Jaeger/Blalock

1
Chapter 10Analog Systems

• Microelectronic Circuit Design
• Richard C. Jaeger
• Travis N. Blalock

Chap10 - 1
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Chapter Goals

• Develop understanding of linear amplification
  concepts such as
• Voltage gain, current gain, and power gain,
• Gain conversion to decibel representation,
• Input and output resistances,
• Transfer functions and Bode plots,
• Cutoff frequencies and bandwidth,
• Low-pass, high-pass, band-pass, and band-reject
  amplifiers,
• Biasing for linear amplification,
• Distortion in amplifiers,
• Two-port representations of amplifiers,
• g-, h-, y-, and z-parameters,
• Use of transfer function analysis in SPICE.

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Example of Analog Electronic System FM Stereo
Receiver

• Linear functions Radio and audio frequency
  amplification, frequency selection (tuning),
  impedance matching(75-W input, tailoring audio
  frequency response, local oscillator
• Nonlinear functions DC power supply(rectification
  ), frequency conversion (mixing),
  detection/demodulation

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Amplification Introduction
A complex periodic signal can be represented as
the sum of many individual sine waves. We
consider only one component with amplitude VS 1
mV and frequency wS with 0 phase (signal is used
as reference) Amplifier output is sinusoidal
with same frequency but different amplitude VO
and phase ?
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Amplification Introduction (contd.)

• Amplifier output power is
• Here, PO 100 W and RL8 W
• Output power also requires output current which
  is
• Input current is given by
• phase is zero because circuit is purely
  resistive.

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Amplification Gain

• Voltage Gain
• Magnitude and phase of voltage gain are
  given by
• 
  and
• For our example,
• Current Gain
• Magnitude of current gain is given by

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Amplification Gain (contd.)

• Power Gain
• For our example,
• On decibel scale, i.e. in dB

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Amplifier Biasing for Linear Operation
VI dc value of vI, vi time-varying
component For linear amplification- vI must be
biased in desired region of output characteristic
by VI.
If slope of output characteristic is positive,
input and output are in phase (amplifier is
non-inverting). If slope of output characteristic
is negative, input and output signals are 1800
out of phase (amplifier is inverting).
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Amplifier Biasing for Linear Operation (contd.)
Voltage gain depends on bias point.
Eg if amplifier is biased at VI 0.5 V, voltage
gain will be 40 for input signals satisfying
If input exceeds this value, output
is distorted due to change in amplifier slope.
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Amplifier Biasing for Linear Operation (contd.)
Output signals for 1 kHZ sinusoidal input signal
of amplitude 50 mV biased at VI 0.3 V and 0.5V
For VI 0.3V
Gain is 20, output varies about dc level of 4 V.
For VI 0.5V
Gain is 40, output varies about dc level of 10 V.
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Distortion in Amplifiers

• Different gains for positive and negative values
  of input cause distortion in output.
• Total Harmonic Distortion (THD) is a measure of
  signal distortion that compares undesired
  harmonic content of a signal to the desired
  component.

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Total Harmonic Distortion
dc
desired output
2nd harmonic distortion
3rd harmonic distortion
Numerator sum of rms amplitudes of distortion
terms, Denominator desired component
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Two-port Models for Amplifiers

• Simplifies amplifier-behavior modeling in complex
  systems.
• Two-port models are linear network models, valid
  only under small-signal conditions.
• Represented by g-, h-, y- and z-parameters.
• (v1, i1) and (v2, i2) represent signal
  components of voltages and currents at the
  network ports.

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g-parameters
Using open-circuit (i0) and short-circuit (v0)
termination conditions,
Open-circuit input conductance
Reverse short-circuit current gain
Forward open-circuit voltage gain
Short-circuit output resistance
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g-parametersExample
ProblemFind g-parameters. Approach Apply
specified boundary conditions for each
g-parameter, use circuit analysis. For g11 and
g21 apply voltage v1 to input port and open
circuit output port. For g12 and g22 apply
current i2 to output port and short circuit input
port.
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Hybrid or h-parameters
Using open-circuit (i0) and short-circuit (v0)
termination conditions,
Short-circuit input resistance
Reverse open-circuit voltage gain
Forward short-circuit current gain
Open-circuit output conductance
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h-parametersExample
ProblemFind h-parameters for the same network
(used in g-parameters example). Approach Apply
specified boundary conditions for each
h-parameter, use circuit analysis. For h11 and
h21 apply current i1 to input port and short
circuit output port. For h12 and h22 apply
voltage v2 to output port and open circuit input
port.
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Admittance or y-parameters
Using open-circuit (i0) and short-circuit (v0)
termination conditions,
Short-circuit input conductance
Reverse short-circuit transconductance
Forward short-circuit transconductance
Short-circuit output conductance
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y-parametersExample
ProblemFind y-parameters for the same network
(used in g-parameters example). Approach Apply
specified boundary conditions for each
y-parameter, use circuit analysis. For y11 and
y21 apply voltage v1 to input port and short
circuit output port. For y12 and y22 apply
voltage v2 to output port and short circuit input
port.
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Impedance or z-parameters
Using open-circuit (i0) and short-circuit (v0)
termination conditions,
Open-circuit input resistance
Reverse open-circuit transresistance
Forward open-circuit transresistance
Open-circuit output resistance
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z-parametersExample
ProblemFind z-parameters for the same network
(used in g-parameters example). Approach Apply
specified boundary conditions for each
z-parameter, use circuit analysis. For z11 and
z21 apply current i1 to input port and open
circuit output port. For z12 and z22 apply
current i2 to output port and open circuit input
port.
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Mismatched Source and Load Resistances Voltage
Amplifier
g-parameter representation (g120) with Thevenin
equivalent of input source
If Rin gtgt Rs and Routltlt RL,
In an ideal voltage amplifier, and Rout 0
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Mismatched Source and Load Resistances Current
Amplifier
h-parameter representation (h120) with Norton
equivalent of input source
If Rs gtgt Rin and Routgtgt RL,
In an ideal current amplifier, and Rin0
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Amplifier Transfer Functions
Av(s)Frequency-dependent voltage gain Vo(s) and
Vs(s) Laplace Transforms of input and output
voltages of amplifier,
(In factorized form)
(-z1, -z2,-zm)zeros (frequencies for which
transfer function is zero) (-p1, -p2,-pm)poles
(frequencies for which transfer function is
infinite)
(In polar form)
Bode plots display magnitude of the transfer
function in dB and the phase in degrees (or
radians) on a logarithmic frequency scale..
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Low-pass Amplifier Description

