AC Circuits 2

7/28/2019 AC Circuits 2

http://slidepdf.com/reader/full/ac-circuits-2 1/304

Introduction to linear circuit analysis and modeling:From DC to RF

L. Moura and I. Darwazeh

2005



Introduction to linear circuit analysis and modeling: From DC to RF Moura and Darwazeh

Powers of ten

femto- pico- nano- micro milli- kilo- mega- giga tera

(f) (p) (n) (µ) (m) (k) (M) (G) (T)

10−15 10−12 10−9 10−6 10−3 103 106 109 1012




Voltage source driving a resistance

Voltage

source

R

I

Ideal conductor

+

Ideal conductor

Resistance +

V

−

−

V

Hydraulic equivalent system

Pipe

ReservoirWater

pump

Water

Water

Reservoir

FlowWater




Ideal voltage source

Symbols

−

+V s

+

−

V s

V –I characteristic

V

V s

I 0




Practical voltage source

V –I characteristic

V

I

0 I x

V s

V x

Electrical model

−

+

V s−

V

Rs

+

I




Ideal current source

Symbol

I –V characteristic

I

I s

V 0




Practical current source

I –V characteristic

I

I x

I s

0

V

V x

Equivalent circuit

I s

Rs V

+

−




Instantaneous and average power

¡

¢ £

£ ¤

¥

¦

§

©

¤

¤

©




Voltage and current in a resistance

!

"

$

!

%

$

!




The capacitor

In an electrical circuit

CapacitorC

i(t)

t

i(t)v(t)

t

+

−

v(t)

Hydraulic analogue

Water pressure

Elastic membrane




The inductance

In an electrical circuit

L

Inductance

−

v(t)

+

v(t)

tt

i(t)

i(t)

Hydraulic analogue

Fly-wheelWater flow

Water pressure




Kirchhoff’s current law

Illustration of the current law

I 2

I 3

I 1

Node

Equivalent representation

I 1

−I 3

I 2Node




Circuit for the application of the current law

&

'

(

&

)

0

1

2

3

4

3

5 3

6

3

7

3

&

8




Kirchhoff’s voltage law

9

@

@

A

B

B

C

D

A

C

E

F

G

H

H

G

A

B

C

I

C

P

Q

R

T

A

B

A

B




Application of the voltage law

U

V

W

X

Y X

a a

b

c

b

d

U

V

U

V

V

U

V

U

U

V

V

U

V

U

e X




Series combination of two resistors

−

V 2

+−

V 1

+I

V s−

+

R1

R2

Equivalent resistance

I

V s

+

−

V s

+

−

Req




Parallel combination of two resistors

(R2)G2

I 2I 1

V

+

−

I s

G1

(R1)

Equivalent resistance (conductance)

I s

+

V

Geq

(Req )

−




Series combination of two capacitors

C 2

C 1

−

v2(t)+

−

v1(t)

+i(t)

+

−

vs(t)

Equivalent capacitor

C eq

vs(t)

−

+

vs(t)+−

i(t)




Parallel combination of two capacitors

C 2

i2(t)i1(t)+

is(t)

v(t)

−

C 1

Equivalent capacitor

C 2

i2(t)i1(t)+

is(t)

v(t)

−

C 1




Series combination of two inductors

−

L2v2(t)

+−

v1(t)

+

L1

i(t)

vs(t)

−

+

Equivalent inductor

vs(t)+

−

i(t)

+

vs(t)

−

Leq

I d i li i i l i d d li F DC RF M d D h




Parallel combination of two inductors

L1

−

is(t)

+i1(t)

v(t) L2

i2(t)

Equivalent inductor

is(t)

+

−

v(t)

Leq

I t d ti t li i it l i d d li F DC t RF M d D h




Resistive circuit

A

B

R1

100 Ω

R2

R5

90 Ω

R380 ΩR4

20 Ω

30 Ω

Calculation of the equivalent resistance

−

+

B

R1

100 Ω

R2

R5

90 Ω

R380 ΩR4

A

I t

V t

20 Ω

30 Ω





Calculation of Req

−

+−

−−

+

− −

+

B

A

I t

V 1 V 2

V 3

I 1

I 2

V t

I 5

I 4

++

V 5+

V 4

I 3

I t





Nodal analysis method

Resistive electrical network

(100 Ω)

R1

(1.6 kΩ)

R3

X Z

Y

R4R5

(7 V)

V X −

+

(1 kΩ)(300 Ω)

I A =?

(10 mA)I Y

R2

(4 kΩ)

Equivalent circuit

−

R4 I 4

V X R1

V Y

R3

I 5R5

V Z

V X

+

0

I A = I 3

I 3

I 1 I Y

X Y Z





Nodal analysis method

f

g

h

i

h

p

q

q

h

p

r

s

h

i

h

s

r

s

q

g

h

s

t

h

p

g

h

i




y g

Resistive voltage divider

V s

I

R1 V o

R2

I

0

−

+

Resistive current divider

R2

I o

I s

R1

0

V




y g

Voltage-controlled voltage source

V i

+

−

−

+V o = Av V i

Voltage-controlled current source

V i

+

−

I o = Gm V i




Current-controlled voltage source

−

+I i V o = Rm I i

Current-controlled current source

I i I o = Ai I i




Circuit containing a voltage-controlled current source

v

w

x

y

w

x

y

v

w

v




Generic DC electrical network

Electrical

NetworkY

X

Thevenin equivalent circuit

+

−

X

Y

RTh

V Th




Electrical network

R1X

R2

(100 Ω) (80 Ω)Y

I r

(30 Ω) (7 V)

R3

V s

(0.1 A)−+

Calculation of the Thevenin voltage

0

I 3

V s

V B

V A

X

R3

Y I r

I 1

R1V C

I 2

R2

−+




Calculation of the Thevenin resistance

0

R3

V tR

2

I 1

R1 I t

I 3

Y

X

+

−

RTh V tI t

Thevenin equivalent circuit

V Th

(49.5 Ω)X

Y

RTh

+

−

(1.1 V)




Generic electrical network

Y

X

Network

Electrical

Norton equivalent circuit

I Nt

RNt

X

Y




Calculation of the short-circuit current

Y

X

V sR1V C

R2I 2

I 1V B

+ −

R3

I Nt

0

I r

Norton equivalent circuit

I Nt RNt

(49.5 Ω)

Y

X

(0.023 A )




Thevenin circuit

V Th

X

Y

−

+

RTh

Equivalent Norton circuit

Y

X

RNt = RTh

I Nt =V Th

RTh




Norton circuit

I Nt

Y

X

RNt

Equivalent Thevenin circuit

+

−

X

V Th = I Nt I Nt

Y

RTh = RNt




Superposition theorem

The contribution from V s

+ −

V A

I

R2 I

V s

R3

R1

0

Y

X

The contribution from I r

R1V C V A

R3I 1

R2

I 2

I 1

0

I r

X

Y




The real axis

π X

axisReal

2√ 3

5

6

1

30−1

2−1

The complex plane

Imaginary axis

z 1

4

3

z 2

−√

3 1

X

Realaxis

0

2

Y




1 multiplied by j

X

Real axis

1−1

Imaginary axis

j × 1

j Y

( j × 1) multiplied by j

j × 1

Real axis

X 1 j2 × 1 = −1

Imaginary axis j Y



Multiplication by− j

~




Addition of z 1 = 3 + j 2 and z 2 = 2− j

ª «

¬

-

®




Subtraction of z 2 = 2− j from z 1 = 3 + j 2

°

±

²

³

²

¶ ·

¹

º ¼

½

¾

¿ º

À

Á

Â

²

Ã

³

Â

Ã

Ä Å ¸

Æ

¿ º

À

Ç

Â

°

²

Ã

È

²




Multiplication of z 1 = 2 + j with z 2 = 2

É

Ê

Ë

Ì

Í

Ì

Î

Ì

Ï

Ð

Ñ

Ó Ô

Õ

Ö

× Ø

Õ

Ù

Ú

Õ

Û ×

Ü

Ý Þ Õ

ß

Õ

Û ×

Ü

Ð

Ð

É




Multiplication of z 1 = 2 + j with z 2 = j 2

à

á

â

á

ã

ä å

æ

è

é

ë

ì

î

á

ï

ð

ñ

ò ó

ô

õ

ö ÷

ô

ø

ù

ô

ú ö

û

ð

ü

ð

ì

ð

ì

ý þ ô

ÿ

ô

ú ö

û

ì




Complex numbers and their conjugates

¡

¢

¤

¥

¡

¤

¡

¦

§

! "

#

§

$

%

§

$

¡

¢

¦

&

%

'

%

§

(

$

§

(

$




Cartesian and polar representations

)

