# Tutorial 11 - Active Filters

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```					The Chinese University of Hong Kong
Department of Electronic Engineering

Tutorial 11 – Active Filters
Revision: Non-inverting amplifier and Inverting amplifier
Assume the open loop gain of the opamps used below is A:
Non-Inverting Amplifier                               Inverting Amplifier

R2

R1

Vout
Vin

Vout = A(V+ − V− )                      Vout = A(V+ − V− )
Vout = A(Vin − β Vout )                 Vout = A(0 − (Vin + β (Vout − Vin )))
Vout (1 + Aβ ) = AVin                   Vout (1 + Aβ ) = A(β − 1)Vin
1                                           1
A⋅                                            A⋅
Vout      β         1                   Vout              β                1
=       = A //                          = (β − 1)       = (β − 1) A // 

Vin    A+
1        β                   Vin            A+
1               β

β                                              β
When there are two resistors R1 and R2 connected in parallel and R1 << R2, we said R1//R2 ≈ R1.
The same concept can be applied when we try to draw the Bode Plot for non-inverting amplifier.

1
β

ω

For inverting amplifier, one more step is required as we need to multiply the factor (β-1). Since the
y-axis of Bode plot is in log-scale, we can simply add the curve of (β-1) to get the overall Bode plot
of the inverting amplifiers.
Page 1
The Chinese University of Hong Kong
Department of Electronic Engineering

Types of Active Filters
Low Pass Filter

ω

 1 
 jωC  // R2
−      
Vout                           − R2
=             =
Vin          R1          R1 (1 + jωR2 C )
High Pass Filter

ω
Vout        − R2      − jωR2 C
=             =
Vin             1    1 + jωR1C
R1 +
jωC
Band Pass Filter

ω

 1 
 jωC  // R2
−       
Vout                             − jωR2 C1
=      2 
=
Vin
R1 +
1       (1 + jωR1C1 )(1 + jωR2 C 2 )
jωC1

You may also construct a Band-Pass Filter by cascading a low pass and a high pass filter.
Page 2
The Chinese University of Hong Kong
Department of Electronic Engineering

Other types of filter: Band Reject Filter, Notch Filter.

The above expression assumes the opamp itself is ideal. For real opamp, the above filter responses
are different to the ideal case.
1. For low pass filter, the cut-off frequency of the desired filter could not be larger than βGB,
where β = R1/(R1+R2).
2. The high pass filter will become a bandpass filter with an upper cut-off frequency at βGB due to
the finite bandwidth of the opamp, where β = R1/(R1+R2).

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