CHAPTER 13 OUTPUT STAGES AND POWER AMPLIFIERS

CHAPTER 13 FILTERS AND TUNED AMPLIFIERS Chapter Outline 13.1 Filter Transmission, Types and Specification 13.2 The Filter Transfer Function 13.3 Butterworth and Chebyshev Filters 13.4 First‐Order and Second‐Order Filter Functions 13.5 The Second‐Order LCR Resonator 13.6 Second‐Order Active Filters Based on Inductor Replacement 13.7 Second‐Order Active Filters Based on the Two‐Integrator‐Loop Topology 13.8 Single‐Amplifier Biquadratic Active Filters 13.9 Sensitivity 13.10 Transconductance‐C Filters 13.11 Switched‐Capacitor Filters 13.12 Tuned Amplifiers

NTUEE Electronics – L. H. Lu

13‐1

13.1 Filter Transmission, Types and Specifications Filter Transfer Function  A filter is a linear two‐port network represented by the ratio of the output to input voltage  Transfer function T(s)  Vo(s) / Vi(s)  Transmission : evaluating T(s) for physical frequency s = j → T(j ) = |T(j )|e j()  Gain: 20 log|T(j)| (dB)  Attenuation: ‐ 20 log|T(j)| (dB) Output frequency spectrum : |Vo(s)| = |T(s)| |Vi(s)|

Types of Filters


13‐2

Filter Specification    

Passband edge : p Maximum allowed variation in passband transmission : Amax Stopband edge : s Minimum required stopband attenuation : Amin Low‐Pass Filter

Band‐Pass Filter

 The first step of filter design is to determine the filter specifications  Then find a transfer function T(s) whose magnitude meets the specifications  The process of obtaining a transfer function that meets given specifications is called filter approximation NTUEE Electronics – L. H. Lu

13‐3

13.2 Filter Transfer Function Transfer Function  The filter transfer function is written as the ratio of two polynomials: T (s) 

   

The degree of the denominator → filter order To ensure the stability of the filter → N  M The coefficients ai and bj are real numbers The transfer function can be factored and expressed as: T (s) 

   

aM s M  aM 1s M 1  ...  a0 s N  bN 1s N 1  ...  b0

aM ( s  z1 )( s  z2 )...(s  z M ) ( s  p1 )(s  p2 )...(s  p N )

Zeros: z1 , z2 , … , zM and ( NM ) zeros at infinity Poles: p1 , p2 , … , pN Zeros and poles can be either a real or a complex number Complex zeros and poles must occur in conjugate pairs


13‐4

Transfer Function Examples  Low‐Pass Filter (with Stopband Ripple) T (s) 

2 l1 3

5

2

zeros poles

0

a4 ( s   )(s   ) s  b4 s 4  b3 s  b2 s 2  b1s  b0 2

| T | (dB)

2 l2

jw wl2 wl1 wp



s

 Low‐Pass Filter (without Stopband Ripple)

wp

w

ws wl1

| T | (dB)

wl2

zeros poles

0

aM T (s)  5 4 s  b4 s  b3 s 3  b2 s 2  b1s  b0

wp wl1 wl2

jw 

wp

5 s

wp wp

 Band‐Pass Filter T (s) 

6

5

ws

| T | (dB)

a5 s ( s   )(s   ) s  b5 s  b4 s 4  b3 s 3  b2 s  b1s  b0 2

w

2 l1

2

2 l2 2

zeros poles

0

jw wl2 wp2



wp1 wl1

w

wp1 wp2 wl1

ws1

ws2


wl2

wl1 wp1 wp2

s

wl2

13‐5

13.3 Butterworth and Chebyshev Filters The Butterworth Filter    

Butterworth filters exhibit monotonically decreasing transmission with all zeros at  =  Maximally flat response → degree of passband flatness increases as the order N is increased Higher order filter has a sharp cutoff in the transition band The magnitude function of the Butterworth filter is: | T ( j ) |

1 1   ( /  p ) 2 N 2

Required transfer functions can be defined based on filter specifications (Amax , Amin , p , s)


13‐6

Natural Modes of the Butterworth Filter  The natural modes lines on a circle.  The radius of the circle is 0   p (1 /  )1/ N  Equal angle space  / N


13‐7

Design Procedure of the Butterworth Filters Design Specifications

Amax Amin p s

Design Procedure 1.

