2
3
4
5
6
7
8
9
Use in a Circuit
May be discrete part
Built in hybrid
Implemented in integrated circuit
When used in circuit
Easier to hold
Ratio rather than absolute value
In hybrid
Can easily automatically trim ratio
Environment
Aging
Humidity
Drift
Temperature
Capacitors
C
A
At the physical level we have
A
C
d
As A increases / decreases
C increases / decreases
d
d increases / decreases
C decreases / increases
Discrete Component Model
We model the capacitor as follows
L  10 nH
C  5 pf
Intuitively
At DC
L is short
C is open
R is finite
At AC - the capacitor has an impedance
L has finite non-zero impedance
C is finite non-infinite impedance
R is finite
z
1
 Ls  R
Cs


 1  LC 2 2   RC  2 

z   
2


 C 


1/ 2
Because of the inductive element
We get a phase shift
The value is given by
  1   2
 RC 

 1  LC 2 
 1  tan 1 
2 

2
Plot of Z(C) vs  for various values of C is given as
C = 1 f, 0.1 f , 0.01 f
6
10
6
1 10
6
10
5
1 10
4
1 10
6
1 10
5
1 10
4
1 10
3
1 10
3
1 10
100
z( w )
z( w )
100
10
10
1
1
0.1
10
0.1
2
0.01
3
1 10
3
10
4
1 10
5
1 10
1 10
w
6
8
1 10
7
1 10
6
10
9
1 10
9
10
10
2
0.01
3
1 10
3
10
4
1 10
5
1 10
1 10
w
6
6
1 10
5
1 10
4
1 10
3
1 10
z( w )
100
10
1
0.1
10
2
0.01
3
1 10
3
10
Typical
Values
Tolerances
Off the shelf values
5%, 10%
Voltage Ratings
Use in a Circuit
May be discrete part
Built in hybrid
Implemented in integrated circuit
Metal Sandwich
Junction Capacitance
Environment
Aging
Humidity
Drift
Temperature
4
1 10
5
1 10
1 10
w
6
7
1 10
8
1 10
9
1 10
9
10
7
1 10
8
1 10
9
1 10
9
10
Simple Circuits
We often use simple first and second order circuits
Model more complex real world circuits
First Order RC
Vin and Vout
Related by simple voltage divider
Vout
 1 


  Cs  Vin
R 1 

Cs 
 1 

V
 RCs  1 in
For Vin a step
Vout 
Vout
Vin 
1 


s  RCs  1


1
1 

 Vin  
s s 1 

RC 
t



Vout  Vin  1  e RC 


Which we plot as
6
6
4.5
Vout( t )
3
1.5
0
0
0
0
0.025
0.05
t
0.075
0.1
0.1
First Order R L
Again we compute output as simple voltage divider
 Ls 
Vout  
V
 R  Ls  in


 s 

 Vin
s R

L
For Vin a step
Vout


Vin  s 



R
s 
s

L
Vout


 1 

 Vin 
s R

L
Vout  e

t
R/ L
Which we plot as
6
6
4.5
Vout( t )
3
1.5
First
0
0
0
0
0.0025
0.005 0.0075
t
0.01
3
10 10
Order Currents
Plots of the first order currents
Will have opposite waveforms
Second Order Series RLC
One variation on second order series RLC appears as
Once again we use simple voltage divider
To compute Vout
Vout
1




Cs
 Vin

 R  LS  1 

CS 




V
1

 in 
1 
LC  2 R
s  s

L
LC 
Expression in denominator on right hand side
Can be written as the characteristic equation
Thus
Vout 

Vin 
1
 2
2
LC  s  2 n s   n 
n 
1
LC
R  L
  
2  C
1/ 2
Recall the value of  determines if circuit is
Underdamped  < 1
Critically damped  = 1
Overdamped  > 1
Q
 L / C1/ 2

R
n L
R

1
2
For Vin a step
sin
v( t )
5  exp
5
1w 
t
2 Q
2
1   4Q
w
2
Q
4 Q
2
1
t
5  exp
2
1w 
1
4 Q
t cos  w 
2 Q
2
Q
1
We can plot the behaviour of the circuit as
7
7
10 10
4.75
4
1 10
3
v( t )
2500
2.5
o( t )
do( t )
1.5 10
4
0.25
2.75 10
2
2
0
0
2.5 10
4
5 10
4
t
7.5 10
4
0.001
3
1 10
4 10
4
4 10
0
4
4
0
7.5 10
4
0.0015 0.00225 0.003
t
.003
Summary
Have looked as some of basic analog issues
Will discuss several of these in greater detail in upcoming weeks
1
t