Analog Circuits and Systems Prof. K Radhakrishna Rao

Lecture 24: Active Filters

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Review

�  RC and RL low pass filters �  First order and second order filters �  Q of second order filters less than half �  RLC second order filters of any Q �  Low pass RLC Butterworth, Chebyschev and inverse Chebyschev

and Elliptic filter designs

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Active Filters

�  Limitations of passive RC filters can be addressed using active elements

�  Approaches in using active elements in designing filters �  Inductor simulation �  The problem of large size of inductor can be resolved using active

devices and RC elements to simulate the inductor in a traditional RLC filter.

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Active Filters (contd.,)

�  Q enhancement by feedback �  Q of a passive second order RC Filter can be enhanced using

feedback and amplification �  Biquad �  Simulate nth order differential equations using n-integrators and

summing amplifiers. A simulator of a second order differential equation is popularly known as Biquad.

�  The traditional approaches to filter design through Q-enhancement and inductor simulation are increasingly replaced by Biquad method because of commercial availability of universal active filter blocks (UAF 42 and UF 10).

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Active Filters: Inductor Simulation

�  All filters used in base-band applications, particularly in telephony, require large valued inductances resulting in large sizes.

�  These filters needed to be designed as active filters simulating large inductances using active devices.

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Miller’s Theorem

�  A voltage amplifier with gain G and an impedance, Z, connected between input and output terminals simulates an impedance at its input port

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i i ii in

i in

V GV I 1 ZI; ; ZZ V Z 1 G−

= = =−

Z1 G−

Simulation of Inductance in series with a Resistance

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Series resistance be Inductance L be

which represents a first order high-pass filter

1 1 2

1in 1 1 1 2

2

2

R ; CR RRZ R sL R sCR R1 GsCRG1 sCR

= = + = +−

=+

Modified L-simulator with only one buffer

�  When the buffer 2 is shorted �  It simulates the same inductance in series with R1+R2

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Bootstrapping Circuit Function

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Simulation of Inductance // Resistance

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1 1

2

2

R R11 G 1sCR

1GsCR

=− +

= −

The circuit simulating inductance in parallel with resistance

Simulation of Inductance // Resistance (contd.,)

�  If the first buffer is shorted the resultant circuit

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It simulates the same inductance in parallel with R1 and R2.

Simulation of Ideal Inductor (Gyrator)

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11 2

2

R sCR R1 G

1G 1sCR

=−

= −

Band Pass Filter

�  Design a second-order band-pass filter with center frequency = 5 kHz and a band-width of 1 kHz

�  For a C of 0.1 mF R = 1590 W

�  L for a resonance of 5 kHz = 9.87 mH

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RQ 5LC

= =

BP Filter with simulated inductance

�  L= CR1R2 = 9.87 mH Let R1 = R2

�  R1R2= 9.87 x 104 �  R1=R2= 314 W

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Frequency response of the BP filter

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Transient response of BP filter for Q = 5

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Transient response of BP filter for Q = 10 (R =3180 W)

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Increasing Q by Negative Resistance

�  Negative resistance is simulated across the simulated inductance

�  As the gain of the first amplifier is 2, and a resistance of RP is connected between its input and output, according to Miller’s theorem, negative resistance gets simulated in shunt with simulated inductance

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Increasing Q by Negative Resistance (contd.,)

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PFor R = 3140 the system becomes unstablePP

R R ;1 2

= − Ω−

Increasing the resonant frequency

�  Frequency is increased by decreasing the value of simulated inductance �  Simulated inductance can be decreased by reducing the values of R1, R2

and/or C with C =0.01 mF and Op Amp 741 with GB of 1 MHz

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The amplitude of oscillation is now limited by slew rate (1V/m sec) and not by saturation

Effect of Active Device Parameters

�  Simulated inductance is influenced by the parameters, DC Gain (A0) and Gain-Bandwidth Product (GB)

�  Gyrator circuit uses non-inverting amplifier of gain (=2) followed by an integrator