Determine e (from Amax) | T ( j p ) |

2.

1

1 

2

 Amax [dB]  20 log 1   2    10 Amax /10  1

Determine the required filter order N (from p , s , Amin)









Attenuation A(s )[dB]  20 log 1 / 1   2 (s /  p ) 2 N  10 log 1   2 (s /  p ) 2 N  Amin

3.

Determine the N natural modes p1 , p2 , .... p N

4.

Determine T (s) K0N T ( s)  ( s  p1 )( s  p2 )...(s  p N )

where 0   p (1 /  )1/ N


13‐8

The Chebyshev Filter     

An equiripple response in the passband A monotonically decreasing transmission in the stopband Odd‐order filter → | T(0) | = 1 Even‐order filter → maximum magnitude deviation at  = 0 The transfer function of the Chebyshev filter is : | T ( j ) |

1 1   2 cos 2 [ N cos 1 ( /  p )]

for    p

| T ( j ) |

1 1   2 cosh 2 [ N cosh 1 ( /  p )]


for    p

13‐9

Design Procedure of the Chebyshev Filters Design Specifications

Amax Amin p s

Design Procedure 1.

Determine e (from Amax) | T ( j p ) |

2.

1

1 

2

 Amax (dB)  20 log 1   2    10 Amax /10  1

Determine the required filter order N (from p , s , Amin)





Attenuation A(s )  10 log 1   2 cosh 2 ( N cosh 1 (s /  p ))  Amin

3.

Determine the N natural modes

4.

Determine T (s)

1 1 1  2k  1   1  2k  1   pk   p sin   cosh sinh 1    sinh sinh 1   j p cos   2 2 N  N N  N

T ( s) 

K pN

 2 N 1 ( s  p1 )( s  p2 )...(s  p N )


13‐10

13.4 First‐Order and Second‐Order Filter Functions Cascade Filter Design  First‐order and second‐order filters can be cascaded to realize high‐order filters  Cascade design is one of the most popular methods for the design of active filters  Cascading does not change the transfer functions of individual blocks if the output resistance is low

First‐Order Filters  Bilinear transfer function

T (s) 

a1s  a0 s  b0


13‐11

First‐Order Filters (Cont’d)


13‐12

Second‐Order Filters  Biquadratic transfer function

a2 s 2  a1s  a0 T ( s)  2 s  (0 / Q) s  02

 Pole frequency: 0  Pole quality factor: Q  Poles: p1 , p2  

0 2Q

 j 0 1 

1 4Q 2

 Bandwidth: BW  2  1 

0 Q


13‐13

Second‐Order Filters (Cont’d)


13‐14

Second‐Order Filters (Cont’d)


13‐15

13.5 The Second‐Order LCR Resonator The Resonator Natural Modes Parallel Resonator

Current Excitation

Voltage Excitation

Current Excitation

Vo 1 1 s/C    2 I i Y 1 / sL  sC  1 / R s  (1 / RC ) s  1 / LC

0  1 / LC

Voltage Excitation

Vo ( R || 1 / sC ) 1 / LC   2 Vi ( R || 1 / sC )  sL s  (1 / RC ) s  1 / LC

Q  0 RC

 The LCR resonator can be excited by either current or voltage source  The excitation should be applied without change the natural structure of the circuit  The natural modes of the circuits are identical (will not be changed by the excitation methods)  The similar characteristics also applies to series LCR resonator


13‐16

Realization of Transmission Zeros  Values of s at which Z2(s) = 0 and Z1(s)  0 → Z2 behaves as a short  Values of s at which Z1(s) =  and Z2(s)   → Z1 behaves as an open

Realization of Filter Functions Low‐Pass Filter T (s) 

Vo 1 / LC  2 Vi s  (1 / RC ) s  1 / LC

High‐Pass Filter T (s) 

