Ch12. Filters and Tuned Amplifiers

Ch12. Filters and Tuned Amplifiers

Ch12. Filters and Tuned Amplifiers

Introduction Introduction

‰ Passive LC filters

‰ Electronic Filter Æ Active filter – Active RC filters

– Switched capacitor circuits

Æ Advantages: No inductors!

Inductors are large and physically bulky for low frequency applications (such as those used in passive LC filters)

Filter Transmission Filter Transmission

‰

Filter - a two port device

▪

Transfer function:

▪

Transfer transmission:

▪

Gain function:

▪

Attenuation function:

▪

Input Output relation:

) (

) ) (

( V s

s s V

T

i

≡ o

dB ) ( log 20 )

(

ω

T j

ω

A ≡ −

) (

) ( )

( j

ω

T j

ω

e ^φ(ω) s j

ω

T = ^j =

dB ) ( log 20 )

(

ω

T j

ω

G ≡

) ( ) ( )

(j

ω

T j

ω

V j

ω

V_o = _i

magnitude

phase

Filter Types Filter Types

‰

Frequency-selection function passing Æ passband:

stopping Æ stopband:

‰

Brick-wall response

=1 T

= 0 T

Filter Specification Filter Specification

▪ _ωp : Passband edge

▪ A_max: Maximum allowed variation in passband tansmission

▪ _ωs : Stopband edge

▪ _Amin: Minimum required stopband attenuation

▪

Passband Ripple Æ range: 0.05 dB ~ 3 dB

factor y

Selectivit :

P

ω

S

ω

Ripple bandwidth

Årange:

20 dB ~ 100 dB

Filter Specification Filter Specification

▪

Filter approximation

— The process of obtaining a transfer function that meets given specifications

— Performed using computer programs(Snelgrove, 1982;Ouslis and Sedra, 1995), filter design table(Zverev, 1967) or closed-form expressions(Section 12.3)

The Filter Transfer Function The Filter Transfer Function

‰

Filter Transfer Function T(s)

▪

− N : Filter order

−

If N

≥ M, stable

−

▪

−

−

Æ

All poles must lie in left half plane.

0 1

1

0 1

)

1

( s b s b

a s

a s

s a

T

_N

N N

M M

M M

+

⋅⋅

⋅ + +

+

⋅⋅

⋅ +

= +

₋

−

− −

numbers real

: ,...,

&

..., _{0 1}

,

0 aM b bN−

a

) (

) )(

(

) (

) )(

) ( (

2 1

2 1

N M M

p s

p s

p s

z s z

s z

s s a

T − − ⋅ ⋅⋅ −

−

⋅⋅

⋅

−

= −

modes natural

poles function

- transfer ,...,

zeros on

transmissi zeros

function -

transfer ,....,

1 1

=

=

=

=

N M

p p

z z

Æ Poles need be in conjugate pairs or negative real numbers

The Filter Transfer Function (zeros) The Filter Transfer Function (zeros)

▪ Since in the stopband the transmission is zero or small

Æ the zeros are usually, placed on the jω axis at stopband frequencies 1. zeros at s =+ jω_l1 & + jω_l2

also at s = − jω_l1 & − jω_l2

— Numerator polynomial

(s + jω_l1)(s − jω_l1)(s + jω_l2)(s − jω_l2)

= (s² + ω_l1²)(s² + ω_l2²) Æ for s = jω,

(s² + ω_l1²)(s² + ω_l2²)

= (− ω² + ω_l1²)(− ω² + ω_l2²)

which is zero at ω= ω_l1 and ω= ω_l2 2. zeros at s = ∞

the numbers of zeros at s = ∞ =N—M

M N

M

s T(s) a

s → ∞, → ₋ Qas

zeros ÆDenominator make zeros at infinity

The Filter Transfer Function (ex. 1 : LPF) The Filter Transfer Function (ex. 1 : LPF)

▪

Number of poles =5

▪

Two pairs of complex-conjugate poles and real-axis pole

Æ

all the poles lie in the vicinity of passband

Æ

high transmission at pass band frequencies

▪

Zeros : s =

± j

ω

_l1

&

± j

ω

_l2

&

∞

▪

0 1

2 2 3 3 4 4 5

2 2 2

2 1 2

4( )( )

)

( s b s b s b s b s b

s s

s a

T ^{l l}

+ +

+ +

+

+

= +

ω ω

▪ It has one or more zeros at s = 0 and one or more zeros at s = ∞

▪ Assuming that only one zero exists at s = 0 & s = ∞ Æ

▪ Number of poles =6 Æ Zeros set the number of poles.

