Lecture Notes on Basic Electronics
for Students in Computer Science
John Kar-kin Zao and Wen-Hsiao Peng
Department of Computer Science, National Chiao-Tung Univeristy
1001 Ta-Hsueh Rd., 30010 HsinChu, Taiwan
August 2006
Contents
1 Preamble
1.1 Goal of This Course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Content of This Course . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Relationship with Other Disciplines . . . . . . . . . . . . . . . . . . . . . .
1
1
1
1
I
2
System and Circuit Analysis
2 Signals and Systems
2.1 Basic Concepts . . . . . . . . . . . . . . . . . . . .
2.2 Types of Systems . . . . . . . . . . . . . . . . . . .
2.3 Axiomatic Properties of Systems . . . . . . . . . .
2.4 Time-Domain Analysis . . . . . . . . . . . . . . . .
2.4.1 Impulse Response . . . . . . . . . . . . . . .
2.4.2 Step Response . . . . . . . . . . . . . . . . .
2.4.3 Sinusoidal Response . . . . . . . . . . . . .
2.4.4 Initial Value/Driving Free/Natural Response
2.4.5 Transient Response . . . . . . . . . . . . . .
2.4.6 Steady-State Response . . . . . . . . . . . .
2.5 Frequency-Domain Analysis . . . . . . . . . . . . .
2.5.1 Phasor . . . . . . . . . . . . . . . . . . . . .
2.5.2 Spectrum and Fourier Transform . . . . . .
2.5.3 System Transfer Function Kv ($) . . . . . . .
2.5.4 Time Domain versus Frequency Domain . .
3 Electrical Circuits
3.1 Basic Concepts . .
3.2 Circuit Elements .
3.3 Circuit Laws . . . .
3.4 Network Theorems
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
ii
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3
3
4
5
6
6
8
8
12
12
12
13
13
13
14
16
.
.
.
.
17
17
17
20
21
CONTENTS
3.4.1
Equivalent Circuits of One-Port Networks . . . . . . . . . . . . . . 23
3.4.2
Equivalent Circuits of Two-Port Networks . . . . . . . . . . . . . . 25
3.5 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.6 Equivalent Circuits in Laplace Domain . . . . . . . . . . . . . . . . . . . . 31
3.7 System Transfer Function in Time and Laplace Domains . . . . . . . . . . 33
3.8 Bode Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
II
Devices — Diode, BJT, MOSFETs
4 Semiconductor
49
50
4.1 Intrinsic Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Doped Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Diode