Vo s2  2 Vi s  (1 / RC ) s  1 / LC


Bandpass Filter T (s) 

Vo (1 / RC ) s  2 Vi s  (1 / RC ) s  1 / LC

13‐17

Notch Filter

Low‐Pass Notch Filter

High‐Pass Notch Filter


13‐18

13.6 Second‐Order Active Filters (Inductor Replacement) Second‐Order Active Filters by Op Amp‐RC Circuits  Inductors are not suitable for IC implementation  Use op amp‐RC circuits to replace the inductors  Second‐order filter functions based on RLC resonator

The Antoniou Inductance‐Simulation Circuit  Inductors are realized by op amp‐RC circuits with negative feedbacks  The equivalent inductance is given by

Z in 

V1  sC4 R1 R3 R5 / R2  sLeq I1

Leq  C4 R1 R3 R5 / R2


13‐19

The Op Amp‐RC Resonator  The inductor is replaced by the Antoniou circuit  The pole frequency and the quality factor are given by 0  1 / C4C6 R1 R3 R5 / R2

Q  0C6 R6  R6

C6 R2 C4 R1 R3 R5

 A simplified case where R1 = R2 = R3 = R5 = R and C4 = C6 = C 0  1 / RC

Q  R6 / R


13‐20

Filter Realization High‐Pass Filter

Low‐Pass Filter

Notch Filter

Bandpass Filter


13‐21

HPN Filter

LPN Filter

All‐Pass Filter


13‐22

13.7 Second‐Order Active Filters (Two‐Integrator‐Loop) Derivation of the Two‐Integrator‐Loop Biquad Ks 2  2 Vi s  (0 / Q) s  02

Vhp

Vhp 

2 1 0 ( Vhp )  ( 20 Vhp )  KVi Q s s

 High‐pass implementation: Vhp  KVi 

1 0 2 Vhp  20 Vhp Q s s

 Bandpass implementation: (0 / s)Vhp Vi



 K0 s  Tbp ( s) s 2  (0 / Q) s  02

 Low‐pass implementation: (02 / s 2 )Vhp Vi



K02  Tlp ( s) s 2  (0 / Q) s  02


13‐23

Circuit Implementation (I) Vhp 

Rf Rf R f 02 R3 R2  (1  )Vi  (1  )( 0 Vhp )  ( Vhp ) R2  R3 R1 R2  R3 R1 s R1 s 2

 R f / R1  1

R3 / R2  2Q  1

K  2 1/ Q

 High‐pass transfer function: Thp ( s) 

Vhp Vi



s 2 [2 R3 /( R2  R3 )] s 2  s[2 R2 /( R2  R3 )]0  02

 Bandpass transfer function: Tbp ( s) 

Vbp Vi



s[2 R3 /( R2  R3 )]0 s  s[2 R2 /( R2  R3 )]0  02 2

 Low‐pass transfer function: Tbp ( s) 

Vbp Vi



[2 R3 /( R2  R3 )]02 s 2  s[2 R2 /( R2  R3 )]0  02

 Notch and all‐pass transfer function: T (s) 

Vo 2 R3 ( RF / RH ) s 2  ( RF / RB )0 s  ( RF / RL )02  Vi R2  R3 s 2  s[2 R2 /( R2  R3 )]0  02


13‐24

Circuit Implementation (II) – Tow‐Thomas Biquad

    

Use an additional inverter to make all the coefficients of the summer the same sign All op amps are in single‐ended mode The high‐pass function is no longer available It is known as the Tow‐Thomas biquad An economical feedforward scheme can be employed with the Tow‐Thomas circuit

T (s) 

Vo  Vi

s2

C1 1 1 r  1   2  s   C C  R1 RR3  C RR2 1 1 s2  s  2 2 QCR C R


13‐25

13.8 Single‐Amplifier Biquadratic Active Filters Characteristics of the SAB Circuits    

Only one op amp is required to implement biquad circuit Exhibit a greater dependence on the limited gain and bandwidth of the op amp More sensitive to the unavoidable tolerances in the values of resistors and capacitors Limited to less stringent filter specifications with pole Q factors less than 10