0

1

2

3

4

5 6 7

8

9 @ 7 A

B

7 C 9

D

E F 7

G

7 C 9

D

H




Solutions of z 5 = 1

I

P

I

Q

I

R

I

S

T

U

W X Y

a b Y c

d

Y e a

f

g

T

T

I

h

g

i

i

p q Y

r

Y

e a

f

s

t

u

v

w




Solutions of z 3 = 4 + j 4

y




AC voltage (current) waveform

Versus time

Period T

(Frequency 1/T )

tTime

V s(I s)

vs(t) (is(t))

Versus phase

2 π = 360o

ω t

Phase

vs(t) (is(t))V s(I s)




Phase difference (φ = π/3)

The current lags the voltage

φφ

A

is(t)

vs(t)

ω t

Phase

The voltage leads the current

A

is(t)

φφ

vs(t)

ω t

Phase




Effective voltage

j

k

l l

m

o




Effective voltage

z

| |




Resistance

+ vR(t)−

i(t)

Current and voltage

t

t

i(t)

vR(t)




Capacitor

i(t)

−+vC (t)

Current and voltage

T

4⇔

π

2

t

t

vC (t)

i(t)




Inductor

−+ vL(t)

i(t)

Current and voltage

i(t)

t

t

vL(t) T

4⇔

π

2




V eff /I eff versus ω for passive elements

ω0

R

ωL

1

ωC

V eff I eff




RL circuit

+

−

ω = 20 krad/s

V s = 4 V

(2 mH)

vs(t) = V s cos(ωt)

+

−

vL(t

)

i(t) (100 Ω)

vR(t)+ −




Complex V –I relationship

Resistance

Capacitance

Inductance

ª




The general impedance

+ −

V ( j ω, t)

I ( j ω, t) Z


Th l h



The complex phasor

The rotating phasor

V s cos(ωt + φ)

ωt

ωt + φφV s sin(ωt + φ)

ω

Imaginary axis

AxisReal

Angular velocity

The stationary phasor

V s sin(φ)φ

V s cos(φ)

Imaginary axis

Real

Axis


AC i i



AC circuit

+

−

(100Ω)

R2

L2

(10 mH)

(3µF)

vs1(t) = V

s1 cos(ωt + π/4)

is2(t) = I

s2 sin(ωt)

V s1 = 7 V

ω = 5 krad/s

I s2 = 25 mA

L1

(30 mH)

R1 (120Ω)

(10µF)

C 1

C 2

Equivalent circuit

(100Ω)

(10 mH)

+

−

( j150Ω)

Z L1

(120Ω)

Z C 2R1

Z R2L2

Z C 1

I A

V S 1 I S 2

I C

V S 1

I D

I B

V X V Y

(100 + j50Ω)

0

(− j66.7 Ω)


AC i i



AC circuit

Thevenin equivalent

+

−

Z Th

V Th

Norton equivalent

Z NtI Nt




AC circuit

+

−

X

Y

(2 kΩ)

(40 mH)

(0.1µF)

R

L

C (3 cos(104 t− π/5) V)

vs(t)

Thevenin voltage

+

−

X

Y

Z L

Z RC

(400− j 800 Ω)

( j 400 Ω)

(3 e− j π/5)

V S




Thevenin Impedance

X

Y

Z L

( j 400 Ω)

Z RC

(400−

j 800 Ω)

Z Th

Equivalent Thevenin circuit

+

−

X

Y

Z Th

(200 + j 600 Ω)

V Th

(4.7 e− j 1.0 V)




Maximum power transfer

Z L

+

−V S

vs(t) = V s cos(ω t)

Source Load

Z S

I S

Z L =

Z ∗

S




Periodic waveforms

« « « « «

« « « « «

« « « « «

« « « « «

« « « « «

« « « « «

« « « « «

¬ ¬ ¬ ¬ ¬

¬ ¬ ¬ ¬ ¬

¬ ¬ ¬ ¬ ¬

¬ ¬ ¬ ¬ ¬

¬ ¬ ¬ ¬ ¬

¬ ¬ ¬ ¬ ¬

¬ ¬ ¬ ¬ ¬

s(t)

0 2T T 3T

T 0 2T 3T

t

t

t3T 2T T 0

x(t)

y(t)

Sine

Rectangular

Triangular




Fourier series

2T T

t

tt

t

y1(t) + y3(t)

x1(t)

x3(t)

x1(t) + x3(t)

y3(t)

y1(t)

2T T


Phasor



Phasor

2π n

T t +∠C n

|C n| cos[2π n

T t +∠C n]

Realaxis

|C n| sin[2π n

T t + ∠C n]

2π n

T

Angular velocity Imaginary axis

Line spectrum

2π n

T ω

∠(C n)

|C n|

ω

A m p l i t u d

e

P h a s e

2π n

T




Rectangular waveform

Line spectrum

−5×2πT

−2πT

−3×2πT

2πT

3×2πT

5×2πT

Phase ∠C n (rad)

−3×2πT

−2πT

2πT

−5×2πT

Amplitude |C n|

2,A

5π

2A3π

2Aπ

2Aπ

2A3π

2A5 π

ω3×2πT

5×2πT (rad/s)

ω

(rad/s)

(volt)

−π2

(−90o)

π2(90o)


RC circuit



+

−

R

vs(t)

(V S )

vc(t)

(V C )

C

(T = 1 s)τ

T = 1

2

+

−

(V a = 1 V)

vs(t)V a

t

τ T 2T −2T −T


Signal bandwidth: Rectangular periodic waveform



Various components

t

slope

Fundamental

+3rd harmonicFundamental

Fundamental+3rd +5th harmonics

Line spectrum

f

|V S 0|

|V S −5

||V S

−3| |V S 3|

−1

T

1

T

−3

T

−5

T

5

T

3

T

|V S 1|

|V S 5|

|V S −1

|




RC circuit

Transfer function

-

® °

-

® ±

-

±

°

®

±

²

®

±

²

µ

®

±

¶

®

±

µ

®

±

®

±

·

®

±

·

®

±

® ±

µ

®

±

²

µ

® ±

²

¹ º »

¼

½ ¾

¹

º »

¼

¿

À ½ ¾

¹

º »

¼

½ ¾

¹ º »

¼

¿ À ½ ¾

¹

º »

¼

½

¿ ¾

® ±

²

µ

® ±

¶

¹

º »

¼

½

¿ ¾

Â

Ã Â

Ã

Â

Ã

Ä Å

Æ

Ç

È

Æ

É Ê Ë

È

Ì

Å

Æ

Ç

È Ì

Ç

Æ

Í Î

È

Ç

Æ

Í Î

È

Ï

Ð

Ò Ó

Ï

µ

Ò Ó

Ï

·

ÒÓ




Distortionless system



Transfer function

Ô

Õ

Ö

Ø

Ù

Ú Õ

Ù

Ù

Û

Ü

Ý Þ

ß

Ù

à Ö

Ø

Ù

Ú

á

á


RC circuit



Bode plot

f (Hz)

f (Hz)

−π4

−π2

−30

−20

−10

0

0

−3

4 × 10−2

100

10−1

|H (f )|

∠H (f ) (rad)1

2πRC 10

2π RC

12πRC

102π RC

110×2πRC

110×2πRC

|H dB(f )| (dB)




CR circuit

â

ã

ä

å

æ

â

ç

è

ä

å


CR circuit

Bode plot



Bode plot

f (Hz)

−30

−20

−10

0

−3

12π RC

102π RC

110×2πRC

|H CRdB(f )| (dB)

f (Hz)

0

π4

π2

12π RC

102π RC

110×2πRC

∠H CR(f ) (rad)



Spectral representations

0

−4

1

T

1

T

1

T

1

T

0 0 3

T

3

T

3

T

3

T

5

T

5

T

−2

0

0.2

0.4

0.6

0

2

4

−1

T

−1

T

−1

T

−3

T

−3

T

−3

T

−5

T

−5

T

−5

T

−5

T

5

T

5

T

−3

T

−1

T

0

0.6

0.4

0.2

4

2

0

−2

−4

f (Hz)f (Hz)

f (Hz) f (Hz)

| V S n |

(V)

| V On |

(V)

(rad)

∠V S n

(rad)

∠ V On




vs(t) and vo(t)

!

"

!

#

$

&

(

0

2

4

6

#

8

@

A

A

B D

!

