High Pass Filters, 2nd Order Filters, Active Filters,Resonances.pdf

Lecture 20 High g Pass Filters,, 2nd Order Filters, Active Filters, Resonances

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Series Resonance

Z s ( f ) = j 2πffL + R − j

1 f 2πfC

For resonance For the reactance resonance of : the inductor and the capacitor cancel: 1 1 2 2πf 0 L = → f0 = 2πf 0 C (2π ) 2 LC 1 f0 = 2π LC ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Series Resonance Quality factor QS Qs ≡ = Substitute L = Qs =

Reactance of inductance at resonance Resistance 2πf 0 L R 1 1 from f 0 = 2 2 (2π ) ( f 0 ) C 2π LC 1 2πf 0 CR

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Series Resonance 1 Z s ( f ) = R + j 2πfL − j 2πfC ⎡ = R ⎢1 + ⎢⎣

⎛ 2πfL 1 ⎞⎤ ⎟⎟⎥ − j ⎜⎜ 2πffRC ⎠⎥⎦ ⎝ R

⎡ ⎞⎤ 2πf 0 L ⎛⎜ f 1 ⎟⎥ = R ⎢1 + j − R ⎜⎝ f 0 (2π ) 2 ff 0 LC ⎟⎠⎥⎦ ⎢⎣ 2πf 0 L 1 Substitute f 0 = and Qs = R 2π LC ⎡ ⎛ f f 0 ⎞⎤ Z s ( f ) = R ⎢1 + jQs ⎜⎜ − ⎟⎟⎥ f ⎠⎥⎦ ⎝ f0 ⎣⎢ ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Series Resonance

Z R

=

2⎛ 1 + Qs ⎜⎜

f0 ⎞ f − ⎟⎟ f ⎠ ⎝ f0

2

⎛ f f0 ⎞ ∠Z s = Tan Qs ⎜⎜ − ⎟⎟ f ⎠ ⎝ f0 −1

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Series Resonant Band-Pass Filter

Vs = I= Zs ( f )

Vs / R ⎛ f f0 ⎞ 1 + jQs ⎜⎜ − ⎟⎟ f ⎠ ⎝ f0 Vs VR 1 VR = RI = → = Vs ⎛ f ⎛ f f0 ⎞ f0 ⎞ 1 + jQs ⎜⎜ − ⎟⎟ 1 + jQs ⎜⎜ − ⎟⎟ f ⎠ f ⎠ ⎝ f0 ⎝ f0 ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Series Resonant Band-Pass Filter VR 1 = Vs 1 + jQs ( f f 0 − f 0 f )

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Series Resonant Band-Pass Filter

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Series Resonant Band-Pass Filter

f0 B = fH − fL = Qs For QS>>1 B f L ≅ f0 − 2

fH

B ≅ f0 + 2

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Example 6.6

f0 =

1 2π LC

=

1 2π (0.15926 H )(0.1592 x10 −6 F )

= 1000 Hz

2πf 0 L 2π (1000 Hz )(0.15926 H ) = = 10 100Ω R f 0 1000 Hz B= = = 100 Hz Qs 10

Qs =

fH

B ≈ f 0 + = 1000 Hz + 50 Hz = 1050 Hz 2

B f L ≈ f 0 − = 1000 Hz − 50 Hz = 950 Hz 2

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Example 6.6

At resonance : Z L = j 2πf 0 L = j 2π (1000 Hz )(0.1592 H ) = j1000Ω 1 1 ZC = − j =−j = − j1000Ω −6 2πf 0 C 2π (1000 Hz )(0.1592 x10 F ) Z s = R + Z L + Z C = 100Ω ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Example 6.6

VS 1∠0 o I= = = 0.01∠0 o Zs 100 VR = RI = (100)(0.01∠0 o ) = 1∠0 o o

VL = Z L I = ( j1000)(0.01∠0 ) = 10∠90

o

VC = Z C I = (− j1000)(0.01∠0 o ) = 10∠ − 90 o ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Example 6.6 VL = Qs Vs

VC = Qs Vs

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Parallel Resonance

1 Zp = (1 R ) + j 2πfC − j (1 2πfL ) At resonance ZP is purely resistive:

j 2πf 0 C = j (1 2πf 0 L ) → f 0 =

1 2π LC

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Parallel Resonance Quality factor QP ⎛ ⎞ Resistance ⎟⎟ QP ≡ ⎜⎜ ⎝ Reactance of inductance at resonance ⎠ R = 2πf 0 L Substitute L =

1 (2π ) ( f 0 ) C 2

2

from f 0 =

1 2π LC

QP = 2πf 0 CR ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Parallel Resonance ZP =

=

1 = (1 R ) + j 2πfC − j (1 2πfL )