Synthesis of the SAB Circuits  Use feedback to move the poles of an RC circuit from the negative real axis to the complex conjugate locations to provide selective filter response  Steps of SAB synthesis:  Synthesis of a feedback loop that realizes a pair of complex conjugate poles characterized by 0 and Q  Injecting the input signal in a way that realizes the desired transmission zeros  Natural modes of the filter: t ( s )  N ( s ) / D( s) L( s)  At ( s)  AN ( s ) / D( s)

The closed‐loop characteristics equation: 1  L( s)  0  t ( sP )  1 / A  0

 The poles of the closed‐loop system are identical to the zeros of the RC network NTUEE Electronics – L. H. Lu

13‐26

RC Networks with complex transmission zeros

1 1  1 1 s 2  s     C1 C2  R3 C1C2 R3 R4 t (s)   1 1 1  1   s 2  s    C1 R3 C2 R3 C1 R4  C1C2 R3 R4

1 1  1 1  s 2  s    R1 R2  C4 R1 R2C3C4 t (s)   1 1 1  1     s 2  s  R2C4 R2C3 R1C4  R1 R2C3C4

Characteristics Equation of the Filter 1 1  1 1  02  s 2  s    Q  C1 C2  R3 C1C2 R3 R4 1  0  C1C2 R3 R4 s2  s

0

 CC R R Q   1 2 3 4 R3 

 1 1      C1 C2 

1

2 Let C1 = C2 = C , R3 = R , R4= R/m  m  4Q

 CR  2Q / 0


13‐27

Injection the Input Signal  The method of injection the input signal into the feedback loop through the grounded nodes  A component with a ground node can be disconnected from the ground and connected to the input source  The filter transmission zeros depends on the components through which the input signal is injected

Vo  Vi

 s( / C1 R4 )  1 1  1 1 s 2  s     C1 C2  R3 C1C2 R3 R4


13‐28

Generation of Equivalent Feedback Loops

Va t Vb

Va  1 t Vc

Equivalent Loop

Characteristics Equation: 1  L( s)  0  1  At ( s)  0

Characteristics Equation: 1


A (1  t )  0  1  At ( s)  0 A 1

13‐29

Generation of Equivalent Feedback Loops (Cont’d)


13‐30

13.9 Sensitivity Filter Sensitivity  Deviation in filter response due to the tolerances in component values  Especially for RC component values and amplifier gain

Classical Sensitivity Function  Definition:

y / y x / x y x S xy  x y S xy  lim

x 0

 For small changes: S xy 

y / y x / x


13‐31

13.10 Transconductance‐C Filters Limitations of Op Amp‐RC Circuits  Suitable for audio‐frequency filters using discrete op amps, resistors and capacitors  Low‐frequency applications limited by the relatively low bandwidth of general‐purpose op amps  Impractical for IC implementations due to:  The need for large capacitors and resistors increases the IC cost  The need for very precise values of RF time constant requires expensive trimming/tuning  The need for op amps that can drive resistive and large capacitive loads

Methods for IC Filter Implementations  Transconductance‐C filters:  Utilize transconductance amplifiers or transconductors together with capacitors for filters  High‐quality and high‐frequency transconductors can be easily realized in CMOS technology  Has been widely used for medium/high‐frequency applications (up to hundreds of MHz)  MOSFET‐C filters:  Replace resistors with MOSFETs in linear region  Techniques have been evolved to obtain linear operation with large input signals  Switched‐capacitor filters:  Replace the required resistor by switching a capacitor at a relatively high frequency  The resulting filters are discrete‐time circuits as opposed to the continuous‐time ones  Is ideally suited for implementation low‐frequency filters in IC form using CMOS technology


13‐32

Transconductors  An ideal transconductor has infinite input resistance and infinite output resistance  The output can be positive or negative depending the current direction  Transconductor can be single‐ended or fully differential


13‐33

Basic Building Blocks  Negative transconductor used to realize a resistance  Transconductor loaded with a capacitor as an integrator