E

A



Fourier transform

F G H I P Q

H R

G S

T U I

V

W I S

X

Y

P Q H

R G S

a

b

a

c

c b c

a

b

a

c

a c

a c

a

c

a

b

a b

a

b

c

c

c

b

b

b

b

b

b

b

c

c

c

a

b

a b

a

b

a

c

a c

a

c

d

e

f

g h i

p q r s

p

q

r

s

a b

s

s

s

r

r

r

q

q

p

p

p

p

q

q

q

r

r

r

s

s

s

a

b

a b

a

b

a b

a

b

a b

a

b

q p

p

t

u

v x

y

t

u

v x

y

t

u

v x

y

t

u

v x

y

u

i

y

u

i

y

u

i

y

u

i

y

a

b

c

c

u

t

y




DC value

( )



t

η → 0

A

w(t)

Fourier transform

η → 0

f

W (f )A/η


Unit-step function

(t)



u(t)

t

1

2sign(t)

t

1

2

−

1

2

t

1

2

1

+

=




The signum function

k

l

k

m

n

o

n

n

o

n

o




Causal exponential




Impulse response of an RC circuit

z

|

z

z

|

z

~




Impulse response: approximation



Output voltage

²

³

µ

·

¹ º

»

¼

¹ ½

»

¼

¹

¾ »

¼

¿

À ¾

À

Á

À

µ

Â

·



RC circuit

+

C

R

+

−

−

V C (f )

V S 2(f )




Region of convergence (unit-step)

Region of

convergence

Imag (s)

Real (s)0



RC circuit

+

C

R

+

−

−

V C (s)

V S 2(s)


RL circuit

(t = 0)

+LR2+



vL(t)

−

L

R1

+

−

V

For t < 0

R2

R1

+

V

L I lo

−

For t ≥ 0

I (s)

−

R1L

+

−

V L(s)V R(s)

+


i(t) versus the time

Ï

Ð

Ì

Ñ Í

Ò

Ó Ô

Ë



Ã Ä

Å Æ

Ç

Å Æ

È

Å

Æ É

Å

Æ Ã

Ê

Ë Ë

Ã Æ Ä Å

Æ Ë

Ì

Í

Î

É

vL(t) versus the time

Õ

Ö

Õ

Ö × Ø

Ö Ù

Ú

Û

Ü

Õ

Ý Þ

ß

Õ

Ý

Þ

à

Õ

Ý

Þ Ù

Õ

Ý Þ ×

á

â

ã

Ú

ä Û ã

å

æ

ç

è é

ä


RC circuit



+

−

V

R4R2

R3

i3(t)

(t = 0)

R1

+

vC (t)

−

For t < 0

+

−V

V A

R4

R1||R3

R2

0

0




LC circuit

(t 0−)



(t = 0−)

(t = 0+)−

+ S 2

S 1

V

Fort≥

0

−

+I (s)

V LC (s)


LC circuit



Hydraulic analogue



i(t)/(C V co ωn)

η = 5

η = 1

η = 0.1

t ωn

14

0.8

0.6

0.4

0.2

0−0.2

−0.4

−0.6




RC circuit



RC circuit

+

R

+

V st

0

−

−

vC (t)(V C (s))

(I (s))i(t) C vS (t)(V S (s) = V s/s)

vS (t)


vC (t) versus the time

1

vC (t)/V s



0.

2

0.4

0.6

0.8

1 2 4 5

t/τ

32.30.1


1 2 3 4 5

0.2

0.4

0.6

0.8

1

0.1

t/τ

i(t)R/V s

2.3


CR circuit

C

+



+R vR(t)

−

(V R(s))vS (t)

−

(V S (s) = V s/s)

vR(t) versus the time

2.3

2 3 41

0.2

0.4

0.6

0.8

5

1

vR(t)/V s

t/τ

0.1


RL circuit



++

R

vL(t)

(V L(s))

−

LvS (t)

(V S (s) = V s/s)−

LR circuit

+

−

+L

vR(t)

(V R(s))

−

RvS (t)(V S (s) = V s/s)


RLC circuit

+

i(t)(I (s))

+ C

L



vC (t)

(V C (s))

−

+

−

C

R

vS (t)(V S (s) = V s/s)

i(t) and v(t)

vC (t)/V s

20 4 6 8 10 12 14

0.5

1.0

1.5

η = 0.1

η = 0.3

η = 0.7

t ωn

η = 1 η = 3

η = 0.1

0.4

0

0.6

−0.2

−0.6

−0.4

0.2

0.8

14

η = 3

η = 0.3

i(t)/(C V s ωn)

η = 0.7

η = 1

t ωn




Settling time versus η



8

6

4

2

0.

5 0.

6 0.

7 0.

8 0.

9 1.

0

(±5%)

(±2%)

η

ts ωn


RLC circuit

+IS(s) =



L

R

C

−

Z eq (s)

t

0

I s/s

I S (s) =

iS (t)

I s

V (s)


iL(t) versus the time



η = 0.1

η = 0.4

η = 0.7

4 8 120

1.0

1.5

0.5 η = 0.9

iL(t)/I s

t ωn




For t < 0I lo



R1

V co

+

−

R3

V S 2

−

+

For 0 ≤ t < 0.25 ms

−

+

− +

+

−

V L(s)

V R1(s)

V C (s)I L(s)


vC (t) (V)

4

6



t (s)

−2

2

×10−4

0

54321

t (s)

−0.3−

0.2

−0.1

0.1

0.2

0.3

iL(t) (A)

×10−41 2 3 4 50




For t ≥ 0.25 ms

+

−

L

V S 1(s)

0

V L(s) V C (s)R2

R1

I L(s)

I R1(s)I R2

(s)

I C (s)


vC (t) (V)



−2

0

2

4

6

0 2 4 6 8 10

t (s)

×10−4




Shunt admittance



Y = R−1

(2 mS)

Z 11 and of Z 21

I 1

V 1

+

Y = R−1

−−

+

I 2 = 0

V 2


T-network

C

L2L1

(2 µH) (1 µH)

(3 nF)



Z 11 and of Z 21

−

V 1

+I 1 L2 (Z 2) +

V 2

−

C

(Z 3)

L1 (Z 1)

I 2 = 0

Z 12 and of Z 22

+ + I 2

−

V 1

−

L1 (Z 1) L2 (Z 2)

V 2

(Z 3)C

I 1 = 0




Y 11 and Y 21

I 1

+

I 2

Two-Port



V 1

+

−

V 2 = 0Network

Output port(Port 2)(Port 1)

Input port

Y 12 and Y 22

I 1

V 1 = 0

I 2

+V 2

−

Two-Port

Network

Input port(Port 1)

Output port(Port 2)


Π-network

(1 nF)C 2C 1

(3 nF)

L (1 µH)



Y 11 and Y 21

+

−

V 1

I 1

C 1

(Y 1) (Y 2)C 2

I 2

L (Y 3)

Y 12 and Y 22

+

−

(Y 3)LV 2

I 2I 1

(Y 1)

C 1 C 2

(Y 2)




Parallel connection

I

2I

1

++



+

V 1−

I 1 I 2

+

−

V 2

−

V

2

I

2−

I 2

V

2V

1

−I 1

−I

1

[Y ] + [Y

]

[Y ]

[Y ]

+

−

V

1

+

+

−

+

+ +

V 2V 1


Chain parameters

A11

I 2 = 0



+

−V 1

+

−

V 2

Output port(Port 2)

Input port(Port 1)

Two-Port

Network

A12

V 1

+

−

V 2 = 0

−I 2

(Port 1)Input port Output port

(Port 2)

Network

Two-Port


A21

I

I 2 = 0



I 1 +

−

V 2

Output port(Port 2)

Input port(Port 1)

Two-Port

Network

A22

© ©

I 1

V 2 = 0

−I 2

Input port

(Port 1)

Output port

(Port 2)

Two-Port

Network


Electronic amplifier model

+

−

Ro

gm

V i

V iRi



A11

+

−

+

V 2

−

V 1+

−

gm V i

V i

Ri Ro

A12

+

−

+V 1

−

Ro

−I 2

gm V i

V iRi


A21

+I+ g V



−

+I 1+

V 2

−

gm V i

V iRi Ro

A22

I 1

−

Ro

−I 2

+

V i

gm V i

Ri


Chain connection

I I

I I I I



+

V 1

−

I 1 I 2

−−−

V 2

+ + +

−

+

I 1 = I 1 I

2I 1

[A] V 1

I 2 = I 2

V 1 [A] V

2

+

−

V 2[A]× [A]


Miller’s Theorem

V 2V 1

+

(I 2 = 0)

+

Y f

[Y a]



V 2

V 2

− −

(I 2 = 0)

Y 2

−

++V 1

−

Y 1

+V 1

+

− −

Y f

[ ]

[Y a]

[Y a]


Miller’s theorem: example

Rf



Z in

Y 1

+

−

Y 2

+

−

Z in

Circuit a

Ri

Ri RL

RL

gm V i

V i

V i

gm V i


Electronic amplifier

R 1

V CCS +R 1



R2

R1

+

−

V CCS

+

−

R3

R 3

V i

gmV i

V i

(20kΩ)

R1

(300Ω)

Ri

R 2gm = 50 mA/V

V s

R2

V o

+++

−−

Ri

(2.5 kΩ)V s

−

V o

+gm V i

−

(5 kΩ)

R3


DC and AC signals: notation




Representation of voltages

Z 2

V

Z s

Z 1

V a V b

V c



+

−

(node 0)

Z 3

Ground terminal

+

−

−

Z 1

Z 2

V s

V s

Z s

−

+

+

−

+

c

V b

Z 3V c

V a


Electronic amplifiers: typical transfer functions

20 log10 |A| 20 log10 |A|



Mid-FrequencyResponseLow-Frequency

g10 | |

3 dB

f H 10 f Lf L

log10 (f )

20log10(AM )

f H

10

Low and Mid-FrequencyResponse

3 dB

f H f H

10

20 log10(AM )

g10 | |

Response

High Frequency

Response

High Frequency

Bandwidth Bandwidth

Response

log10 (f )


Voltage amplifier



!