R

⎛ R ⎞ ⎟⎟ 1 + j 2πfRC − j ⎜⎜ ⎝ 2πfL ⎠ R R = ⎞ ⎛ f ⎞ f 1 1 ⎟ 1 + jQP ⎜ − ⎟ − 2 2 ⎜ f ⎟ f 0 (2π ) f 0 fLC ⎠⎟ ⎝ 0 (2π ) f 0 fLC ⎠

⎛ 1 + j 2πf 0 RC ⎜ ⎜ ⎝ R = ⎛ f f0 ⎞ ⎜ 1 + jQP ⎜ − ⎟⎟ f ⎠ ⎝ f0

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Parallel Resonance

Vout

IR = IZ P = ⎛ f f0 ⎞ 1 + jQP ⎜⎜ − ⎟⎟ f ⎠ ⎝ f0

Vout for constant current, varying the frequency

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Parallel Resonance

Fi d the Find th phasor h currents t IR, IC andd IL for: f I = 10 −3 ∠0 o , R = 10kΩ, L = 159.2 μH , C = 159.2 pF R 10 4 f0 = = 1x10 Hz QP = = = 10 6 2πf 0 L 2π (1x10 Hz )(159.2μH ) 2π LC 1

6

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Parallel Resonance −3

o

Vout = IR = 10 ∠0 (10 ) = 10∠0 4

o

Vout Vout 10∠0 o −3 o = 4 = = 10 ∠ 0 IR = 4 R 10 10 Vout Vout 10∠0 o −2 o IL = = = = 10 ∠ − 90 ZL j 2πf 0 L j10 3 Vout Vout 10∠0 o −2 o IC = = = = 10 ∠ 90 −j ZC − j10 3 2πf 0 C ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

I C = QP I

I L = QP I

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Ideal Filters

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Noise Rejection j

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Second Order Low-Pass Filter

−j 2πfC

Vout

−j ZC 2πfRC = Vin = Vin = Vin j Z R + Z L + ZC ⎞ 2πf 0 L ⎛ f 1 R + j 2πfL − ⎜⎜ − ⎟⎟ 1+ j 2πfC R ⎝ f 0 2πff 0 LC ⎠

Vout Vin

−j − jQS ( f / f 0 ) 2πfRC = = H( f ) = ⎞ ⎛ f 2πf 0 L ⎛ f f 1 ⎜⎜ − ⎟⎟ 1 + jQS ⎜⎜ − 0 1+ j f R ⎝ f 0 2πff 0 LC ⎠ ⎝ f0

⎞ ⎟⎟ ⎠

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Second Order Low-Pass Second-Order Low Pass Filter − jQs ( f 0 f ) = 1 + jQs ( f f 0 − f 0 f )

t u o

=

n i

V H(f )= V

Qs ( f 0 f )∠ − 90 o

1 + QS2 ( f f 0 − f 0 f )2 ∠Tan T −1Qs ( f f 0 − f 0 f )

H(f ) =

Qs ( f 0 f )

2 1 + QS

(f

f0 − f0 f )

2

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Second Order Low-Pass Filter

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Second Order High-Pass g Filter At low frequency the capacitor i an open circuit is i i At high frequency the capacitor is a short and the inductor is open

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Second Order Band-Pass Filter At low frequency the capacitor i an open circuit is i i At high frequency the inductor is an open circuit

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Second Order Band-Reject j Filter At low frequency the capacitor i is i an open circuit At high frequency the inductor is an open circuit

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Example 6.8

Design a filter that passes frequency components higher than 1 kHz and rejects components lower than 1 kHz. Chose L L=50 50 mH f 0 = 1kHz =

1 2π LC

→C =

1 (2π )

2

f 02 L

=

1 −3

(2π ) (1x10 ) (50 x10 ) 2

3 2

= 0.507 μF

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Example 6.8

To avoid amplifying the signal at f0 choose Qs=1

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Example 6.8 2πf 0 L 2πf 0 L 2π (1kHz )(50 x10 −3 H ) Qs = →R= = = 314.1Ω R Q 1

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Exercise 6.20

Design a filter with QS=1 that passes frequency components lower than 5 kHz and rejects components higher than 5 kHz. Chose L L=55 mH f 0 = 5kHz =

1 2π LC

→C =

1 (2π )

2

f 02 L

=

1 −3

(2π ) (5 x10 ) (5 x10 ) 2

3 2

= 0.2026μF

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Exercise 6.20 2πf 0 L 2πf 0 L 2π (5kHz )(5 x10 −3 H ) Qs = →R= = = 157.1Ω R Q 1

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Exercise 6.21

Design a filter that passes frequency components between fL=45 kHz and fH=55 kHz. Chose L=1 mH f 0 = 50kHz =

1 2π LC

→C =

1 (2π )