First‐Order Gm‐C Low‐Pass Filter

Vo Gm1 G /G    m1 m 2 Vi sC  Gm 2 1  sC / Gm 2


13‐34

Second‐Order Gm‐C Low‐Pass Filter

V2 

Gm 2 V1 sC2

sGm 4 / C1 V1  2 Vi s  sGm3 / C1  Gm1Gm 2 / C1C2

BP center - freq gain 

Gm 2Gm 4 / C1C2 V2  2 Vi s  sGm3 / C1  Gm1Gm 2 / C1C2

LP dc gain  

Gm 4 Gm3

Gm 4 Gm1


0  Q

Gm1Gm 2 C1C2 Gm1Gm 2 Gm 3

C1 C2

13‐35

Simplified Circuit  Gm1 = Gm2 = Gm  C1 = C2 = C Gm C G Q m Gm3

0 

Fully Differential Circuit


13‐36

13.11 Switched‐Capacitor Filters Basic Principle  A capacitor switched between two nodes at a sufficiently high rate is equivalent to a resistor  The resistor in the active‐RC integrator can be replaced by the capacitor and the switches  Equivalent resistor: iav 

v T C1vi  Req  i  c iav C1 Tc

 Equivalent time constant for the integrator = Tc(C2/C1)


13‐37

Practical Circuits  Can realize both inverting and non‐inverting integrator  Insensitive to stray capacitances  Noninverting switched‐capacitor (SC) integrator

 Inverting switched‐capacitor (SC) integrator


13‐38

Filter Implementation  Circuit parameters for the two integrators with the same time constant Tc

C2 C  Tc 1  C1  C2  C  C3  C4  KC C4 C3

0 

1 1  C1C2 R3 R4 TC

C3 C4 K  C2 C1 Tc

Q

C R5 C4   C5  0Tc R4 C5 Q


13‐39

13.12 Tuned Amplifiers Characteristics of a Tuned Amplifier  Center frequency: 0  3‐dB bandwidth: B  Skirt selectivity: S (the ratio of the 30‐dB bandwidth to the 3‐dB bandwidth)  The 3‐dB bandwidth is less than 5% of 0 in many applications  narrow‐band characteristics  The circuits studied are small‐signal voltage amplifiers in which transistors operate in class A mode

The Basic Principle for Single‐Tuned Amplifiers  Parallel RLC circuit is used as the load for an amplifier  The amplifier exhibits a second‐order bandpass characteristic.  g mVi  g mVi  YL sC  1 / R  1 / sL Vo gm s  2 Vi C s  s(1 / RC )  1 / LC

Vo 

0  1 / LC B  1 / RC Q  0 / B  0 RC Vo ( j0 )   gm R Vi ( j0 )


13‐40

Inductor Losses Q

L rs

 Q0 

Y ( j0 ) 

0 L rs

1 1 1 1 1  j (1 / Q0 )   rs  j0 L jL 1  j (1 / Q0 ) jL 1  (1 / Q0 ) 2

1  1  1 1 1  j    For Q0  1  Y ( j0 )  j0 L  Q0  j0 L 0 LQ0

Q0  0 L / rs  1 RP  0 L  / rs 2

RP  0 LQ0  0 L  / rs 2

       

Power loss in a practical inductor is represented by a series resistance rs Rather than specifying the value of rs, a inductor is specified by the Q factor at the frequency at interest It is desirable to replace the series connection of L and rs with an equivalent parallel connection of L and RP RP is expressed in terms of rs, L and the frequency of interest 0 under the assumption that Q0 >>1 In fact, the value of resulting RP varies with frequency, however, it is approximated as a constant resistance at frequencies around 0 to simplify the analysis High inductor Q factor means low inductor loss rs as small as possible for a given L RP as large as possible for a given L NTUEE Electronics – L. H. Lu

13‐41

The Use of Transformers  In many cases, the required values of inductance and capacitance are not practical  use a transformer to effect an impedance change  use a tapped coil (autotransformer) n2

n=n2 /n1

C’=C /n2 R

L

C

n1

R

L


L’=n2L

13‐42

Amplifiers with Multiple Tuned Circuits  



 