"

#

$

%

'

(

0

1 1

2

%

'

"

3

4

0 6 7

8

"

9

A

"

A

B

6

7

"

0

1

C

D

F 1 1

7

B

G

D

#

H I P

Q R S

T I

U

V W

X

S

Y `

C

U a b

S

Q

c I W d

A

B

1 1

7


Transfer function

1601

Avs

Low f



100

101

102

103

104

105

106

107

108

109

100

100

100

102

102

102

104

104

106

106

106

108

108

108

0.5

150

50

0

100

40

80

1200

1

0.5

0 0

104

f Hz

f Hz

f Hz

f Hz

High f

Mid f


Voltage dividers

vi

C i

Rs

Ri

vs



C Bvi

i

Rsvs

Ri


Additional losses: Z C i

1

1

1 +Rs||Ri

Z C i



100

1

0.9

0.8

0.7

101 102

Rs

C i

Ri

vivs

|Z C i|R

s||R

i


Additional losses: Z C B

1

1

1 +Z C B

Ri + Rs



10−20.7

0.8

0.9

1

100

|Z C B|Rs + Ri

RsC B

vi

Ri

vs




e

f

g

e

h

g

i

e

p

q

p

r

q

f

q

s

t u v

w

x y

e

p

g

r

e

f

q

h

u

u

y

y

w


Voltage amplifier

vi

+

Ri

+Ro

vo

+

io



i

−

i−

−Av vi

Current amplifier

ii vo

Ro+

−

Ri

Ai ii

io


Transimpedance amplifier

Riii+

+Ro

vo

io



ii−

vo

−

Rm ii

Transconductance amplifier

vi

+

−

Ri vo

+

io

−

Ro

Gm vi


Current amplifier



j k l

m n o

j j

l

z

l

j


Operational amplifier

Circuit symbol

3

1

+

−

vovi

4

V CC



2 5V EE

Equivalent circuit

−

+

Ro→ 0

Av →∞

Ri →∞

31

+

vi Ri

−

2

Av vi

Ro

+

−

vo


Non-inverting amplifier

Block diagram

+

−

vi

vs+

−

R1

−

vo+



R2

R1

Electrical model

+

−−

+

Feedback network

vf R2

−

−

+Ri

(∞)

Av vi

(0)

RoOp-amp

vs vi vo

−

+

R1+


Inverter amplifier



|

~

~

~


Integrator amplifier



Differentiator amplifier

ª


The adder amplifier

«

¬

«

¬

®

°

®

±

®

³

®

µ

¶

°

¶

µ

¶

³

·

µ

·

°

·

³

¹



«

¬

®

¶

·

The difference amplifier

º

»

º

¼

½

¼

½

¾

¿

¼

À

¼

Á Â

¼

Á Ã

Ä

Å

Ä

Å

Ä

Æ

Ä

Æ

Ç

Æ

Ç

Å

»


Instrumentation amplifier

1vsa

+− +



+−

vo

v

2

3

vsb

va

vb

i

i

vsa

vsb

R3

R3

R4

R1

R1

R2

R2

+

+

−

−

−


The diode

Geometry ( p –n junction)

p n



CathodeAnode

Symbol

+ −V D

CathodeAnode

I D


Junction diode: DC characteristic

Ê

É

Ë

Ì Î

Ï

Ô Ñ



È

É

Ð

Ñ

Ò

Ó

Ð

Ô Ò Ñ

Ð

Ô Ò

Ó

Ð

Õ Ò

Ñ

Ñ Ò

Ó

Ñ

Ò

Ö

×

Ø

Ù

Õ


NPN bipolar transistor

Collector(C)

C

I C



Base

p

n

(E)

B

E

n+

(B)

I B

I E

Emitter


PNP bipolar transistor

Collector(C)

C p

I C



Base

n

(B)

(E)

B

E

p+

I B

I E

Emitter


Ebers-Moll model: NPN bipolar transistor

Ú Ú

Ú Ú

Û Û

Û Û

−

V+

C I C

+−

V BC

F ID

C



Ú Ú

Ú Ú

Û Û

Û Û

Ü Ü

Ü Ü

Ü Ü

Ü Ü

Ý Ý

Ý Ý

Ý Ý

Ý Ý

− −

+

V BC

V BE

I C

B

E

+

I E

I B

V CE

+

−−

V BE

I DE

I B

αF I DE

I DC

αR I DC

I E

V CE

E

+B


Ebers-Moll model: PNP bipolar transistor

Þ Þ

Þ Þ

ß ß

ß ß

VCB

C+

−

I C

F ID

V CB

C+

−



Þ Þ

Þ Þ

ß ß

ß ß

à à

à à

à à

à à

á á

á á

á á

á á

E

B

I B

I C

I E V EB

V CB

++

−

−

V EC

I B

αF I DE

αR I DC

I E

I DC

I DE

V EC

E

V EB

B−

−

++


NPN bipolar transistor

DC characteristic



+

−

I B

I C

V BE

meterVolt-

meterCurrent-

V CE


NPN transistor

DC characteristic

I C regionSaturation

Active region(mA)

10

I B = 90 µA (V BE = 0.731 V)



0.20.1 0.3 2 4 6 8

2

4

6

8

I B = 70 µA (V BE = 0.725 V)

I B = 50 µA (V BE = 0.716 V)

I B = 30 µA (V BE = 0.703 V)

I B = 10 µA (V BE = 0.676 V)

V CE V

cut-off region I B 0 (V BE < 0.65 V)


I C versus V BE (linear region)

I C I E (mA)30

25



0.64 0.66 0.68 0.72 0.74 0.760.7

V BE (V)

∆V BE

∆V BE

20

15

10

5

0

∆I C 1

∆I C 2


Common-emitter amplifier

+

(10 V)V CC

−

+

V

(5 kΩ)

RL



+

−

V i

V o

−

V CC

V i

V o

RL


Common-emitter

Amplification process

V be (V i)

1 mA

I c

15 mV

VBE Q0 68 V

I C Q

V CC

Rc

Q

V BE = 0.695 V



V o (V ce)8 V5 V2 V

V CE Q V CC

V cet

t

3 V

1 mA V 0.68 VQ

V BE = 0.665 V


Clipping

25 mV

V CC

Rc

I cV be (V i)

