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Chapter 15 Active Filter Circuits, Slides of Design

Design

Filter is circuit that capable of passing signal from input to output that has frequency within a specified band and attenuating all others outside the band.

Typology: Slides

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Chapter 15

Active Filter Circuits

_____________________________________________

15.0 Introduction

Filter is circuit that capable of passing signal from input to output that has

frequency within a specified band and attenuating all others outside the band.

This is the property of selectivity.

They are four basic types of filters. They are low-pass, high-pass, band-

pass, and band-stop. The all-pass filter circuit that can be designed.

The basic filter is achieved by with various combinations of resistors,

capacitors, and sometimes inductors. It is called passive filter. Active filters use

transistors or operational amplifier and RC circuit to provide desired voltage

gains or impedance characteristics. Inductance is not preferred for active filter

design because it is least ideal, bulky, heavy, and expensive and does not lend

itself to IC-type mass production.

Each type of filter response can be tailored by circuit component values

that have Butterworth, Chebyshev, or Bessel characteristics. Each of these

characteristics is identified by the shape of its response curve and each has an

advantage in certain application.

Butterworth characteristic has very flat amplitude in the pass band and a

roll-off rate of -20dB/decade/pole. The phase response is not linear. However,

the phase shift of the signals passing through the filter varies nonlinearly with

frequency. Therefore, a pulse applied to a filter with Butterworth response will

cause overshoots on the output because each frequency component of the

pulse's rising and falling edges experiences a different time delay.

Chebyshev has characteristic response that roll-off greater than -

20dB/decade/pole. The circuit has characteristic of overshoot and ripple

response in the pass band.

Bessel has a linear phase characteristic, which shall mean that the phase

shift increases linearly with frequency. Thus, Bessel response is used for

filtering pulse waveform without distorting the shape of the waveform.

Partial preview of the text

Download Chapter 15 Active Filter Circuits and more Slides Design in PDF only on Docsity!

Chapter 15 Active Filter Circuits

_____________________________________________

15.0 Introduction

Filter is circuit that capable of passing signal from input to output that has

frequency within a specified band and attenuating all others outside the band.

This is the property of selectivity.

They are four basic types of filters. They are low-pass, high-pass, band-

pass, and band-stop. The all-pass filter circuit that can be designed.

The basic filter is achieved by with various combinations of resistors,

capacitors, and sometimes inductors. It is called passive filter. Active filters use

transistors or operational amplifier and RC circuit to provide desired voltage

gains or impedance characteristics. Inductance is not preferred for active filter

design because it is least ideal, bulky, heavy, and expensive and does not lend

itself to IC-type mass production.

Each type of filter response can be tailored by circuit component values

that have Butterworth, Chebyshev, or Bessel characteristics. Each of these

characteristics is identified by the shape of its response curve and each has an

advantage in certain application.

Butterworth characteristic has very flat amplitude in the pass band and a

roll-off rate of -20dB/decade/pole. The phase response is not linear. However,

the phase shift of the signals passing through the filter varies nonlinearly with

frequency. Therefore, a pulse applied to a filter with Butterworth response will

cause overshoots on the output because each frequency component of the

pulse's rising and falling edges experiences a different time delay.

Chebyshev has characteristic response that roll-off greater than -

20dB/decade/pole. The circuit has characteristic of overshoot and ripple

response in the pass band.

Bessel has a linear phase characteristic, which shall mean that the phase

shift increases linearly with frequency. Thus, Bessel response is used for

filtering pulse waveform without distorting the shape of the waveform.

The damping factor ξ of the active filter determines which characteristic

the filter exhibits. Refer to Fig. 15.6, the circuit has a RC filter element at

positive feedback and negative feedback circuit, which comprises resistors R 1

and R 2.

The characteristics of Butterworth, Chebyshev and Bessel filters are shown

in Fig. 15.1.

Figure 15.1: The characteristic of Butterworth, Chebyshev and Bessel filters

15.1 Transfer Function

Filters are implemented with devices exhibiting frequency-dependent

characteristic like inductor and capacitor. The behavior of a circuit is uniquely

characterized by its transfer function T(s). Using simple laws such as KVL,

KCL, and the superposition theorem, the transfer function T(s), which is the

ratio of output voltage or current and input voltage or current, is mathematically

expressed for voltage as

T(s) =

V (s)

V (s) in

out (15.1)

Once the function T(s) is known, the output voltage response Vout( t ) to a given

input voltage response V in ( t ) can be determined from the inverse Laplace’s

function containing the transfer function and the input voltage, which is

expressed as

Vout( t ) = L -1{T(s)Vin(s)} (15.2)

Solution

The Transfer function is =

sLC sRC 1

sRC V (s)

T( s) V (s) 2 in

out s (R/L)s 1 / LC

s L

R

Substitute the known values for L, C, and R, the transfer function becomes T(s)

[s ( 1 2 ) 10 ]x[s ( 1 2 ) 10 ]

2 x 10 s 3 3

3 − − + j − − − j

. Thus, the function has a scaling factor

of 2x10^3 V/V, a zero at the origin, and conjugate pole pair at – 1 ± j 2 complex

KNp/s. Its pole-zero plot is shown below.