▪

The Filter Transfer Function (ex. 2 : BPF) The Filter Transfer Function (ex. 2 : BPF)

0 5

5 6

2 2 2

2 1 2

5

...

) )(

) (

( s b s b

s s

s s a

T ^{l l}

+ + +

+

= +

ω ω

∞

=

=

±

=

±

= j s j s s

s

ω

_l1,

ω

_l2, 0,

The Filter Transfer Function (ex. 3 : all

The Filter Transfer Function (ex. 3 : all - - pole LPF) pole LPF)

▪

It is possible that all zeros are at s =

∞

▪

The more selective the required filter response is, the higher its order must be, and the closer its natural modes are to the j ω axis.

0 1

1

) 0

( s b s b

s a

T _N

N

N + +⋅ ⋅⋅+

= ₋

−

Î all-pole filter

Butterworth and

Butterworth and Chebyshev Chebyshev Filters Filters

▪ In this section, we present two functions that are frequently used in approximating the transmission characteristics of low-pass filters.

: Closed-form expressions

Butterworth Filters : Filter Shape and Butterworth Filters : Filter Shape and

‰

The Butterworth Filter

▪ Monotonically decreasing transmission

▪ All the transmission zero at ω ^{= ∞} Æ all-pole

▪ The magnitude function for an N^th-order Butterworth filter with a passband edge ω_P is

Æ at ω = ω_P,

▪ Thus, the parameter ε determines the maximum variation in passband transmission,

▪ Conversely, given A_max,

N

P

j

T 2

1 2

) 1 (

⎟⎟⎠

⎜⎜ ⎞

⎝ + ⎛

=

ω ε ω ω

1 2

) 1

(

ω ε

= + j p

T

2 max = 20log 1+

ε

A

1 10 ^max/10 −

= ^A

ε

)

( j ω

T

The Butterworth Filter : N effects The Butterworth Filter : N effects

▪ In the Butterworth response the maximum

deviation in passband transmission occurs at the passband edge, ω_P, only

▪ The first 2N—1 derivatives of |T| relative to ω are zero at ω = 0

Æ very flat near ω = 0 (maximally flat response)

▪ The degree of passband flatness increases as the order N is increased

Æ as the order N is increased the filter response approaches the ideal brick-wall

▪ At the edge of the stopband, ω = ω_S, attenuation is

▪ The required filter order = the lowest integer value of N that yields A(ω_S)≥A_min

[

^{S P N}

] [ S P N ]

A(

ω

S) = −20log1 1+

ε

²(

ω ω

)² =10log1+

ε

²(

ω ω

)²

The Butterworth Filter : Poles The Butterworth Filter : Poles

▪ The natural modes of an N^th-order Butterworth filter can be determined from the graphical construction above.

▪ Natural modes lies on a circle of radius ω_P(1/ε)^1/N Æ same frequency of ω₀= ω_P(1/ε)^1/N

▪ Space by equal angles of π/N, with the first mode at an angle π/2N from the +jω axis.

▪ Transfer function is

Æ K

is a constant dc gain of the filter

)

( ) )(

) ( (

2 1

0

N N

p s p

s p s s K

T = − −

ω

⋅ ⋅⋅ −

The Butterworth Filter The Butterworth Filter

▪ How to find a Butterworth transfer function 1. Determine ε.

2. Determine the required filter order as the lowest integer value of N that results in A(ω_S) ≥ A_min.

3. Determine the N natural modes.

4. Determine T(s) 1 10 ^max/10 −

= ^A

ε

[

^{S P N}

] [ S P N ]

A(

ω

S) = −20log1 1+

ε

²(

ω ω

)² =10log1+

ε

²(

ω ω

)²

) (

) )(

) ( (

2 1

0

N N

p s p

s p s s K

T = − −

ω

⋅ ⋅⋅ −

The The Chebyshev Chebyshev Filter Filter

▪ Equi-ripple response (A_max = the peak ripple) in the passband and a monotonically decreasing transmission in the stopband.