Q 0.68 V



V CC V ce

Saturation

Cut-off

t

25 mV

V o (V ce)

t


Ideal DC voltage source

V



I

∆V = 0

∆ I


Hybrid-π model

é

ì



â â

ã

ä

å

æ

ç

è

ê

ë

ç

è

ã

è


The Early effect

ð

ñ

ò

í

ñï

ó

ð

î



í

î

ï

ô

õ

ö

÷

ø

í

ù


Common-emitter

AC equivalent circuit

vi

vo

RL



Small-signal equivalent circuit

Cvo

gm vπ

RL

EE

vi B

−

vπ

+

rπro


n-channel FET

Symbol

V DS

V GD

V GS

I DS

S

D

G



Geometry

ú ú ú ú ú ú

ú ú ú ú ú ú

ú ú ú ú ú ú

ú ú ú ú ú ú

ú ú ú ú ú ú

ú ú ú ú ú ú

ú ú ú ú ú ú

ú ú ú ú ú ú

ú ú ú ú ú ú

ú ú ú ú ú ú

ú ú ú ú ú ú

ú ú ú ú ú ú

ú ú ú ú ú ú

û û û û û û

û û û û û û

û û û û û û

û û û û û û

û û û û û û

û û û û û û

û û û û û û

û û û û û û

û û û û û û

û û û û û û

û û û û û û

û û û û û û

û û û û û û

ü ü ü ü ü

ü ü ü ü ü

ü ü ü ü ü

ü ü ü ü ü

ü ü ü ü ü

ü ü ü ü ü

ü ü ü ü ü

ü ü ü ü ü

ü ü ü ü ü

ü ü ü ü ü

ü ü ü ü ü

ü ü ü ü ü

ü ü ü ü ü

ü ü ü ü ü

ý ý ý ý ý

ý ý ý ý ý

ý ý ý ý ý

ý ý ý ý ý

ý ý ý ý ý

ý ý ý ý ý

ý ý ý ý ý

ý ý ý ý ý

ý ý ý ý ý

ý ý ý ý ý

ý ý ý ý ý

ý ý ý ý ý

ý ý ý ý ý

ý ý ý ý ý

þ þ þ þ þ þ þ þ þ

þ þ þ þ þ þ þ þ þ

ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ

ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ

¢

¢

¢

n+ n+L p-type substrate

W

Metal

DrainGate

Metal

Source

SiO2


p-channel FET

Symbol

S

D

G V DS

V GS

V GD

I SD



Geometry

£ £ £ £ £ £

£ £ £ £ £ £

£ £ £ £ £ £

£ £ £ £ £ £

£ £ £ £ £ £

£ £ £ £ £ £

£ £ £ £ £ £

£ £ £ £ £ £

£ £ £ £ £ £

£ £ £ £ £ £

£ £ £ £ £ £

£ £ £ £ £ £

¤ ¤ ¤ ¤ ¤ ¤

¤ ¤ ¤ ¤ ¤ ¤

¤ ¤ ¤ ¤ ¤ ¤

¤ ¤ ¤ ¤ ¤ ¤

¤ ¤ ¤ ¤ ¤ ¤

¤ ¤ ¤ ¤ ¤ ¤

¤ ¤ ¤ ¤ ¤ ¤

¤ ¤ ¤ ¤ ¤ ¤

¤ ¤ ¤ ¤ ¤ ¤

¤ ¤ ¤ ¤ ¤ ¤

¤ ¤ ¤ ¤ ¤ ¤

¤ ¤ ¤ ¤ ¤ ¤

¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥

¦ ¦ ¦ ¦ ¦

¦ ¦ ¦ ¦ ¦

¦ ¦ ¦ ¦ ¦

¦ ¦ ¦ ¦ ¦

¦ ¦ ¦ ¦ ¦

¦ ¦ ¦ ¦ ¦

¦ ¦ ¦ ¦ ¦

¦ ¦ ¦ ¦ ¦

¦ ¦ ¦ ¦ ¦

¦ ¦ ¦ ¦ ¦

¦ ¦ ¦ ¦ ¦

¦ ¦ ¦ ¦ ¦

¦ ¦ ¦ ¦ ¦

¦ ¦ ¦ ¦ ¦

§ § § § § § § § §

§ § § § § § § § §

¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨

¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨

©

©

SourceGate

Drain

Metal

W p+ p+

L n well p-type substrate

Metal SiO2


DC curves of n-channel FET

!

" # $ % & '

# '

(

$ % )

0

1

4 9

4 9

5

4 9 7

2

@ B

3

1

C

D

F G

H I P Q

# I P

$ % ) #

'

(

$ % )

1

R

D

F G



1

2

3 4

! 5

6

7

8 9

8 9

4 9

8

9

7

8 9

5

8

8

S

T U V

S

W X

Y

a b

c

d e

f


I DS versus V GS in the saturation region

4

5

(mA)

I DS



1

2

3

1 2 3 4 5 6 7 8 V GS (V)V GS Q

QI DS Q

0


FET small-signal equivalent circuit

i

p

s



g

h

q

r

g

t

u

v

w

x

v

w

x


High-frequency small signal models

BJT

rxB

C µC

E

C πro

+

−

vπrπ

gm vπ



FET

−

+

G D

vgs

S

C gs

ro

gm vgs

C gd


Calculation of f T for a BJT




Common-emitter amplifier: DC analysis

RC

I C I E = 1 mAR1

V CC

C L

R1

(9 kΩ)

RC

(5 kΩ)

V CC (10 V)



+

−

V E = 0.3 V

RE I E = 1 mAR2

V C = 5 V

I B 0

V B = 1 V

(1µF )

RE

C E

(10µF )

(100 Ω)

C B

(5µF )

R2

(1 kΩ) (300 Ω)

(15 kΩ)

RL

Rs

vs

I R1= 1 mA

V BE 0.7 V


Small-signal circuit (low-frequency)

C B

vo

C Lv

ovin

+vπrπ

= vinZ ina

gm vπ

Rs



+

−

Z in Z ina

vs

C E

RC RL

−

vπ

RE

(R1||R2)

RB

rπ


Calculation of Z ina

+

vo

C L

gm vπ

v

o

it

rπvπ



+

−

−

vt RLRC

C E

rπ

Z ina=

vtit

RE


Voltage gain: low-frequency range

l m m

l m

l m

z



j

k

l m

n

l m

l m

l m

l m

l m

m

m

m

m

m


Short-circuit time constants method

Resistance seen by C B

+

Rs

it

gm vπ

RC

RL

− +

−

rπ(R1||R2)

RB vπ

vt

Resistance seen by C E

gm vπvπ

rπ

+



+

−

vtRE

RC RL−

Rs

ie2

ie1 it

rπ

vπrπ

(R1||R2)

RB

Resistance seen by C L

R→∞ it

RL

RC

gm vπ

−

rπ

+ − +

Rs RB

(R1||R2)vπ

vt


C-E: Mid-frequency range

~



|

~

|


C-E: High-frequency range

+

vinRs

C µvo



+

−vs

−

vπ

rπ

C πRB(R1||R2)

gm vπ

RC

RL


Voltage gain for the high-frequency range




Application of Miller’s theorem to C µ

+

−

C π

rπ

−

vπ

+Z 1

vinRs

vs

C µ(Z µ = ( j ωC µ)−1)

Z 2

RC

vo

gm vπRB

(R1

||R2

)

RL



+

−

Z 2 =AvBC

Z µAvBC

−1

Rsvin

+

vπ

−rπ

vs

gm vπ

vo

RC

C µAvBC

AvBC −1

C π + C µ(1 − AvBC )

AvBC = vovin

Z 1 = Z µ1−AvBC

RB

(R1||R2)

RL


Common-base amplifier

«

¬

³

µ

-

¶ ¹

±

³

º

-

» ¹

±

¼

º º

-

® ½ ¾

±

¼

Â



ª

-

®

°

±

«

²

³

«

¿

³

À

-

»

°

±

³

¿

-

®

½ ½

¹

±

-

®

½

°

±

Á

À

-

®

»

¹

±

³

¬

-

®

¹

± -

Ã ½ ½

¹

±





−

+

vo

vt

itgm vπ

rπ



+

+

−

R

Lvπ

Z ina


Calculation of the current gain Ai

vin vo

−

gm vπ

Rs

iois =

vsRs



R

L

+

vπ

Z in

rπRE


C-B: Equivalent circuit at high frequencies

Ä

Å

Æ

É Ê

Æ

Ë

Ì

Í

Æ

Î

È

Ð

Î

Ä

Ó



Æ

Å

Ç

È

Ä

Ï

Ç

Æ

Î

Ñ

Ò

Ñ

Î

Õ


Common-collector amplifier

Ý

ß

á

â

ã ã

Ø

Ù Ú î

Ü

Ø

ð Ú ì

Û

Ü



Ö

×

Ø

Ù

Ú Ú

Û

Ü

Þ

ß

à

ß

ä

å

ä

ß

æ

Ø

Ù Ú

ç

è

Ü

Ø

é ê

ë ì

Û

Ü

Ø

Ù

í ì

Û

Ü

Ý

ï

ß

ï

Ø

ð

Ú ì

Û

Ü

Ø

í

ç

è

Ü


DC analysis

ñ

ó

ô

õ õ

ö

ý þ ÿ

ü

ö

¡

þ ú

û

ü

¦

ó

¤

ý §



ñ

ò

ö

÷ ø ù ú û

ü

ñ

ö

¡ þ ú

û

ü

ô

£

¤

¥ ÿ

¦

ó

¤

ý §

¦

¤

ý §

¦

£

©

þ

©

þ ø

ÿ


C-C: AC equivalent circuit

!

"

# #

$

%

(

)

6

7

8

7



&

&

'

0

1

3

4

0

!