55
5.1 Physical Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.1.1
The sq Junction Under Open Circuit . . . . . . . . . . . . . . . . . 55
5.1.2
The sq Junction Under Reverse-Bias . . . . . . . . . . . . . . . . . 57
5.1.3
The sq Junction in the Breakdown Region . . . . . . . . . . . . . . 59
5.1.4
The sq Junction Under Forward Bias Conditions . . . . . . . . . . 59
5.2 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2.1
Forward Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2.2
Reverse Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2.3
Breakdown Region . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3.1
Large Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3.2
Small Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3.3
Circuit Analysis with Diodes . . . . . . . . . . . . . . . . . . . . . . 70
5.4 Special Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.4.1
Zener diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.4.2
Switching Controlled Rectifier (SCR) . . . . . . . . . . . . . . . . . 72
5.4.3
LED/Varactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.5.1
Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.5.2
Rectifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.5.3
Limiting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.5.4
Clamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.5.5
Digital Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
iii
CONTENTS
6 MOS Field-Egect Transistors (MOSFETs)
6.1 Structure . . . . . . . . . . . . . . . . . . . .
6.1.1 Physical Structure . . . . . . . . . . .
6.2 Characteristics of NMOS Transistor . . . . . .
6.2.1 Cut-og Region (yJV ? Yw ) . . . . . . .
6.2.2 Triode (yJV A Yw , 0 ? yGV ? yJV Yw )
6.2.3 Saturation (yJV A Yw > yGV A yJV Yw )
6.3 Characteristics of PMOS Transistor . . . . . .
6.3.1 Body Egect . . . . . . . . . . . . . . .
6.3.2 Internal Capacitances . . . . . . . . . .
6.3.3 Temperature Egect . . . . . . . . . . .
6.3.4 Summary . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
7 Bipolar Junction Transistor (BJT)
7.1 Structure . . . . . . . . . . . . . . . . . . . . . .
7.1.1 NPN Transistor . . . . . . . . . . . . . . .
7.1.2 PNP Transistor . . . . . . . . . . . . . . .
7.2 Operations of NPN Transistor . . . . . . . . . . .
7.2.1 Active Mode . . . . . . . . . . . . . . . . .
7.2.2 Reverse Active Mode . . . . . . . . . . . .
7.2.3 Ebers-Moll (EM) Model . . . . . . . . . .
7.2.4 Saturation Mode . . . . . . . . . . . . . .
7.3 Operations of PNP Transistor . . . . . . . . . . .
7.3.1 Active Mode . . . . . . . . . . . . . . . . .
7.3.2 Reverse Active Mode . . . . . . . . . . . .
7.3.3 Saturation Mode . . . . . . . . . . . . . .
7.3.4 Summary of the lF > lE > lH Relationships in
7.4 The l y Characteristics of NPN Transistor . . .
7.4.1 Common Base (lF yFE ) . . . . . . . . .
7.4.2 Common Emitter (lF yFH ) . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
Active
. . . .
. . . .
. . . .
8 Comparisons of BJT and MOSFET
8.1 NMOS and NPN Transistors . . . . . . . . . . . . .
8.1.1 NMOS Triode v.s. NPN Saturation . . . . .
8.1.2 NMOS Saturation v.s. NPN Forward Active
8.1.3 NMOS Cut-og v.s. NPN Cut-og . . . . . . .
8.2 PMOS and PNP Transistors . . . . . . . . . . . . .
8.2.1 PMOS Triode v.s. PNP Saturation . . . . .
8.2.2 PMOS Saturation v.s. PNP Forward Active
8.2.3 PMOS Cut-og v.s. PNP Cut-og . . . . . . .
iv
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
Mode
. . . .
. . . .
. . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
81
82
82
83
83
83
87
90
91
92
93
93
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
95
95
95
97
98
98
102
103
105
106
106
108
108
108
109
109
110
.
.
.
.
.
.
.
.
115
. 115
. 116
. 117
. 118
. 118
. 119
. 120
. 121
.
.
.
.
.
.
.
.
.
.
.
1
Preamble
1.1
Goal of This Course
• Analysis and design of electronic circuits.
1.2
Content of This Course
• Electric circuit (Passive Circuit ) analysis.
— Only containing Resistor (U), Capacitor (F), and Inductor (O).
— No signal amplification.
• Electronic circuit (Active Circuit ) analysis.
— Containing transistor.
— Single transistor circuit.
— Signal amplification.
• Operational amplifier and application.
1.3
Relationship with Other Disciplines
1
Part I
System and Circuit Analysis
2
2
Signals and Systems
2.1
Basic Concepts
Table 2.1: Signals and Systems in Time and Frequency Domains.
Signals
Systems
Time Domain
Waveforms, {(w) Impulse Response, k(w)
Frequency Domain Spectrum, [($) Frequency Response, K($)
Definition 2.1 System stands for the transformation of signal from one to another. It
can be viewed as a process in which input signals are transformed by the system or cause
the system to respond in some way, resulting in other signals as outputs.
Objects
Abstraction
Systems
Composition
Systems
Decomposition
Analysis
Synthesis
Components
System characteristics
/brhaviors
Formalization
Models
Figure 2.1: System from engineering perspective.
• Approach
— Abstraction
Representing a real object by its special characteristics; that is, the relation
between its inputs and outputs, which becomes a system.
3
Sec 2.2. Types of Systems
— Decomposition
Dividing a system into several smaller systems (components) and studying
them to understand the large system.
— Composition
Putting several systems together to form a larger system and studying it.
• Model
Inputs
LTI Systems
Outputs
States
• Relative
Input
States
Output
• Perspectives
— Time domain.
— Frequency domain.
2.2
Types of Systems
• Temporal characteristic
— Continuous-Time Systems
Inputs/outputs of the systems are functions defined at continuous time.
Input x(t) Continuous-Time Output y(t)
System
Magnitude
t
Figure 2.2: Continuous-time systems.
— Discrete-Time Systems
4
Lecture 2. Signals and Systems
Input x[n] Discrete-Time Output y[n]
System
Magnitude
6 7 8 9
t
0 1 2 3 4 5
Figure 2.3: Discrete-time systems.
Input x(t)
Output y(t)
Digital System
Magnitude
t
Figure 2.4: Digital system.
Inputs/outputs of the systems are functions defined at discrete time.