▪ The odd-order filter, |T(0)|=1

The even-order filter exhibits its maximum magnitude deviation at ω = 0.

▪ Total number of passband maxima and minima equals the order of the filter, N.

▪ All the zeros are at ω = ∞. Æ all-pole filter

Even-order Odd-order

The The Chebyshev Chebyshev Filter : Equation Filter : Equation

▪ The magnitude of the transfer function with a passband edge ω_P is

▪ Maximum passband ripple :

conversely,

▪ The attenuation at the stopband edge(ω = ω_S) is

ÆRequired order N calculation by

finding the lowest integer value of N that yields A(ω_S) ≥ A_min.

▪ Increasing the order N of the Chebyshev filter causes its magnitude function to approach the ideal brick-wall low-pass response.

))]

( cosh (

cosh 1

log[

10 )

( _S ^{2 2} N ¹ _{S P}

A ω = +ε ⁻ ω ω

P P

P P

P P

j T

N j

T

N j

T

ω ε ω

ω

ω ω ω

ω ω ε

ω ω ω

ω ω ε

+ =

= + ≥

=

+ ≤

=

−

−

for 1

) 1 (

for ]

) (

cosh [

cosh 1

) 1 (

for ]

) (

cos [

cos 1

) 1 (

2

1 2

2

1 2

2

) 1

log(

10 ²

max = +ε

A

1 10 ^max/10 −

= ^A

ε

The The Chebyshev Chebyshev Filter Filter

▪ The poles are

▪ The transfer function is

▪ How to find the transfer function 1. Determine ε

2. Determine the order required, A(ω_S) 3. Determine the poles, p_k

4. Determine the transfer function, T(s)

N N k

N j k

N N

p_{k P} k _P 1 1,2, ,

1 sinh 2 cosh

1 cos 2

sinh 1 sinh 1

2 1

sin 2 ^{1 1} ⎟ = ⋅ ⋅⋅

⎠

⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

⎛ −

⎟+

⎠

⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

⎛ −

−

= ^{− −}

ε ω π

ε ω π

) (

) )(

( ) 2

(

2 1

1

N N

N P

p s p

s p s s K

T = ₋ − − ⋅ ⋅⋅ −

ε

ω

First

First - - Order and Second Order and Second - - Order Filter Functions Order Filter Functions

▪

Nth-order response is very hard to visualize Æ Simple filter transfer functions

—

first and second order

▪

Cascade design.

—

possible for the design of active filters (utilizing op amps and RC circuits).

—

OP-amp output : low impedance

▪

High-order transfer function T(s) can be factored into the product of first-

order and second-order functions.

First

First - - Order Filter Function Order Filter Function

‰

First-Order Filter Function

Æ

bilinear transfer function

—

natural mode at

—

transmission zero at

—

high-frequency gain = a

₁

—

The numerator coefficients, a

₀

and a

₁

, determine the type of filter

0 0

)

1

( + ω

= + s

a s

s a T

ω

0

−

= s

1 0

a s = − a

First

First - - Order Filter Function Order Filter Function

Poles can only be at real axis

First

First - - Order Filter Function Order Filter Function

▪

Although the transmission is constant, its phase shows frequency selectivity

▪

All-pass filters are used as phase shifters and in systems that require phase

shaping

Second

Second - - Order Filter Function Order Filter Function

‰

Second-Order-Filters

▪

Where ω

₀

and Q determine

the natural modes (poles) according to

—

ω

₀

= pole frequency

▪

Q determines the distance of the poles

from the j ω axis: the higher the value of Q, the closer the poles are to the j ω axis

Æ

more selective

▪

Q < 0 Æ poles are in the RHP Æ oscillations

▪

Q = pole quality factor = pole Q

20 0

2

0 1

2 2

) / ) (

( + ω + ω

+

= +

s Q s

a s

a s

s a T

) 4 / 1 ( 2 1

, ₂ ⁰ ₀ ²

1 j Q

p Q

p = −

ω

±

ω

−

Pole freq.