3

# #

5

%



+

−

+gm vπ

vπ

−vt rπ

it =vπrπ



(RE ||RL)R

E

io


Calculation of the output impedance

+

−

ie

RB

rπ vπ

gm vπ

it

vπrπ



vt

ie

+−

it

io R

E

(RB||Rs)

(RE ||RL)


Calculation of the current gain

9

@

A

B

C

D

9

@

E

@

P

Q

R

Q

S

P

T U

V

T U

W

e

f

F

f

F

g

X

F

h

Y Y

F

i

a



F

G

I

X

F

I

Y Y

F

a

b

c d

b

c d

W

e

p


Differential pair

Implementation with IGFETs Common-mode operation

V o1 V o2

RD

I Q2

I Q2

RD

V DD

(12 kΩ)

V o2V o1

V DD

RD

(12 kΩ)

RD

(10 V)



+

−

+

−

I Q

V c

Q1

V GS Q

V c

V GS Q

Q2

−V SS

V in2

Q1

V in1

(1 mA)

I Q

Q2

−V SS

(−5 V)


Differential-mode operation

V o1 V o2

Q1

RD

Q2Q

V o1

V o2

RDRD

Q

RD

V DDV DD I Q2+ ids1

I Q2+ ids2

⇔



Q1

I Q

V GS Q −vs2

Q2

−V SS

vs2 −

vs2

Q1

I Q

vs

Q2

−V SS

−

+

−

−

+

V GS Q +vs2

+


Differential pair

Large signal operation

q r

v

w r

q

y

y

y

y



q r s

q

r t

q r

u

q

r s

q

r

u

w r

q

w r t

x

q

r s

x

q

r

u

x

w r

q

x

w r t


Current mirrors

−V SS (-5 V)

+

−

V GS 2

Q1

+

−

V GS 1

Q2

I REF

I o = I REF



−V SS (-5 V)

Q2

+

− −

+V GS 2V GS 1

Q1

RREF

I REF

3 kΩ

I o = I REF


Improved current mirror

k

l

m

n

o

m

m

n

o



j


Propagating sine wave

λ = 10× l

0

2

4

6

8

10

0

0.2

0.4

0.6

0.8

1

−1

−0.5

0

0.5

1

DISTANCE

TIME

2.1 ns

A M P L I T U D E

l



λ = l

0

2

4

6

8

10

0

0.2

0.4

0.6

0.8

1−1

−0.5

0

0.5

1

DISTANCE

l

A M P L I T U D E

2.1 ns

TIME


Ideal transmission line

|

|

|



z

|


Transmission line

Z L

lx = l

N sections

x = 0



Z L

1 2

L∆xL∆x

C ∆xC ∆x Z 2

Z 1

Z L

N

Equivalent model


Transmission of a square pulse

+−

x = 0 x = l

Transmissionline (lossless)

Load

Z o

Z o

Source

Z o

vs(t)



line (lossless)

t

T

V A

vs(t)


Transmission line with load Z L

Source incident wave

fl t d



reflected wave


Voltage patterns

0

2

4

6

8

10

0

0.2

0.4

0.6

0.8

1

−1

−0.5

0

0.5

1

−0.5

0

0.5

1

1.5

22V A

(Load)

(source)

dt

Z L

d = ld = 0(source)(Load)

d = 0

V A

|V (d)|

d = l

|V (d)|

2V A

Matched transmission line

V A

v(t, d)

v(t, d)Open-circuit transmission line

−V A

0

0

0

0

Matched

Open-circuit



0

2

4

6

8

10

0

0.2

0.4

0.6

0.8

1

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

0

2

4

6

8

10

0

0.2

0.4

0.6

0.8

1

−2

−1.5

−1

−2V A

2V A

−2V A

Short-circuited transmission line

(Load)

(Load)

(source)

(source)

dt

t d

d = ld = 0

|V (d)|

2V A

d = 0 d = l

v(t, d)

0

0

0

0

0

Short-circuited


Open-circuit transmission

V s(r−1)2

(1+r)3 2V s(r−1)2

( )3

V s(r−1)(1+r)2

2V s(r−1)

(1+r)2

V s(r−1)(1+r)2

V s

1+r

2V s1+r

V s1+r

T i m e

+−

Z oZ s

V s

t = 0 x = 0 x = l



(1+r)3

Voltage versus the distance at t = T +P

ª


Load voltage at x = l

5 10 15

V s

V (x = l)

t/T P

V s

V (x = l)

Z s = 4Z o

Z s = Z o



5 10 15

V (x = l)

15105

t/T P

t/T P

s

V s

Z s = Z o/4


Waveforms monitored at the input of a faulty cable

5 10

V

t (µs)

Transmitted

t (µs)

10

5

V



5 10

2 T P

Reflected

10


Load matching using a quarter-wave transformer

Ζ L «

¬ -

«

¬

²

³

«

¬ -

«

µ

¶

·



«

¬ -

®

°


Transient analysis

¹ º

¹ »

¼

½ ¾

¼

½ ¾

Â Ã

Ä

Å

Æ

È É

Ê

Ë

Æ

¼

½ ¾

¿

¾Á

¿

¾ ½

Â Ã

Ä

Å

Æ

È É

Ê

Ë

Ì

¼

½ ¾

¿

¾ Á

Â Ã

Ä

Å

È É

Ê

Æ

È

Í

Ë

¿

½ ¾

Ï

Ð

Ò

Ó

Æ

¼

½ ¾

¿

¾ Á

¼

¾ ½

Ê

Ô

Õ Ê

Ô

Ö

×

Ø

Ù

Ú

Û



¼

½ ¾

¿

À

¾ ½

¿

À

¾ Á

Â Ã

Ä

Å

Æ

È É

Ê

Ë

Æ

¼

½ ¾

¿

¾ ½

¿

À

¾Á

Â Ã

Ä

Å

È É

Ê

Æ

È

Í

Ë

¼

½ ¾

¿

À

¾ ½

¿

Î

¾ Á

Â Ã

Ä

Å

È É

Æ

È

Í Ê

Ë

¼

¾ ½

¼

½ ¾

¿

¾ ½

¿

À

¾ Á

Æ

¼

¾ ½

¼

½ ¾

¿

À

¾ ½

¿

Î

¾Á


Lossy transmission line: electrical model

l

N sections

Z L



R∆x

G∆x

L∆x

1 N

C ∆x

Z L


Attenuation constant

è

é ê

ë

Ý Þ

é

á

ì

í

ä

ê

á

í

ì

ç

Ü

ï



Ü

Ý Þ

ß

á

â

ã

ä

æ

á

ã

â

ç

î

Ü

Ü

è

æ

ß

Propagation constant


Signal distortion in a lossy transmission line



ð

ñ ò

ó

ô õ

ö

ñ ÷ ø


Geometry of microstrip lines

Stripsubstrate

Dielectric

conductor

£



ù

ú

û ü

ý

þ

ÿ

¡

¢


Incident and reflected waves

¤

¥

¥

¤

§

§



§ §


S -parameters

Z o1

a1(x1)

b1(x1)

a1(l1)

b1(l1)

a2(l2) a2(x2)

b2(x2)b2(l2)Z o2

2-Port

Circuit



x2 = l2x1 = l1x1 = 0 x2 = 0


Two-port circuit

S 11 and S 21

Circuit+

−

Z o1

x2 = l2 x2 = 0

Z o2

S 11

V s

Z o1

x1 = 0 x1 = l1

2-Port a2(l2) = 0



S 12 and S 22

+−

Circuit

Z o1

x2 = l2 x2 = 0x1 = 0 x1 = l1

2-Port

S 22

a1(l1) = 0

Z o1Z o2

Z o2

V s


S 11 and S 21 of a series impedance Z

!