• Magnitude
— Analog systems
Inputs/outputs of the system have continuously varying values.
— Digital systems
Inputs/outputs of the system have discrete values.
2.3
Axiomatic Properties of Systems
• Linearity
— If an input consists of the weighted sum of several signals, then the output is
the superposition of the responses of the system to each of those signals.
|1(w) = K{{1(w)}
|2(w) = K{{2(w)}
d × |1(w) + e × |2(w) = K{d × {1(w) + e × {2(w)}
5
(2.1)
Sec 2.4. Time-Domain Analysis
y(t)
x(t)
Inverse
System
System
x(t)
Figure 2.5: Inverse system.
• Time-Invariant
— The behavior and characteristics of the system are fixed over time.
— For example, the magnitudes of resistors and capacitors of a circuit are unchanged over time.
|(w) = K{{(w)}
|(w ) = K{{(w )}
(2.2)
• Causality
— The output of the system depends only on the inputs at the present time and
in the past.
Z w
{( )g =
— For example, |(w) =
0
• Invertability
— Distinct inputs of the system lead to distinct outputs, and an inverse system
exists.
— For example, a system which is |(w) = 2{(w), for which the inverse system is
|(w) = 12 {(w).
• Stability
— If the input of the system is bounded, then the output must be bounded.
2.4
Time-Domain Analysis
Definition 2.2 The analysis of a LTI system that is based on the relationship between
time-varying inputs and their corresponding time-varying outputs.
• Inputs/outputs are time functions (waveforms).
2.4.1
Impulse Response
• Output of the system with a fictitious input of Direc-Delta function (w).
— For LTI systems, the system characteristic in time domain is the system impulse response.
6
Lecture 2. Signals and Systems
• Direc-Delta function (w).
(
(w) =
Z
0 li w 6= 0
=
4 li w = 0
(2.3)
W
(w)gw = 1> ;W A 0=
(2.4)
3W
f
Magnitude
1/T
Magnitude
t
-T/2 T/2
t
0
Figure 2.6: Direc-Delta function
— Signal sampling using Direc-Delta function.
Z
"
{(w)(w )gw
{( ) =
(2.5)
3"
G (t W )
x (W )
1
T
³ x(t )G (t W )dt
lim T o0 x (W ) u
1
uT
T
x (W )
x (t )
t
W
T
2
W
T
W
2
Figure 2.7: Sampling using Direc-Delta function.
— Signal reconstruction: any real time-functions can be represented by using the
integrals of Delta functions as in Eq. (2.6).
Z
"
{( )( w)g
{(w) =
3"
— Convolution
7
(2.6)
Sec 2.4. Time-Domain Analysis
x (W 1 )G (t W 1 )
Magnitude
x (W 2 )G (t W 2 )
x (t )
t
W1
W2
Figure 2.8: A time function represented by a set of delta functions.
Outputs of the LTI system is the convolution of the input and the system
impulse response.
Z
"
|(w) = {(w) k(w) =
{( )k(w )g
(2.7)
3"
2.4.2
Step Response
• Output of the system w.r.t. an input {(w) of step function.
P( t )
1
t
0
(
{(w) = x(w) =
0 li w ? 0
=
1 li w 0
gx(w)
= (w)=
gw
2.4.3
(2.8)
(2.9)
Sinusoidal Response
• Output of the system w.r.t. sinusoidal function input {(w).
{(w) = cos(2w) with $ = 2i
(2.10)
i is frequency.
$ = 2i is the angular frequency.
W = i1 is period.
— For a real LTI system with a sinusoidal input function, the output is also a
sinusoidal function but with changes in both magnitude and phase.
—
8
Lecture 2. Signals and Systems
x (W 1 )G (t W 1 )
Magnitude
x (W 2 )G ( t W 2 )
x (t )
t
W1
W2
Magnitude
1
LTI
System
h(t )
Impulse Response
t
0
Time-Invariant
W1
t
W2
Linearity
y (t )
x (W 2 )h (t W 2 )
x(W 1 ) h (t W 1 )
W1
t
W2
Figure 2.9: Continuous time convolution operation.
k(w)
{(w) = cos $w $ |(w) = kK($)k cos ($w + ]K($)) >
Z "
where K($) =
k( )h3m$ g = kK($)k hm]K($)
3"
†Advanced Topics
Proof.