15.2 General Two-Pole Active Filter

A general two-pole active filter is shown in Fig. 15.2 with Y 1 through Y 4 are

admittances and with an ideal voltage follower.

Figure 15.2: An unity gain general two-pole active filter

A KCL equation at node Va shall be

(Vin -Va)Y 1 = (Va - Vb)Y 2 + (Va -Vout)Y 3 (15.5)

A KCL equation at node Vb produces

(Va - Vb)Y 2 = VbY 4 (15.6)

Since Vb is also equal to Vout therefore

Va = V

Y Y

Y V^

Y Y

b out Y

2 4 2

Thus, substitute equation (15.7) into equation (15.5) and multiply it by term Y 2 ,

it yields the transfer function T(s) for the filter, which is

T(s) =

V s V s

Y Y

Y Y Y Y Y Y

out in

( ) =^ + ( + + )

1 2 1 2 4 1 2 3

For non-unity gain filter, the transfer function of the general two-pole filter

follows equation (15.9).

T(s) =

ZZ Z Z ZZ ZZ ZZ ( 1 A )

A ZZ

V (s)

V (s) 1 2 2 3 3 4 1 3 1 4 V

V 3 4 in

out

- - - −

AV is the pass-band gain of the filter, which is defined as AV =

2

1 R

1 + R and Z

Z 2 , Z 3 , and Z 4 are impedances. The equation can be derived using KCL law and

the circuit shown in Fig. 15.3.

Figure 15.3: A non-unity gain general two-pole active filter

The transfer function T(s) of the filter shall be

T(s) =

V s V s

sC

sC R^

sRC

out in

The magnitude of the function is |T(s)| =

1 + ωRC 2

. If considering the

negative feedback gain AV of the amplifier as shown in Fig. 15.6, then the

transfer function T(s) shall be T s

R

R sRC

1 2

and its magnitude shall be

T(s)| =

1 2 2

R

R (^) ωRC

. The phase is φ = -tan-1(ωRC).

Figure 15.6: Single-pole low-pass active filter

The magnitude of the transfer function |T(s)| =

1 + ωRC 2

can be re-written as

|T(s)| =

2

f f (^) C

. If the internal critical frequency of the operational amplifier

is much larger than f c of the low-pass filter, then the transfer function T(s)

which is also the voltage gain AV will roll-off at the rate of -20dB/decade/pole

which is shown in Fig. 15.7. Its Bode plot function shall be

|T(s)|dB = 20

log (^10 )

f  f (^) C

Equation (15.6) can be re-written as 20 log 10 (1) - 20 log 10 1

2

f

f C^ = 20

log 10 (1) - 10 log 10

1 +^2

f C

f .

Figure 15.7: The voltage gain response of low-pass active filter

As it has been mentioned earlier, Butterworth filter exhibits very flat amplitude

in its pass band. For this reason it is also called maximally flat filter.

Butterworth filter utilizing two RC networks, which is also called two-pole or

second order filter , is shown in Fig. 15.8. Its roll-off rate is -40dB/decade.

Substituting Y 1 = 1/R 1 , Y 2 = 1/R 2 , Y 3 = sC 1 , and Y 4 = sC 2 into the general

transfer function T(s) shown in equation (15.1), the transfer function of the filter

is T(s) =

V s V s

R R

R R sC R R sC

out in

1 2 1 2 2 1 2 1

. This transfer function can

be re-arranged as

1 2 1 2 1 2 1 2

2 2 1 2

1 2 1 2

RR C C

s^1 RR CC

s C (R R )

RR CC

T(s)

=. The transfer function can

further be arranged as the standard second-order low-pass network equation,

which is

Figure 15.8: Second order Butterworth low-pass filter showing cut-off frequency of 1.0kHz

When ω = 0, equation (15.18) yields C 1 = 2C 2. This shall mean that equation

(15.15) becomes

4 1 4 ( 2 )

| T(s)|^1

ωτ

Substituting C 1 = 2C 2 in to equation (15.16), it yields critical frequency f c =