Multiple tuned circuits are used if high selectivity is required  tuned circuit at both input and output Radio‐frequency choke (RFC) is frequently used for bias circuit  RFC behaves as a short‐circuit at dc  provide dc biasing  RFC exhibit high impedance at frequency of interest  eliminate the loading effect of dc bias circuit The Miller capacitance Cm may results in many problems in the multiple tuned circuit  The Miller impedance at the input is complex since the load is not simply resistive  The reflected impedance will cause detuning of the input circuit  Skewing of the response of the input circuit  May results in circuit oscillation Using additional feedback circuits to provide current cancellation can neutralize the effect of Cm An alternative approach is to change the circuit configuration to avoid Miller effect


13‐43

The Cascode and CC‐CB Cascade 



Two amplifier configurations do not suffer the Miller Effect and are suitable for multiple tuned circuits:  Cascode configuration  CC‐CB cascade configuration The CC‐CB cascade is usually preferred in IC implementations because of its differential structure


13‐44

Synchronous Tuning  Assume the stages do not interact  the overall response is the product of the individual responses  Synchronous tuning  cascading N identical resonant circuits  The 3‐dB bandwidth B of the overall amplifier is related to that of the individual tuned circuit: B

0 Q

21/ N  1

 Bandwidth‐shrinkage factor: 21/ N  1 1/ N  Bandwidth decreased by the factor of 2  1  Q factor increased by the factor of 21/ N  1  Given B and N, we can determine the bandwidth required of the individual stages


13‐45

Stagger‐Tuning  Stagger‐tuned amplifiers exhibit overall response with maximal flatness around the center frequency f0  Such a response can be obtained by transforming the response of a Butterworth low‐pass filter to 0  The transfer function of a second‐order bandpass filter can be expressed in terms of its poles as: T (s) 

a1s  1 1     s  0  j0 1   s  0  j0 1  2  2Q 4Q  2Q 4Q 2 

jw

   

+ jw0

 For Q >> 1, and for values of s in the neighborhood of +j0 the transfer function is approximated as (narrow‐band approximation): T (s) 

a1 a1 / 2 a1 / 2   s  0 / 2Q  j0 2 j0  s  0 / 2Q  j0 s  j0   0 / 2Q

s

(s  p2) ≈ 2jw0

 jw0

 The response of the 2nd‐order bandpass filter in the neighborhood of its center frequency s = j0 is identical to a first‐order low‐pass filter with a pole at (0/2Q) in the neighborhood of p = 0  The transformation p = s  j0 can be applied to low‐pass filters of order greater than one


13‐46

 The second‐order narrow‐band bandpass filter:

s = p + jw0

s = p + jw0

 The fourth‐order stagger‐tuned narrow‐band bandpass amplifier:

s = p + jw0

s = p + jw0


13‐47

CHAPTER 13 OUTPUT STAGES AND POWER AMPLIFIERS

CHAPTER 13 OUTPUT STAGES AND POWER AMPLIFIERS

Suggest Documents

High-Power Integrated Stimulator Output Stages With

Chapter 13 Energy and Power Study Guide

CHAPTER 2 OPERATIONAL AMPLIFIERS

RF Power Amplifiers - QSL.net

CMOS output stages for low-voltage power supplies - IEEE Xplore

Power Amplifiers - McGraw-Hill

CHAPTER 7 DIFFERENTIAL AND MULTISTAGE AMPLIFIERS

Feedforward Amplifiers Incorporate Parallel Output ...

Radio Frequency Power Amplifiers - CiteSeerX

TUNED POWER AMPLIFIERS - Radau5.ch

Chapter 13

Chapter 13

Chapter 13

Chapter 13

Chapter 13

Chapter 13

Chapter 13

Chapter 13

1 3 Tuned Power Amplifiers

1 3 Tuned Power Amplifiers

CHAPTER 13

Chapter 13

Chapter 13

CHAPTER 13