Microelectronic Circuits II Ch 11 : Filters and Tuned Amplifiers

Microelectronic Circuits II

Ch 11 :

Filters and Tuned Amplifiers

11.1 Filter Transmission, Types, and Specification 11.2 Filter Transfer Function

11.4 First-order and Second-order Filter Functions

CNU EE 11.1-2

§Important building block of communications, instrumentation systems, electronic filter - Passive LC filters : use of inductors & capacitors; work well at high-frequencies

Inductor : large, physically bulky, nonideal characteristic, No monolithic form - Inductorless filters à active-RC filters : use of op amps, resistors & capacitors;

switched-capacitor filters : fully integrated monolithic filters - Tuned amplifier : radio & TV receiver à bandpass filter

§Filter Transmission

- Filter : linear circuit, general two-port network

- Filter transfer function T(s) :

- Filter transmission by s=jw à magnitude & phase :

Magnitude of transmission in decibel à gain function : Attenuation function :

- Filter output V_o(jw) :

Filter Transmission, Types & Specification

( ) ( ) ( )

V s s V

T

i

= o

( )

( )

T =

( )

( )

( )

( )

Aw º-20log w ,

( )

( ) ( )

V_o = _i

§Filter Types

- frequency selection function : passing signals whose frequency spectrum lies within a specified range, and stopping whose frequency spectrum falls outside this range

- Filter passband : a frequency band (or bands) over which the magnitude of transmission is unity - Filter stopband : a frequency band (or bands) over which the magnitude of transmission is zero - Major filter types : (a) low-pass (LP), (b) high-pass (HP), (c) bandpass (BP) &

(d) bandstop (BS) or band-reject :: ideal vertical edge characteristics à brick-wall responses

Filter Transmission, Types & Specification

CNU EE 11.1-4

§Filter Specification

- Realistic specifications for the transmission characteristics of a low-pass filter - Upper deviation bound of the passband transmission, A_max (dB) : 0.05 ~ 3 dB

- Stopband signals to be attenuated by at least A_min (dB) relative to the passband signals : 20 ~ 100 dB - Transition band from the passband edge w_p to the stopband edge w_s

- Selectivity factor w_s/w_p : a measure of the sharpness of the low-pass filter response

Filter Transmission, Types & Specification

Filter Transmission, Types & Specification

§Low-pass Filter Specification - Passband edge w_p

- Maximum allowed variation in passband transmission A_max - Stopband edge w_s

- Minimum required stopband attenuation A_min - Ideal filter spec. : Lower A_max , higher A_min ,

selectivity ratio w_s/w_p closer to unity à higher order & more complex & expensive

§Transfer function whose specification meets the specification

- Since the peak ripple is equal to A_max, passband ripple A_max & ripple bandwidthw_p - Minimum stopband attenuation is equal to A_min, with the ripple peaks all equal

à equiripple in both the passband & the stopband

- Filter approximation : The process of obtaining a transfer function that meets given specification

CNU EE 11.1-6

Filter Transmission, Types & Specification

§Transmission specification for a bandpass filter - approximation function does not ripple in the passband

- The transmission decreases monotonically on both sides of the center frequency

- The transmission attains the maximum allowable deviation at the two edges of the passband

Filter Transfer Function

- Filter transfer function T(s)

degree of denominator, N : filter degree ; stable filter circuit :

numerator coefficients, a₀, a₁,…., a_M & denominator coefficients, b₀, b₁,…., b_N-1 : real number - Factored polynomials form T(s)

numerator roots, z₁, z₂,…, z_M : transfer function zeros, or transmission zeros denominator roots, p₁, p₂,…, p_N : transfer function poles, or natural modes

- Since in the filter stop band the transmission is required to be zero or small, the filter transmission zeros are usually placed on the jw axis at stop band frequencies

- low-pass filter has infinite attenuation (zero transmission) at two stopband frequencies : w_l1 & w_l2 à transmission zeros at s = +jw_l1 & s = +jw_l2 à the other transmission zeros at s = -jw_l1 & s = -jw_l2 since complex zeros occur in conjugate pairs ànumerator polynomial factors (s²+w_l1 ²)(s²+w_l2 ²)

- In the low pass filter, the transmission decreases toward – as w approaches à one or more transmission zeros at s = à the number of transmission zeros at s = is N – M

- For a filter circuit to be stable, all its poles must lie in the left half of the s plane, and thus p₁, p₂,…, p_N must all have negative real parts

( )

0 1

1

0 1

1

b s

b s

a s

a s s a

T _N

N N

M M M M

+

×

×

× + +

+

×

×

× +

= + _-

-

- -

N M £

( ) ( )( ) ( )

( )( ) (

)