π τ 2 2 π (^22)

RC

. Likewise the critical frequency is also equal to f c =

π 2 RC 1

after substituting C 1 = 2C 2. Substituting f c =

2 π 2 τ (^2)

into equation

(15.19), its yields the magnitude of the transfer function as | T s( )| =

f^4 f (^) c

Substituting C 1 = 2C 2 and R 1 = R 2 = R into equation (15.13), which is

1 2 1 2

LP^2212 2 RR C C

ξ =R^ C +RC , it yields damping factor ξ

LP of^2

1 = 0.707 for Butterworth

low-pass active filter.

Let’s now consider a Sallen-Key equal component low-pass active filter

shown in Fig. 15.9. Equal component shall mean the value of R 1 = R 2 = R and

C 1 = C 2 = C. Based on the general two-pole equation (15.9), which is T(s) =

ZZ Z Z ZZ ZZ ZZ ( 1 A )

A Z Z

V (s)

V (s) 1 2 2 3 3 4 1 3 1 4 V

V 3 4 in

out

- - - −

= , after substituting R and C, the

transfer function T(s) becomes

R R/sC 1 /sC R/sC R/sC( 1 A )

A 1 /sC V (s)

V (s) V

2 2 2

V^22 in

out

- - - −

V

2

V^22 s s( 3 A )/CR 1 /C R

A /sC

− +

. As the standard second-order low-pass network

equation, the transfer function shall be 2

C C

2

V^2 C s 2 s

T( s) A

ξω +ω

= ω , where the critical

angular frequency ωc =

RC

1 and 2ξω

c = (3-Av)/CR. This will give rise to

damping factor ξLP = 0.5(3 – Av). This shall mean that the pass-band gain AV is

equal to AV = 3 - 2ξLP. Also values of R 3 and R 4 are determined to be R 3 =

2RAV and R 4 =

A 1

R

V

3 −

. R 3 is determined from R 3 ||R 4 = R 1 + R 2 = 2R that used

for offset current error.

Figure 15.9: Sallen-Key equal component low-pass active filter

A third order Butterworth low-pass filter is shown in Fig. 15.10 respectively.

For R 1 = R 2 = R 3 , the critical frequency of the third order Butterworth low-pass

filter is

f c =

2 π 3 τ τ τ 1 2 3

The output response for different frequency is shown in Fig. 15.12. For input

frequency f much smaller then critical frequency f c, the output is approximately

zero volt.

Figure 15.12: The expected output signal of the basic high-pass filter with input of various frequency

The basic active high-pass circuit and the output response with respect to

frequency are shown in Fig. 15.13.

(a) Basic high-pass circuit (b) frequency response Figure 15.13: (a) A basic high-pass filter and (b) its output-frequency response

The transfer function T(s) of the filter shall be

T(s) =

V s V s

sRC sRC sRC

out in

( ) =^1 + = +

The magnitude of the transfer function T(s) shall be

|T(s)| =

V s V s R C

out in

ω^2 2

^

f^2 f

c

The phase is φ = tan-^

ωRC 0

^

^ -tan

-1(ωRC) = 90 0 - tan-1(ωRC). The critical

frequency f c shall be equal to

2 πRC.

Substituting Y 1 = sC 1 , Y 2 = sC 2 , Y 3 = 1/R 1 , and Y 4 = 1/R 2 into the general

transfer function T(s) shown in equation (15.1), the transfer function of for two-

pole Butterworth active high-pass filter is T(s) = =

V (s )

V (s) in

out

sC sC 1 /R (sC sC 1 /R )

sC sC 1 2 2 1 2 1

1 2 ⋅ + + +

⋅ . This transfer function can be re-arranged as

1 2 1 2 1 2 1 2

2 1 1 2

2

RR C C

s^1 RR CC

s R (C C )

T(s) s

=. The transfer function can further arrange as

the standard second-order high-pass network equation, which is

2 C C

2

2 s 2 s

s T( s)

ξω +ω

where

1 2 1 2

C RR CC

ω =^1 is the critical angular frequency and the damping factor

ξHP is equal to

1 2 1 2

HP^1112 2 RR C C

ξ = R^ C +RC (15.24)

4

1

|T(s)|^1



ωτ

Substituting 2R 1 = R 2 into equation (15.28) yields critical frequency f c =

π τ 1 2 π (^21)

R C

. Likewise the critical frequency is also equal to f c =

π 2 R C 2

after substituting 2R 1 = R 2. Substituting f c =

π τ

into equation (15.29), its

yields the magnitude of the transfer function as | T s( )| =

^

f^4 f

c

. Substituting

C 1 = C 2 and 2R 1 = R 2 = R into equation (15.24), which is

1 2 1 2

HP^1112 2 RR C C

ξ = R^ C +RC , it

yields damping factor ξHP of

1 = 0.707 for Butterworth high-pass active filter.