• Amplifies signals over a range of frequencies
  including dc.
• Most operational amplifiers are designed as low
  pass amplifiers.
• Simplest (single-pole) low-pass amplifier is
  described by
• Ao low-frequency gain or mid-band gain
• wH upper cutoff frequency or upper half-power
  point of amplifier.

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Low-pass Amplifier Magnitude Response

• For wltltwH
• For wgtgtwH
• 
  
• For wwH
• Gain is unity (0 dB) at wAowH , called
  gain-bandwidth product
• Bandwidth (frequency range with constant
  amplification ) wH (rad/s)

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Low-pass Amplifier Phase Response

• If Ao positive phase angle 00
• If Ao negative phase angle 1800
• At wC phase 450
• One decade below wC phase 5.70
• 
  One decade above wC phase 84.30
• Two decades below wC phase 00
• Two decades above wC phase 900

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RC Low-pass Filter

• Problem Find voltage transfer
• function
• Approach Impedance of the where
• capacitor is 1/sC, use voltage
• division

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High-pass Amplifier Description

• True high-pass characteristic impossible to
  obtain as it requires infinite bandwidth.
• Combines a single pole with a zero at origin.
• Simplest high-pass amplifier is described by
• wH lower cutoff frequency or lower half-power
  point of amplifier.

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High-pass Amplifier Magnitude and Phase Response

• For wgtgtwL
• For wltltwL
• 
  For wwL
• Bandwidth (frequency range with constant
  amplification ) is infinite
• Phase response is given by

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RC High-pass Filter
Problem Find voltage transfer function A
pproach Impedance of the
where capacitor is 1/sC, use voltage
division

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Band-pass Amplifier Description

• Band-pass characteristic obtained by combining
  highpass and low-pass characteristics.
• Transfer function of a band-pass amplifier is
  given by
• Ac-coupled amplifier has a band-pass
  characteristic
• Capacitors added to circuit cause low frequency
  roll-off
• Inherent frequency limitations of solid-state
  devices cause high-frequency roll-off.

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Band-pass Amplifier Magnitude and Phase Response

• The frequency response shows a wide band of
  operation.
• Mid-band range of frequencies given by
  , where

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Band-pass Amplifier Magnitude and Phase Response
(contd.)

• At both wH and wL, assuming wLltltwH,
• Bandwidth wH - wL.
• The phase response is given by

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Narrow-band or High-Q Band-pass Amplifiers

• Gain maximum at center frequency wo and decreases
  rapidly by 3 dB at wH and wL.
• Bandwidth defined as wH - wL, is a small fraction
  of wo with width determined by
• For high Q, poles will be complex and
• Phase response is given by

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Band-Rejection Amplifier or Notch Filter

• Gain maximum at frequencies far from wo and
  exhibits a sharp null at wo.
• To achieve sharp null, transfer function has a
  pair of zeros on jw axis at notch frequency wo ,
  and poles are complex.
• Phase response is given by

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All-pass Function

• Uniform magnitude response at all frequencies.
• Can be used to tailor phase characteristics of a
  signal
• Transfer function is given by
• For positive Ao,

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Complex Transfer Functions
Amplifier has 2 frequency ranges with constant
gain. Midband region is always defined as region
of highest gain and cutoff frequencies are
defined in terms of midband gain.
Since wH w4 and wL w3,
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Bandwidth Shrinkage

• If critical frequencies arent widely spaced, the
  poles and zeros interact and cutoff frequency
  determination becomes complicated.
• Example for which ,
  Av(0) Ao
• Upper cutoff frequency is defined by
  or

Solving for wH yields wH 0.644w1.The cutoff
frequency of two-pole function is only 64 that
of a single-pole function. This is known as
bandwidth shrinkage.
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