�  Ideal value for G

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0G 1sω⎛ ⎞= −⎜ ⎟⎝ ⎠

Effect of Active Device Parameters (contd.,)

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With finite gain A and DC gain of of Op Amps0

0

0 0

0

0 0 0

A

1s 3G 1 1

s A sA12 s1 1A A

3 31 1A sA s A sA

ω⎛ ⎞−⎜ ⎟ ω ω⎛ ⎞ ⎛ ⎞⎝ ⎠= − − −⎜ ⎟ ⎜ ⎟ω⎛ ⎞ ⎝ ⎠ ⎝ ⎠+⎜ ⎟⎛ ⎞+ +⎜ ⎟⎜ ⎟⎝ ⎠ ⎜ ⎟⎜ ⎟⎝ ⎠ω ω ω⎛ ⎞= − − − − −⎜ ⎟

⎝ ⎠

;

Effect of Active Device Parameters (contd.,)

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where

20 0 0 0

2

20 02

0 00 0

0

0 0

in 0 00 00

00 0

3G 1s sA sA s A

1 31 1 atDCs A AA

1 21 1A s A

R R 1 2Z 111 G A1 21 RA sRA s A

ω ω ω ω= − − + +

⎛ ⎞ω ω= − − + −⎜ ⎟ω ⎝ ⎠⎛ ⎞ ⎛ ⎞ω

= − − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞′= = = ω = ω −⎜ ⎟′ω− ⎛ ⎞ω ⎝ ⎠++ −⎜ ⎟

⎝ ⎠

Effect of Active Device Parameters (contd.,)

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Inductance is shunted by a

negative resistance

0

0

02

0 0 0 02

0 0

Q ARL

RA

3G 1s sA sA s A

2s1s GB GB

=

=′ω

ω ω ω ω= − − + +

ω ω− − +;

Effect of finite gain bandwidth

product is to slightly increase

the inductance and add a negative resistance

in shunt with the Inductance

00

0

RL1GB

RGB2

=ω⎛ ⎞ω −⎜ ⎟⎝ ⎠

ω

Example

�  Band-pass filter with simulated inductance �  For C = 0.1 mF, R = 1 kW

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=o

i

V 1V

Effect of finite GB �  Q =10 f0=1.59 kHz ;With GB = 1 MHz

the negative resistance = 314 kW; Gain changes to 1.033

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Effect of finite GB (contd.,)

�  BP filter with Q = 100 �  Q =100 f0=1.59 kHz; With GB = 1 MHz

the negative resistance = 314 kW; Gain changes to 1.47

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Effect of increased frequency

�  Q =100 f0=15.9 kHz �  The circuit oscillates. Negative resistance simulated is

31.4 kW < positive resistance of 100 kW used in the circuit

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Effect of increased frequency (contd.,)

�  Amplitude of oscillations gets limited by the slew rate of the Op Amp which is 1V/m sec

�  Filter designed with simulated inductor will require usage of an Op Amp with GB >>f0Q

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Sensitivity Sensitivity 0A

A0

Q 0 0GB GB

QQ2 Q1GB

2 QS ; SGB GB

ω

=ω⎛ ⎞−⎜ ⎟⎝ ⎠

ω ω= − = −

Q-enhancement

�  due to finite GB of the active device

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Sensitivity

Sensitivity

A

0

0A 0A

00

Q 0GB

0GB

QQ ;2 Q 11

GBGB2 QSGB

SGB

ω

ω= ω =

ωω⎛ ⎞ −−⎜ ⎟⎝ ⎠ω

= −

ω= −

Generalization of Gyrator

�  Positive Impedance Inverter

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1 3 5in

2 4

Z Z ZZZ Z

=

Generalization of Gyrator (contd.,)

�  When Z1=Z2=Z3=Z5=R and Z4 = 1/sC the resultant inductance simulator

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Generalization of Gyrator (contd.,)

�  When Z1=Z3=Z4=Z5=R and Z2 = 1/sC the resultant inductance simulator

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Conclusion

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