Fig. 15.14 shows the two-pole Butterworth high-pass filter where it has

R 2 value equals to twice the value of resistor R 1 and same capacitor values for

C 1 and C 2.

Figure 15.14: Two-pole high-pass active filter

Let’s now consider a Sallen-Key equal component high-pass active filter shown

in Fig. 15.15. Equal component shall mean the value of R 1 = R 2 = R and C 1 = C 2

= C. Based on the general two-pole equation (15.9), which is T(s) =

ZZ Z Z ZZ ZZ ZZ ( 1 A )

A Z Z

V (s)

V (s) 1 2 2 3 3 4 1 3 1 4 V

V 3 4 in

out

- - - −

= , after substituting R and C, the

transfer function T(s) becomes

1 /sC R/sC R R/sC R/sC( 1 A )

A R

V (s)

V (s) V

2 2 2

V^2 in

out

- - - −

V

2

V^2 s s( 3 A )/CR 1 /C R

A s

− +

. As the standard second-order low-pass network

equation, the transfer function shall be 2

C C

2

2 V s 2 s

A s T( s)

ξω +ω

= , where the critical

angular frequency ωc =

RC

1 and 2ξω

c = (3-Av)/CR. This will give rise to

damping factor ξHP = 0.5(3 – Av). This shall mean that the pass-band gain AV is

equal to AV = 3 - 2ξHP. Also values of R 3 and R 4 are R 3 = RAV and R 4 =

A 1

R

V

3 −

R 3 is determined from R 3 ||R 4 = R 2 = R that used for offset bias current.

Figure 15.15: Sallen-Key equal component high-pass active filter

A three-pole Butterworth high-pass filter with C 1 = C 2 = C 3 , the critical

frequency f c shall be

f c = 1

2 π 3 R R R C C C 1 2 3 1 2 3

Filter

Order

Section

Order ξ^ and^ κLP^ Response Type

Bessel Butterworth Chebyshev

2 2 ξ^ 0.866^ 0.707^ 0.

κ 0.785^1 1.

2 ξ^ 0.7385^ 0.5^ 0.

κ 0.687^1 1.

ξ -^ -^ -

κ 0.753^1 2.

2 ξ^ 0.958^ 0.924^ 0.

κ 0.696^1 1.

ξ 0.620^ 0.3825^ 0.

κ 0.621^1 1.

2 ξ^ 0.5455^ 0.31^ 0.

κ 0.549^1 1.

2 ξ^ 0.8875^ 0.81^ 0.

κ 0.619^1 1.

ξ -^ -^ -

κ 0.321^1 3.

2 ξ^ 1.959^ 0.966^ 0.

κ 0.621^1 2.

2 ξ^ 0.818^ 0.707^ 0.

κ 0.590^1 1.

ξ 0.4885^ 0.259^ 0.

κ 0.523^1 1.

Figure 15.17: The damping factor ξ and low-pass frequency correction factors κLP of higher order low-pass active filter

The frequency correction factor for a high-pass filter κHP is the reciprocal of the

low-pass frequency correction factor κLP. i.e.

κHP = 1/κLP (15.31)

The corrected frequency f o is equal to the ratio of cut-off frequency f c and the

frequency correction factor κ as shown in equation (15.32).

f o = f c/κ (15.32)

Example 15.

Design a fourth-order low-pass Sallen-Key Chebyshev active filter with critical

frequency at 10kHz.

Solution

This fourth-order low-pass filter can be achieved by cascading two second-order

Sallen-Key low-pass active filter.

The parameter for first stage shall be ξ = 0.6375 and κLP = 1.992.

Thus, the pass-band gain of first stage shall be AV1 = 3 - 2ξLP = 1.725V/V.

The corrected cut-off frequency is f o = 10kHz/1.992 = 5.02kHz.

Let C = 0.01 μF, then R =

2 C

π f (^) o

= 3.17kΩ.

R 11 = 2RAV1 = 2x3.17kΩ(1.725) = 10.9kΩ

R 12 = =

9 k A 1

R

V 1

11 15.0kΩ.

The parameter for second stage shall be ξ = 0.1405 and κLP = 1.06.

