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)