Passive filters
Use Passive components (R, L, C) Does not provide gain Bulky inductors for low frequencies (not suitable for integration) RC filters cannot realize Q > 0.5 Filters parameters are coupled (changing one component can change different filter parameters) Cannot realize ideal integrator
Integrated Circuits
Chip micrograph
Wi-Fi Receiver17mm2
Integrated Inductors
Used in GHz range (L in the range of nH) Low quality factor (need Q-enhancement) Value of L (also R & C) not well controlled
Operational Amplifier Model: Basic
Represented by:
A= open-circuit voltage gain
vid = (v+-v-) = differential input signal voltage
Rid = amplifier input resistance
Ro = amplifier output resistance
Signal developed at amplifier output is in phase with the voltage applied at + input (non-inverting) terminal and 1800 out of phase with that applied at - input (inverting) terminal.
Operational Amplifier Model: With Source and Load
RL = load resistanceRS = Thevenin equivalent resistance of signal source
vs = Thevenin equivalent voltage of signal source
LRoRLRA
idv*
ov
•Op amp circuits are mostly dc-coupled amplifiers. Signals vo and vs may have a dc component representing a dc shift of the input away from Q-point. •Op-amp amplifies both dc and ac components.
LRoRLR
SR
idRidR
vA svov
SR
idRidR
svidvand
Problem: Calculate voltage gain
Given Data: A=100, Rid =100k, Ro = 100, RS =10k, RL =1000Analysis:
Ideal amplifier’s output depends only on input voltage difference and not on source and load resistances. This can be achieved by using fully mismatched resistance condition (Rid >> RS or infinite Rid and Ro << RL or zero Ro ).
A = open-loop gain (maximum voltage gain available from the device)
dB3.386.820100100
1000k100k10
k100100
svov
LRoRLR
SR
idRidR
vA
idvov A AvA
idvov
Ideal Operational Amplifier
Ideal op amp is a special case of ideal differential amplifier with infinite gain, infinite Rid and zero Ro .
If A is infinite, vid is zero for any finite output voltage.
Infinite input resistance Rid forces input currents i+ and i- to be zero.
Ideal op amp has following assumptions: Infinite common-mode rejection, power supply rejection, open-loop bandwidth, output voltage range, output current capability and slew rate Zero output resistance, input-bias currents and offset current, input-offset voltage.
Aov
idv 0
idvlim
A
Inverting Amplifier: Configuration
Positive input is grounded.Feedback network, resistors R1 and R2 connected between inverting input and signal source and amplifier output node respectively.
Inverting Amplifier:Voltage Gain
Negative voltage gain implies 1800 phase shift between dc/sinusoidal input and output signals.Gain greater than 1 if R2 > R1
Gain less than 1 if R1 > R2
Inverting input of op amp is at ground potential (not connected directly to ground) and is said to
be at virtual ground.
0ov22i
1isv RRs
1
svsi R
But is=i2 and v-=0 (since vid=v+-v-=0)
and
1
2
svov
R
RvA
Non-inverting Amplifier: Configuration
• Input signal is applied to the non-inverting input terminal.• Portion of the output signal is fed back to the negative input
terminal.
• Analysis is done by relating voltage at v1 to input voltage vs and output voltage vo .
1
21121
svov121
svov
R
R
RRR
vA
RRR
Unity-gain Buffer
A special case of non-inverting amplifier, also called voltage follower with infinite R1 and zero R2. Hence Av =1.
Provides excellent impedance-level transformation while maintaining signal voltage level.Ideal voltage buffer does not require any input current and can drive any desired load resistance without loss of signal voltage.Unity-gain buffer is used in may sensor and data acquisition systems.
Alternative realization
Prone to parasitic capacitance Voltage swing on the input terminals
Differentiator
• Input resistor R1 in the inverting amplifier is replaced by capacitor C.
• Derivative operation emphasizes high-frequency components of input signal, hence is less often used than the integrator.
Rov
Ri dtsdvCsi
Since iR= is
dtsdvRCov
Output is scaled version of derivative of input voltage.
.11
1
)21
)(2
//1
(1.
2
21
invov)(
RsC
CCRRs
R
RRsvA
Passive realization
Non-inverting realization
.)21
)(2
//1
(111
1.21
2
invov)(
CCRRs
RsC
RR
RsvA
How to implement RHP zero?
.22
111
1
1
2
invov)(
RsC
RsC
R
RsvA
All-pass filter
Low-pass Frequency Response
For Q=0.707,magnitude response is maximally flat (Butterworth Filter: Maximum bandwidth without peaking)For Q>0.707, response shows undesired peaking.For Q<0.707: Filter’s bandwidth capability is wasted.
At <<o, filter has unity gain.At >>o response exhibits two-pole roll-off at -40dB/decade.At =o, gain of filter =Q.
High-pass Frequency Response
For Q=0.707,magnitude response is maximally flat (Butterworth Filter response). Amplifier gain is constant at >o, the lower cutoff frequency of the filter.
Band-pass Frequency Response
Response peaks approximately at o.
At <<o or >>o, filter response corresponds to single-pole high-pass or low-pass filter changing at a rate of 20dB/decade.
Single amplifier Biquad (SAB)
Enhanced Positive Feedback (EPF) Enhanced Negative Feedback (ENF)
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