Thus, the pass-band gain of first stage shall be AV2 = 3 - 2ξLP = 2.719 V/V.

The corrected cut-off frequency is f o = 10 kHz/1.06 = 9.43 kHz.

Let C = 0.01 μF, then R =

2 C

π f (^) o

= 1.68kΩ.

R 21 = 2RAV2 = 2x1.68 kΩ(2.719) = 9.1kΩ.

R 22 = =

1 k A 1

R

V 2

21 5.3kΩ.

The overall pass-band gain is AV1xAV2 = 1.725x2.719 = 4.69V/V.

dBG = 20 log(4.69) = 13.4dB.

15.6 Band-Pass Filter

Chapter 15 Active Filter Circuits, Slides of Design

Related documents

Partial preview of the text

Download Chapter 15 Active Filter Circuits and more Slides Design in PDF only on Docsity!

Chapter 15

Active Filter Circuits

_____________________________________________

15.0 Introduction

Filter is circuit that capable of passing signal from input to output that has

frequency within a specified band and attenuating all others outside the band.

This is the property of selectivity.

They are four basic types of filters. They are low-pass, high-pass, band-

pass, and band-stop. The all-pass filter circuit that can be designed.

The basic filter is achieved by with various combinations of resistors,

capacitors, and sometimes inductors. It is called passive filter. Active filters use

transistors or operational amplifier and RC circuit to provide desired voltage

gains or impedance characteristics. Inductance is not preferred for active filter

design because it is least ideal, bulky, heavy, and expensive and does not lend

itself to IC-type mass production.

Each type of filter response can be tailored by circuit component values

that have Butterworth, Chebyshev, or Bessel characteristics. Each of these

characteristics is identified by the shape of its response curve and each has an

advantage in certain application.

Butterworth characteristic has very flat amplitude in the pass band and a

roll-off rate of -20dB/decade/pole. The phase response is not linear. However,

the phase shift of the signals passing through the filter varies nonlinearly with

frequency. Therefore, a pulse applied to a filter with Butterworth response will

cause overshoots on the output because each frequency component of the

pulse's rising and falling edges experiences a different time delay.

Chebyshev has characteristic response that roll-off greater than -

20dB/decade/pole. The circuit has characteristic of overshoot and ripple

response in the pass band.

Bessel has a linear phase characteristic, which shall mean that the phase

shift increases linearly with frequency. Thus, Bessel response is used for

filtering pulse waveform without distorting the shape of the waveform.

The damping factor ξ of the active filter determines which characteristic

the filter exhibits. Refer to Fig. 15.6, the circuit has a RC filter element at

positive feedback and negative feedback circuit, which comprises resistors R 1

and R 2.

The characteristics of Butterworth, Chebyshev and Bessel filters are shown

in Fig. 15.1.

Filters are implemented with devices exhibiting frequency-dependent

characteristic like inductor and capacitor. The behavior of a circuit is uniquely

characterized by its transfer function T(s). Using simple laws such as KVL,

KCL, and the superposition theorem, the transfer function T(s), which is the

ratio of output voltage or current and input voltage or current, is mathematically

expressed for voltage as

T(s) =

out (15.1)

Once the function T(s) is known, the output voltage response Vout( t ) to a given

input voltage response V in ( t ) can be determined from the inverse Laplace’s

function containing the transfer function and the input voltage, which is

expressed as

Vout( t ) = L -1{T(s)Vin(s)} (15.2)

Solution

The Transfer function is =

R

Substitute the known values for L, C, and R, the transfer function becomes T(s)

. Thus, the function has a scaling factor

of 2x10^3 V/V, a zero at the origin, and conjugate pole pair at – 1 ± j 2 complex

KNp/s. Its pole-zero plot is shown below.

A general two-pole active filter is shown in Fig. 15.2 with Y 1 through Y 4 are

admittances and with an ideal voltage follower.

A KCL equation at node Va shall be

(Vin -Va)Y 1 = (Va - Vb)Y 2 + (Va -Vout)Y 3 (15.5)

A KCL equation at node Vb produces

(Va - Vb)Y 2 = VbY 4 (15.6)

Since Vb is also equal to Vout therefore

Va = V

Y Y

Y V^

Y Y

Thus, substitute equation (15.7) into equation (15.5) and multiply it by term Y 2 ,

it yields the transfer function T(s) for the filter, which is

T(s) =

Y Y

Y Y Y Y Y Y

( ) =^ + ( + + )

For non-unity gain filter, the transfer function of the general two-pole filter

follows equation (15.9).