(

) 0

(

1

(

1

5

2

6

3

2

7

%

'

(

1

(

(

1



"

#

$

%

'

#

2

%

3

2

#

$

%

3

$

#

2

%

'

(

1

9

2

6

3

2

7

@

A


RC low-pass filter

V 1(x1)

+ +

− −

V 2(x2)

I 2(x2)I 1(x1)

Z o Z o

x1 = l1 x2 = l2 x2 = 0

R

C

x1 = 0

S 11 and S 21

V 1(x1)

+ +

− −

I 2(x2)I 1(x1)

Z o

R

V 2(x2)( jωC )−1

Z o

Z o

Z



Z oZ IN 1

S 12 and S 22

+

−

V 2(x2)

I 2(x2)I 1(x1)

R

Z o

Z o

Z oV 1(x1)

+

−

( jωC )−1

Z IN 2

Z o


Measurement of the S -parameters

Z o1

a1(x1)

b1(x1)

a1(l1)

b1(l1)

a2(l2) a2(x2)

b2(x2)b2(l2)Z o2

θ1 = β l1 θ2 = β l2

2-Port

Circuit



x2 = l2x1 = l1x1 = 0 x2 = 0


S -parameters and travelling waves

Circuit

2-PortI 1(x1)

V 2(x)V 1(x)

Z o1−

+I 2(x2)

Z o2

−

+



x1 = 0 x2 = l2 x2 = 0x1 = l1


Calculation of S 11 and of S 21

Circuit+−

+

−

−

+

x1 = 0 x1 = l1 x2 = l2 x2 = 0

b2(l2)

Z o2

I 2(x2)

V 2(0) Z o2

Z o1

V s V 1(0)Z o1

I 1(x1)2-Port

Calculation of S 12 and of S 22



Circuit

2-Port+

−

+

−

+−

x1 = 0

x1 =

l1

x2 =

l2

x2 = 0

V s

Z o2I 2(x2)

Z o2 V 2(0)V 1(0)Z o1

I 1(x1)Z o1


Impedance voltage divider

B

C

D

E

G

H

I

G

P

H

P



F


Calculation of S p of a two-port circuit

V 1 V 2 ++

Z 1 Z 2

I 2I 1

2-port

Circuit



V s1 V s2−−


Series impedance

I 2

Z 2

Z in

Vs1

+

Z 1

10 ΩV 2

Z A j 25 Ω

I 1

Z A j 25 Ω

V 1

40 + j 50 Ω

S p11



V s1−

Z in

Z 2

V s2V 2

Z A j 25 Ω

I 1 I 2

Z 1

V 1

10 Ω

40 + j 50 Ω

−

+

S p21


Constant resistance impedances

Γ-plane

r4r3r2 r5r1

x

r

z -plane

Constant resistance circles



r2 r3 r4 r5r1

U

V Constant resistance circles


Constant reactance impedances

−x3

−x2

x3

x2

x1x

r

−x1

Vx2

Γ-plane

z -plane

Constant reactance circles



x3x1

−

x3

−x2

−x1

U

V 2


Constant reactance and constant resistance impedances

x2V

r1 r2 r3 r4 r5

r

x

−x3

−x2

−x1

x3

x2

x1

z -plane

Γ-planeThe Smith chart



r1 r2 r3r4

−x3

x3

2

x1

−x1

−x2

U

r5


Smith chart: impedance representation

0 0.5 1 2 4 8

20

8

0.5

1

2

3

4

56

8

20

z 3

z 2 z 1

∞

U

V

y8z 7

y

z 5 z 6



8

6

5

4

3

2

1

0.5

z 4

z 8

y7


Transmission line

d = 0d = l

Source Load

Z LZ o



Z in(d = l)


Input impedance and reflection coefficient of transmission line

0 0.5 1 2 4 8

20

8

6

5

0.5

1

2

3

4

5

6

8

20

T o w a r d s L

T o w a r d s

S o u r c

e Y

a

b

c d

WX

e

g

h

g

i

q r s s

Y

t u

v

w

T

x

y

x

T



5

4

3

2

1

0.5

0.40.30.20.1 0.60.5 0.7 0.8 0.9

Lo

a d

Q

R

Q

T

V

W X X


Constant Γ circles

0 0.5 1 2 4 8

20

8

6

0.5

1

2

3

4

5

68

20

|Γ| = 0.6

|Γ| = 1

|Γ| = 0.

3



5

4

3

2

1

0.5

|Γ|

0.2 0.4 0.6 0.8


Electrical lengths of short-circuited and open-circuit transmission lines

0 0.5 1 2 4 8

20

8

65

0.5

1

2

3

4

56

8

20



5

4

3

2

1

0.5


Representation of impedances versus frequency f b > f a

0 0.5 1 2 4 8

20

8

6

5

0.5

1

2

3

4

5

6

8

20

j

k

j



5

4

3

2

1

0.5

k


Representation of impedances versus frequency f b > f a

0 0.5 1 2 4 8

20

8

6

5

0.5

1

2

3

4

5

6

8

20

l l

l l

m m

m m

n

n

o

n



5

4

3

2

1

0.5

n

o




L-section circuits

Z L

Z L

Z LZ L

Z L

Z L



Z L Z L


Impedance matching with L-section circuits

+

−

Z LV sL-Section

RL

LL

Rs



Rs


Impedance matching using L-section circuits

0 0.5 1 2 4 8

20

8

6

0.5

1

2

3

4

5

6

8

20

Solution a)

z L

z a

y = j2.0

1

z L

z = j0.2Solution b)

Solution a)

a)b1)



54

3

2

1

0.5

Solution b)

y = − j2.0

z b

z = − j0.6

z L

1

b)a1)


Impedance matching using L-section circuits

0 0.5 1 2 4 8

20

8

6

5

0.5

1

2

3

4

5

6

8

20

z

|

z

~

z

~

~



4

3

2

1

0.5


Impedance matching using transmission lines

0 0.5 1 2 4 8

20

8

65

4

3

0.5

1

2

3

4

56

8

20



3

2

1

0.5


Impedance matching circuit

ª

«

¬

«

-

®

°

±



ª

«

°

²


Current measured in N resistances

A m p

l i t u d e

A m

p l i t u d e

³

³

¶

¶

· ¹ ¹ º » ¼

½

¾ º ¼ º ¹



A m p l i t u d e

³

³

³

µ

¶


Histogram

i1

NoiseAmplitude

0

Relative numberof occurrences

Histogram

i1

Relative number

NoiseAmplitude

0

of occurrences



Gaussian Probability Density Function

0

i1

pI 1(i1)

Gaussian PDF

NoiseAmplitude


Gaussian PDF

σ = 1

σ = 0.5

pI 1(i1)

0.4

0.

6

0.8

0.2

i

σ = 2



2 4 60−2−4

i1

−6


Calculation of probabilities

i ib

pI 1(i1)

i



ia ib i1


Uniform PDF

¿

À

Á

Â

Ä

Å

Æ

È

Â



Å


Poisson distribution

pX (x)



x0


Calculation of the mean of a PDF

xx

pX

(x

) pX (x)

x pX (x)

µ

f (x) = xf (x) = x

x pX (x)



xxA−A−

A+ A+

A+ + A− = µA+ + A− = 0


Calculation of the variance of a PDF

xx

f (x) = x2 f (x) = x2

pX (x)

x2 pX (x)

pX (x)



x x

x2 pX (x)


Central limit theorem

Exact PDFGaussian PDF

−A

P Z(z )

2A−2A

σ2 = 2 (2A)2

12

P Y (y)

A

Y = X 1 + X 2

X 1

P X 1(x1)

σ2 = (2A)2

12

x1

y



−3A 3Az

Z = X 1 + X 2 + X 3

Z ( )

σ2 = 3 (2A)2

12


Correlation between X and Y

y y

yy

xx

ρXY = 0.7

ρXY = −0.8

ρXY = 1

ρXY = 0



x x


Autocorrelation function

A m p l i t u d e

Negative area

i1(t + τ 1)i1(t)

τ 1

τ 1

t

t

t

i1(t + τ 1)i1(t)

Negative area

t

i1(t) i1(t + τ 1) i1(t) i1(t + τ 1)

A m p l i t u d e

A m p l i t u d e

A m p

l i t u d e

Positive areaPositive area



τ 1 τ τ 1 τ

Ri(τ ) Ri(τ )


Power spectral density

f −B B 2B−2B

Rx(τ )

S xx∗(f )

τ T = 1

B

B = 3B

τ T = 1

B




1

B

1

3B−

1

3Bτ −

1

B


Linear system

É

Ê

Ë

Í

Î

Ê

Ë

Í

Ï

Ê

Ë

Í




Power spectral densities

Ð

Ñ

Ó

Ô

Õ Ð

Ö

Ø

Ù Ù

Ú

Ó

Ô

Õ

Ø

Û Û

Ú

Ó

Ô

Õ

Ô

Ô

Ü

Ý



Ô


White noise

Power Spectral density

Rn(τ )