M M

p s p

s p s

z s z

s z s s a

T - - ××× -

-

×

×

× -

= -

2 1

2 1

¥ ¥

¥ ¥

CNU EE 11.1-8 - sixth order (N=6) bandpass filter T(s)

transmission zeros at s = +jw_l1 & s = +jw_l2, one or more zeros at s = 0 &

Filter Transfer Function

- Typical pole & zero locations for fifth order (N=5) low-pass filter T(s)

five poles : two pairs of complex-conjugate poles + one real-axis pole

All poles lies in the vicinity of the passband à high transmission at passband frequencies

five transmission zeros at

( ) ( )( )

0 1 1 2 2 3 3 4 4 5

2 2 2 2

1 2 4

b s b s b s b s b s

s s

s a

T ^{l l}

+ +

+ +

+

+

= +w w

¥

¥

=

±

=

±

= j s j s

s w_l1, w_l2,

( ) ( )( )

0 5

5 6

2 2 2 2

1 2 5

b s

b s

s s

s s a

T ^{l l}

+

×

×

× + +

+

= +w w

Filter Transfer Function

- a fifth order (N=5) low-pass filter having all transmission zeros at infinity T(s)

No finite values of w at which the attenuation is infinite (zero transmission) à all zeros at s = à all-pole filter

- General filters : Transmission zeros are on the jw axis, in the stopband(s), w = 0 & w =

¥

( )

0 1

1 0

b s

b s s a

T _N

N

N + +×××+

= _-

-

CNU EE 11.1-10

First-order & Second-order Filter Function

§First-Order filters

- The general first-order transfer function (bilinear transfer function)

a natural modes at s = -w₀ , a transmission zero at s = -a₀/a₁, & a high-frequency gain that approaches a₁ The numerator coefficients, a₀ and a₁, determines the type of filter (i.e., LP, HP, etc.)

- Passive (RC) and active (op amp - RC) realizations

- The Output impedance of the active circuits is very low (ideally zero) à cascading does not change the transfer functions of the individual blocks

( )

0 0 1

w +

= + s

a s s a

T

CNU EE 11.1-12

First-order & Second-order Filter Function

- all-pass filter : the transmission zero and the natural modes are symmetrically located relative to the jw axis (mirror-image symmetry with respect to the jw axis)

à the transmission of the all-pass filter is (ideally) constant at all frequencies à its phase shows frequency selectivity

à phase shifters

First-order & Second-order Filter Function

§Second-Order (biquadratic) filter functions - The general second-order transfer function :

a natural modes (poles) by w₀ & Q :

Q > 0.5 : complex-conjugate natural modes

- Location of the pair of complex-conjugate poles in the s plane

pole frequency w₀ : radial distance of the natural modes (from the origin) pole quality factor (or pole) Q : distance of the poles from the jw axis

The higher the value of Q, the closer the poles are to the jw axis, and the more selective the filter response becomes

à infinite value of Q = poles on the jw axis = sustained oscillation

à negative value of Q = poles in the right half of the s plane = oscillations

- The numerator coefficients, a₀ , a₁ and a₂, determines the type of filter (i.e., LP, HP, etc.) - low-pass (LP) case : two transmission zeros at s = ; peak occurs only for

- high-pass (HP) case : both transmission zeros at s = 0; peak occurs only for

- bandpass (BP) case : one transmission zero at s = 0 (dc);& the other at s = ; magnitude response peaks at w = w₀ , center frequency ; selectivity of the filter by 3-dB bandwidth, w₂ - w₁ at which the

magnitude response is 3dB below its maximum value (at w )

( )

(

)

0 1 2 2

w

w +

+

+

= +

s Q s

a s a s s a

T

(

)

0 0

2

1 1 1 4

, 2 j Q

p Q

p = - w ± w -

¥ Q >1 2

2 1

>

Q

¥

CNU EE 11.1-14

First-order & Second-order Filter Function

§Second-Order (biquadratic) filter functions - notch filter, bandstop (BS) case :

If the transmission zeros are on jw axis, at the complex-conjugate locations ,then the magnitude response exhibits zero transmission at w = w_n à notch occurs notch frequency w_n

three cases : regular notch when w_n = w₀, low-pass notch when w_n > w₀, high-pass notch when w_n < w₀ No transmission zeros at either s = 0 or s = à transmission at dc & at s = is finite¥ ¥

jwn

±

CNU EE 11.1-16

First-order & Second-order Filter Function

§Second-Order (biquadratic) filter functions -All-pass (AP) filter case :

two transmission zeros are in the right half of the s plane, at the mirror-image locations of the poles flat gain : the magnitude response is constant over all frequencies

frequency selectivity is in its phase response