k(w)
{(w) = cos $w $ |(w) = kK($)k cos ($w + ]K($))
9
(2.11)
Sec 2.4. Time-Domain Analysis
x [ 1]G [ n 1]
x [ 2 ]G [ n 2 ]
x [ 0 ]G [ n ]
5
x [ 3 ]G [ n 3 ]
4
Magnitude 4
3
0 1
2
Input
x [n ]
n
3
Magnitude
4
3
LTI
System
h [n ]
2
1
Impulse Response
0 1 2 3
n
16
12
8
x [ 0 ]G [ n ] o x [ 0 ]h [ n ]
4
n
0 1 2 3
20
15
10
x [1 ]G [ n 1 ] o x [1 ] h [ n 1]
5
n
0 1 2 3 4
16
12
x [ 2 ]G [ n 2 ] o x [ 2 ] h [ n 2 ]
8
4
n
0 1 2 3 4 5
12
9
6
x [ 3 ]G [ n 3 ] o x [ 3 ]h [ n 3 ]
3
n
0 1 2 3 4 5 6
39 38
32
Output
22
16
10 3
y[n]
n
0 1 2 3 4 5 6
¦ x [ k ]h [n k ]
k
Figure 2.10: Discrete time convolution operation.
10
Lecture 2. Signals and Systems
x(t)=cos(Ȧt)
1
0.5
0
-5
-2.5
0
2.5
5
t
-0.5
-1
Figure 2.11: Sinusoidal waveform.
|(w) = {(w) k(w)
Z "
{( )k(w )g
=
3"
Z "
cos($ )k(w )g
=
3"
Z
1 " m$
(h + h3m$ )k(w )g
=
2 3"
Z
Z
1 " 3m$
1 " m$
h k(w )g +
h
k(w )g
=
2 3"
2 3"
0
By defining = w > the equation above can be written as follows:
1
|(w) =
2
Z
"
Z
"
0
m$(w3 )
h
3"
0
1
k( )g +
2
0
0
Z
"
0
h3m$(w3 ) k( 0 )g 0
3"
0
h3m$ k( )g 0 = kK($)k hm]K($) as a complex function of $= Its
3"
Z "
0 W
0
W
complex conjugate is K ($) =
hm$ k ( )g 0 = kK($)k h3m]K($) =Since k(w) is a real
Define K($) =
W
0
0
3"
function, k ( ) = k( )= Thus, |(w) can be formulized as follows:
Z
Z
1 m$w " 3m$ 0
1 3m$w " m$ 0
0
0
|(w) =
h
h
k( )g + h
h k( 0 )g 0
2
2
3"
3"
1 m$w
1
=
h × kK($)k hm]K($) + h3m$w × kK($)k h3m]K($)
2
2
= kK($)k cos ($w + ]K($))
• More generally, it is the output of the system w.r.t. complex exponential function
input {(w).
(2.12)
{(w) = hm$w = cos($w) + m sin($w)
11
Sec 2.4. Time-Domain Analysis
— Complex exponential function hm$w is the eigenfunction of any LTI systems.
k(w)
{(w) = hm$w $ |(w) = K($)hm$w
(2.13)
= kK($)k cos ($w + ]K($)) + m kK($)k sin ($w + ]K($))
Z
"
k(w)h3m$w gw=
K($) =
3"
h(t )
f
x (t )
Input time function
LTI y (t ) ³f x (W )h (t W )dW
System Output time function
h(t)
x( t ) e
jZt
jZt
LTI y (t ) De ,D
System
H (Z )
³ h ( t )e
jZt
dt
h (t )
x ( t)
X (Z) e
jZt
jZt
LTI y(t ) D ' e , D ' H (Z) X (Z)
System
Figure 2.12: The output of a LTI system with exponential complex function.
2.4.4
Initial Value/Driving Free/Natural Response
• Output |(w) w.r.t. null input {(w) = 0 and possibly non-zero initial system states.
Equivalently, it is solution of Homogeneous System Equation.
2.4.5
Transient Response
• The part of system output that will disappear (die down) as time progress.
|W (w) $ 0 as w $ 4=
(2.14)
— For Linear Time-Invariant (LTI) circuits, |W (w) = impulse response (with necessary scaling and time-shifting).
2.4.6
Steady-State Response
• The part of system output that will remain after transient response dies down.
|V (w) = |(w) |W (w)=
12
(2.15)
Lecture 2. Signals and Systems
— For Linear Time-Invariant (LTI) circuits, |W (w) = impulse response (with necessary scaling and time-shifting).