S nn∗(f )

f

η

2




τ

η

2


The equivalent noise bandwidth

A2

|H (f )|2



f BN


Noisy resistor

R

Noisy resistor

Thevenin equivalent

−+

R

Equivalent voltagethermal noise source

Noiselessresistor

un

Norton equivalent



R

Noiseless

resistor

currentthermalnoise

source

Equivalent in


Noisy p –n junction

+ −V D

I DC Noisy diode

Model for shot noise

incurrentnoiseEquivalent

I DC

+

Noiselessdiode VD



noisesource

V D

−


1/f noise and white noise

Noise sources

η2

η2

f cf

Noise power spectral density

A 2 / H

z )



log10(f )f c

η2

η2

f cf

P S D

(


Noisy resistor

R

Noisy resistor

Series voltage sources

−

+−

+

untunf R

Noiselessresistor

Parallel current sources

Noiseless



int inf

Noiseless

resistor

R




Ideal inductor

Ideal inductor

L

Practical inductor

Ideal inductor

Rs

L

Noise model

Ruu



−+−+

Rsuntunf

Noiselessresistor

Ideal inductor

L


FET

Noiseless small-signal model

+

vgs

−

G

C gd

C gs

gm vgs

ro

D

S

Noise model

G

+i

D

C gd



ro

C gs

−

vgs

ing

+

inf

indgm vgs

S


BJT

Noiseless hybrid-π model

ro

C

gm vπ

vπ

+

−

C π rπ

C µB

E

rx

Noise model

inc

B C

inbC π

inf +

C µ−+

gm vπ

unB rx



E

π

rπ

vπ

−

ro


Noisy amplifier

AmplifierNoisy

vo

vi

+

−

Equivalent model

−

+

un

+



Amplifier

Noise-freevi

in vo

+

−




Output noise voltage

Contribution from unL

−

vπ

rπ

+

C π

RLGm vπ

vovi = 0

−

+

unL

Contribution from inc

−

vπ

rπ

+

C πGm vπ

vi = 0 vo

inc

RL

Contribution from inb



−

vπ

rπ

+

inbC π

RLGm vπ

vovi = 0


Output noise current

Contribution from unL

+

−−

vπ

rπ

+

C π

RL

unL

Gm vπ

io

ii = 0

Contribution from inc

−

vπ

rπ

+

C π

io

ii = 0

Gm vπ

inc

RL

Contribution from inb



−

vπ

rπ

+

inbC π

RLGm vπ

io

ii = 0


Equivalent input noise spectral densities

10−12

10−11

10−10

10−8

10−9

V o l t a g e

s p e c t r a l d e n s i t y

( V / √ H z )

C u r r e n

t s p e c t r a l d e n s i t y ( A / √ H z )

I C = 0.1 mA

I C = 1.0 mA

I C = 0.1 mA

I C = 1.0 mA

I C = 10 mA



106 108 1010 106 108 1010 1012

f (Hz)

1012

10−1310

−10f (Hz)

V

I C = 10 mA

C


Contributions to the equivalent input current noise spectral densities

Þ

ß

à

á

ã

ä

å

æ

è

ê

ë

í

î ï

ð

å

ñ

á

ó

ô

§

"

§

Þ

à

ñ #

$

%

&

ä

ß

á

ó

ô

ý

ÿ

'

ñ

¡

ó

¢

£

¢

(

)

'

¤

¥

£

¢

(

)

Þ

1

2

3

4

5

2

6

3 4

5

8

G

I

Q

S

E

V

I

B

a

c

E

G

Q

g

I

i

p

r

t

v

x

y



Þ

õ

ö ÷ ø

ù

ú

ß

û

ê

ë

í

î ï

ð

è

æ

ü

á

ó

ý

ÿ

í

ñ

¡

ó

¢

£

¢

¤

¥

£

¢

¦

§

©

§

§

§

û

ü

§

!

9

@

B

B

E


Noisy amplifier

vi Z sNoise-freeAmplifier voins

+

− in

un+ −is

AmplifierSource

Equivalent noise model

Noise-freeZ sineq

is



Noise-freeAmplifier vo


Current spectral density of ineq

I ( A)

×10−11

1.4

1.5

1.6

C u

r r e n t s p e c t r a l d e n s i

t y ( A / √ H z )



100

101

102

I C (mA)1.3


Noisy amplifier

vsZ s

ununs

in vo

Noise-freeAmplifier

−+−+

AmplifierSource

+

−

Equivalent noise model

uneq Z s+ −



+

−vs

vo

Noise-freeAmplifier


Chain of three noisy amplifiers





vo

(100 Ω)RE

(5 kΩ)RL

V CC (10 V )

viC i

(1 µF )

RB

(840 kΩ)

AC equivalent circuit

vi

C i

RB

vo

RL



RE


Two-port noise representations

=+

+=

[A] [S ]

[Y ] [Z ]

−+

−+ −+

V 1

I 2I 1

+

−−

+

V 2i

u

a1 bn2bn1

b1

a2

b2

i1

++

− −

V 1 V 2

I 2I 1

i2

+ +

−−

V 2V 1

I 2I 1u1 u2

I 1

I 2

Y 11 V 1 i1

i2Y 21 Y 22 V 2

Y 12 u1

u2

I 1

I 2

Z 12

Z 22Z 21

V 1

V 2

Z 11



+== +

A11

A22A21

A12

−I 2

uV 2

iI 1

V 1 bn1

bn2a2b2

b1

S 22

S 11

S 21

S 12 a1


Noisy shunt admittance

+

−

+

−

−+

+

−

io

+

vo

−

u

un

Y

u u

Y

Y

Y



noiseless

Y in


Noisy two-port circuits in parallel

[CY2]

[Y 2]

[CY1]

[Y 1]

Equivalent two-port circuit

[Y ] + [Y ]



[CY1] + [CY2]

[Y 1] + [Y 2]


Two-port circuits in series

[CZ2]

[Z 2]

[CZ1]

[Z 1]


[Z ] + [Z ]



[Z 1] + [Z 2]

[CZ1] + [CZ2]


Chain of two-port circuits

[CA1]

[A2][A1]

[CA2]


[A

1][A

2][A1] [CA2

][A1]+ +[CA1

]




Transformations between noise representations

Original representation

R e s u l t i n g r e p r e s e n t a t i o n

Admittance Impedance

A d m i t t a n c e

1 0

0 1

Y 11 Y 12

Y 21 Y 22

I m p e d a n c e

Z 11 Z 12

Z 21 Z 22

1 0

0 1

C h a i n

0 A12

1 A22

1 −A11

0−A21

t e r i n g

1+S 112√ Y o

S 12

2√ Y o

S S

1−S 112√ Z o

−S 122√ Z o

S S



S c a t t

S 21

2√ Y o

1+S 222√ Y o

−S 212√ Z o

1−S 222√ Z o


Transformations between noise representations. (Cont.)

Original representation

R e s u l t i n g r e p r e s e n t a t i o n

Chain Scattering

A d m i t t a n c e

−Y 11 1

−Y 21 0

Y o+Y 11√ Y o

Y 12√ Y o

Y 21√ Y o

Y o+Y 22√ Y o

I m p e d a n c e

1−Z 11

0 −Z 21

Z o+Z 11√ Z o

Z 12√ Z o

Z 21√ Z o

Z o+Z 22√ Z o

C h a i n

1 0

0 1

√ Z o

−(A12+A11Z o)√ Z o

−1

√ Z o

−(A22+A21Z o)

√ Z o

t e r i n g

1−S 112√ Z o

−(1+S 11)√ Z o

2

S S√ Z

1 0



S c a t t

−S 212√ Z o

−S 21√Z o

2

0 1



−

+

C π

−

+

− −

+

−

+

+

−

CB

inB

vi vo

RB

C µ

gm vπ

vπ

inbe

unE

RE

C i

Yµ

Ci

VCCS

RL

RE

unL

inc

inB

voC

inb inf

BC i

vi

RB

C µ

rπvπ

gm vπ

inc

RL

unL

RE

E

unE

+




The BJT and RB

C π

inB

+inf

B C

vπ

−

rπ

inb

inc

gm vπ

RB

C µ

E

+

CB

inB

RB

C µ

VCCS

Yµ

B C



+

−

gm vπ

vπ

inbeE

E

BJT

inc



−

+

−

+

unE

RE

C

B

E

RB

C i

vo

BJT’

vi

unL

RL

CiRE

Equivalent model

+ −

N i f

uce



Noise-freeC-E

Amplifierice

vo

vi


100

200

104

108

108

104

100

50

00

40

30

20

10

f (Hz) f (Hz)

( i c e

i ∗ c

e ) 1

/ 2

c e

u ∗ c

e ) 1

/ 2

10−12

10−11

10−8

10−6

( A / √ H z )

( V / √ H z )

C u r r e n t G a i n ( m

a g n i t u d e )

V o l t a g e G a i n ( m a g n i t u d e )

100

150



100

10810

4

100

104

108 f (Hz)f (Hz)

( u

10−1310−10


Noise figure

N o i s e fi g u r e

( d B ) 25

20

10

30

15

5

0100 102 104 106 108 f (Hz)

Y s = 10−2 S

Y s = 10−3 S

Y s = 10−4 S

Y s = 10−5 S



100 102 104 106 108 f (Hz)

AC Circuits 2

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Transcript of AC Circuits 2