2

f 02 L

=

1 −3

(2π ) (50 x10 ) (1x10 ) 2

3 2

= 10.13nF

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Exercise 6.21 B = f H − f L = 10kHz f 0 50kHz Q= = =5 B 10kHz 2π f 0 L R= Q 2π (50kHz )(1x10 −3 H ) = = 62.83Ω 5

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Active Filters Ideally, an active filter circuit should: 1.. Co Contain ta few ew components co po e ts 2.. Have ave a ttransfer a s e function u ct o that t at iss insensitive se s t ve to component tolerances

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

3. Place modest demands on the op p amp’s p ggain– bandwidth product, output impedance, slew rate, and other specifications 4. Be easily adjusted 5. Require a small spread of component values 6. Allow a wide range of useful transfer functions to be realized

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Active First-Order Low-Pass Filter Zf Vo H( f ) = =− Vi Zi 1 1 = + Zf Rf Zf =

j 2πfR f C f 1 1 = + 1 Rf Rf j 2πfC f Rf

1 + j 2πfR f C f

⎛ Rf ⎞ 1 ⎟ H( f ) = − = −⎜⎜ ⎟ 1 + j 2πfR C Zi R i f f ⎝ ⎠ ⎛ R f ⎞⎡ ⎤ 1 ⎟⎢ = −⎜⎜ ⎟ 1 + j( f / f ) ⎥ R i B ⎦ ⎝ ⎠⎣ 1 fB = 2πR f C f Zf

A low-pass filter with a dc gain of -Rf/Ri ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

1 vo (t ) = − RC

t

∫

vin (t )dt

0

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Active First-Order High-Pass g Filter Zf vo H( f ) = =− vi Zi Z i = Ri + H( f ) = −

Zf Zi

1 j 2π f Ci

Z f = Rf Rf

=− Ri +

j 2π f R f Ci

1 j 2π f Ci

⎛R = −⎜⎜ i 1 + j 2π f Ri Ci ⎝ Ri ⎛ R f ⎞⎡ j ( f / f B ) ⎤ ⎟⎢ = −⎜⎜ ⎥ ⎟ ⎝ Ri ⎠ ⎣1 + j ( f / f B ) ⎦ 1 fB = 2π Ri Ci =−

⎞ j 2π f R f Ci ⎟ ⎟ 1 + j 2π f R C i i ⎠

A high-pass filter with a high frequency gain of -Rf/Ri ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Differentiator Circuit

ddvin vo (t ) = − RC dt ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Higher g Order Active Filters

H ( f ) = H1 ( f ) H 2 ( f ) ∗ ∗ ∗ H n ( f ) Rf n⎛ = (−1) ⎜⎜ ⎝ Ri

⎞ ⎟ ⎟ ⎠

n

⎡ ⎤ 1 ⎢ ⎥ ⎣1 + j ( f / f B ) ⎦

n

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Butterworth Transfer Function Butterworth filters are characterized by having a particularly flat pass-band.

H(f ) =

H0

1 + ( f fB )

2n

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Sallen Key Circuits Sallen–Key

1 fB = 2πRC

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Sallen–Key y Circuits v− =

Rf R f + ( K − 1) R f

vo

vi = v − ( K − 1) R f vo = Av = 1 + =K vi Rf

N i Non-inverting ti amplifier lifi with ith a gain i off K ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

K Values for Low-Pass and High-Pass Butterworth Filters of Various Orders

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Low-Pass Active Filter Design g

Design a fourth fourth-order order low low-pass pass Butterworth filter having a frequency cut-off of 100 Hz ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Low-Pass Active Filter Design g Choose C = 0.1 0 1μF 1 1 R= = 2π Cf B 2π (0.1x10 −6 )(100 Hz ) = 15.92kΩ From the table a fourth order filter requires K values of 1.152 and 2.235. The DC gain is (1.152)(2.235) = 2.575

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Low-Pass Active Filter Design g

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Low-pass Active Filter Design g

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Low-pass Active Filter Design g

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Review for Next Lecture! • Magnetic fields (B flux density, density H magnetic field intensity, magnetic permeability μ) • Right hand rule • Forces on charges and current carrying wires moving i in i a magnetic i field fi ld • Faraday’s Law • Lenz’s Law • Ampere Ampere’ss Law ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Equal Component Sallen-Key LowPass Active Filter 1 fB = 2πRC

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Equal Component Sallen-Key LowPass Active Filter va

va − vin va − vo va − v− + + =0 R ZC R ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Equal Component Sallen-Key LowPass Active Filter va − vin va − vo va − v− + + =0 R ZC R Rf

1 v− = vo = vo R f + ( K − 1) R f K va − vin va − vo va − vo / K + + =0 R ZC R va − vin v a − vo / K + ( v a − v o ) j ωC + =0 R R ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

Equal Component Sallen-Key LowPass Active Filter va

v− − va va + =0 R ZC vo / K − v a + v a j ωC = 0 R ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.