2.5
Frequency-Domain Analysis
Definition 2.3 Determination of system output(s) w.r.t complex sinusoidal inputs at different frequencies and with specific initial system state. Results are often displayed along
frequency axis or expression as functions of angular frequency ($).
• Input X and Output Y are complex functions of angular frequency.
Input
Output
x (Z)
y( Z)
LTI Systems
States
2.5.1
Phasor
• An electrical-engineering representation of sinusoidal signals in frequency domain.
• A constant complex number that encodes the magnitude and the phase of the
sinusoidal signals.
Example 2.4 Given sinusoidal signal {(w) = N cos($w + !)> S kdvru [ = Nhm! , where
° °
°[ ° = N and phase ][ = !.
• {(w) = <{[hm$w } = N cos($w + !)=
Im[X]
K
X
M
0
2.5.2
Re[X]
Spectrum and Fourier Transform
• Any finite-energy signal can be represented by sinusoidal functions (including both
sin and cos waveforms) of digerent frequencies.
13
Sec 2.5. Frequency-Domain Analysis
— The principle of Inverse Fourier Transform.
Z
1
{(w) =
2
Z
1
=
2
Z
1
=
2
"
3"
"
3"
"
[($)hm$w g$=
k[($)k hm][($) hm$w g$=
(2.16)
{k[($)k cos ($w + ]K($)) + m k[($)k sin($w + ][($))} g$=
3"
— [($) is a complex function of angular frequency $, which expresses the values
of the signal phasor in digerent frequencies.
[($) = k[($)k hm][($) =
(2.17)
k[($)k is called magnitude spectrum, specifying the magnitude for digerent sinusoidal components.
][($) is called phase spectrum, specifying the phase for digerent sinusoidal components.
— [($) can be obtained by taking the Fourier Transform of {(w).
Definition 2.5 Given i (w), its Fourier Transform, which is defined as follows, is a complex function of the angular frequency $.
Z
I ($) =[i (w)] =
"
i (w)h3m$w gw=
(2.18)
3"
• Fourier Transform of i(w) is the projection of i(w) on the basis functions hm$w =
Definition 2.6 Correspondingly, the Inverse Fourier Transform is as follows:
1
i(w) = [I ($)] =
2
31
2.5.3
Z
"
I ($)hm$w g$=
(2.19)
3"
System Transfer Function Kv ($)
• A ratio between the spectra of input and output signals of a linear time-invariant
circuit.
14
Lecture 2. Signals and Systems
Time-Domain Analysis
h(t )
x(t )
Input time function
x( t)
1
2S
LTI
System
y (t )
³ X (Z )e
dZ
LTI
System
f
f
x (W )h (t W )dW
Output time function
h(t )
jZt
³
y (t )
1
2S
³ X (Z) H (Z )e
jZ t
dZ
H (Z )
X (Z)
Input Spectrum
LTI Y (Z) H (Z ) X (Z )
System Output Spectrum
Frequency-Domain Analysis
Figure 2.13: Relationship between time-domain analysis and frequency-domain analysis.
— It is also known as the frequency response of a LTI system.
\ ($)
[($)
k\ ($)k hm]\ ($)
=
k[($)k hm][($)
k\ ($)k m(]\ ($)3][($))
h
=
k[($)k
= kKv ($)k hm]Kv ($)
Kv ($) (2.20)
kKv ($)k = k\ ($)k @ k[($)k is the magnitude response of the system.
]Kv ($) = ]\ ($) ][($) is the phase response of the system.
• From Eq. (2.13) and Eq. (2.16), each sinusoidal component [($)hm$w produces an
output signal of \ ($)hm$w = [($)K($)hm$w , as shown in Figure 2.13. Thus, |(w) can
be written as follows.
Z "
Z "
1
1
m$w
\ ($)h g$ =
[($)K($)hm$w g$=
(2.21)
|(w) =
2 3"
2 3"
• The system transfer function Kv ($) = K($), which is the Fourier Transform of the
system impulse response k(w).
15
Sec 2.5. Frequency-Domain Analysis
2.5.4
Time Domain versus Frequency Domain
• The frequency response of a LTI system is the Fourier transform of the system
impulse response.
K($) = = {k(w)}
(2.22)
• The convolution of two signals in time domain is equivalent to the multiplication of
their representations in frequency domain.
=
|(w) = {(w) k(w) #$ K($)[($)
16
(2.23)