Second

Second - - Order Filter Functions Order Filter Functions

2

= 1 Q

Butterworth

Second

Second - - Order Filter Functions Order Filter Functions

Second

Second - - Order Filter Function Order Filter Function

Second

Second - - Order Filter Function Order Filter Function

‰

LP case: The peak occurs only for

Æ

Butterworth, or maximally flat

‰

HP case: Transmission zeros at s=0 Dual to LP

‰

BP case: Transmission zeros at s=0 and s=

∞

Magnitude response peaks at ω = ω

_o

=center frequency 3dB: ω

₁

, ω

₂

=

BW= ω

₂—

ω

₁

= :as Q

↑

BW

↓

more selective

2

> 1 Q

Q Q

o

o 4 2

1 1 ₂

ω

ω

+ ±

Q

ω

o

2

= 1 Q

The Second

The Second - - order LCR Resonator : Natural Modes order LCR Resonator : Natural Modes

‰

The Resonator Natural Modes

▪ The natural modes can be determined by applying an excitation that does not change the natural structure of the circuit

CR LC Q

CR Q

LC

LC s CR

s

C s

sC R sL

Y I

V

o o

o o

ω ω ω

ω

= = = =

+ +

= +

+

=

=

1 ; 1 ;

; 1

) 1 ( 1

1 1

1 1

0 2

2

The Second

The Second - - order LCR Resonator : Adding Zeros order LCR Resonator : Adding Zeros

‰

Realization of Transmission Zeros

▪

∞

≠

∞

=

≠

=

= +

=

) ( while )

(

0 ) ( while 0

) ( :

) ( )

(

) ( )

( ) ) (

(

2 1

1 2

2 1

2

s Z s

Z

s Z s

Z zeros

s Z s

Z

s Z s

V s s V

T

i o

The Second

The Second - - order LCR Resonator order LCR Resonator

‰

Realization of the Low-Pass Function

▪

LC CR

s s

LC

sC R sL

sL Y

Y Y Z

Z Z V

s V T

s sC R

sL

i o

/ 1 ) /

1 (

/ 1

1 1

1 )

(

at zeros

two

0 1 )

( 1

zeros

2

2 1

1 2

1 2

+

= +

+ +

+ = + =

=

≡

∞

=

⇒

= +

∞

=

The Second

The Second - - order LCR Resonator order LCR Resonator

‰

Realization of the High-Pass Function

▪

1

) ( )

(

Inductor :

0

Capacitor :

0 zeros

2

2 2

2 2

=

+ +

=

≡

⎜⎝

⎛ ==

a

s Q s

s a V

s V T

s s

o i o

o

ω ω

The Second

The Second - - order LCR Resonator order LCR Resonator

‰

Realization of the Band-Pass Function

▪

at ω0,

LC-tuned circuit exhibits an infinite impedance Æ no current flows the center freq. gain is unity

⎜⎝

⎛ == ∞

Capacitor :

Inductor :

zeros 0 s s

LC ) ( CR )

s(

s

CR ) s(

sL sC R

R Y

Y Y

T(s) Y

C L

R

R

1 1

1

1 1

1

2 + +

=

+ +

+ =

= +

Realization of the Notch Functions Realization of the Notch Functions

▪ The impedance of the LC circuit becomes infinite at

Æ zero transmission

▪ The resistor does not introduce zeros.

▪ To obtain arbitrary ω_n

▪ L₁C₁ tank will introduce a pair of zeros at , provided the L₂C₂ tank is not resonant at ω_n.

▪ The natural modes have not been altered,

▪ It is obtained from the original LCR resonator by lifting part of L and part of C off ground

LC /

0 =1

ω

(

₀

)

₀²

2

2 0 2

2 /

)

(

ω ω

ω

+ +

= +

Q s

s a s s

T

1 2 1

1

n

C

L ⁼

ω

j

ω

n

±

L L

L C

C

C₁ + ₂ = ; ₁ || ₂ =

Realization of the All

Realization of the All - - Pass Function Pass Function

▪

The all-pass transfer function

▪

All pass realization with a flat gain of 0.5

2

2 ( )

) 5 (

. 0 ) (

o o

o

Q s

s

Q s s

T

ω ω

ω

+

− +

=

2 2

2 2

2 2

) (

) (

1 2

) (

) ) (

(

o o

o

o o

o o

Q s

s

Q s

Q s

s

Q s

s s T

ω ω

ω

ω ω

ω ω

+

− +

=

+ +

+

= −

bandpass function with a center-frequency gain of 2