The document discusses different types of filters including low-pass, high-pass, band-pass, and band-stop filters. It describes the characteristics of each type of filter including their passbands, cutoff frequencies, and responses. The document also covers Butterworth filters and how to design first-order filters using op-amps and passive components.

1. Filters
• Filters are circuits that are capable of
passing signals with certain selected
frequencies while rejecting signals with
other frequencies.
– This property is called selectivity.

2. Active Filters
• Active filters use transistors or op-amps combined
with passive RC, RL, or RLC circuits.
• The active devices provide voltage gain. and the
passive circuits provide frequency selectivity.
• In terms of general response, the four basic categories
of active filters are
– low-pass
– high-pass
– band-pass
– band-stop.

3. Passband, Critical Frequency &
Stopband
• The passband of a filter is the range of frequencies that are
allowed to pass through the filter with minimum attenuation
(usually defined as less than - 3 dB of attenuation)
• The critical frequency (also called the cutoff frequency) defines
the end of the passband and is normally specified at the point
where the response drops - 3 dB (70.7%) from the passband
response.
• Following the passband is a region called the transition region
that leads into a region called the stopband. There is no precise
point between the transition region and the stopband.

4. Low-Pass Filter & Response
• A low-pass filter is one that passes
frequencies from dc to fc and significantly
attenuates all other frequencies.
• The passband of the ideal low-pass filter is
shown in the blue-shaded of fig (a)
• The bandwidth of an ideal low-pass filter is
equal to fc .
– BW = fc
• Actual filter responses depend on the number
of poles, a term used with filters to describe
the number of RC circuits contained in the
filter.
– The most basic low-pass filter is a
simple RC circuit consisting of just one
resistor and one capacitor; the output is
taken across the capacitor as shown in
fig (b).
– This basic RC filter has a single pole,
and it rolls off at - 20 dB/decade beyond
the critical frequency. The actual
response is indicated by the blue line in
fig (a)
– The -20 dB/decade roll-off rate for the gain of a basic
RC filter means that at a frequency of 10 fc ,the output
will be - 20 dB (10%) of the input.

5. Low-Pass Filter Response
• The critical frequency
of a low-pass RC filter
occurs when Zc = R,
where fc = 1/2πRC
• Output at the critical
frequency is 70.7% of
the input. This
response is equivalent
to an attenuation of - 3
dB.

6. High Pass Filter & Response
• A high-pass filter is one that significantly
attenuates or rejects all frequencies below fc
and passes all frequencies above fc.
• The critical frequency is, again, the
frequency at which the output is 70.7% of the
input (or - 3 dB) as shown in Fig (a).
• The ideal response, indicated by the blue-
shaded area. has an instantaneous drop at fc,
which, of course, is not achievable.
• Ideally, the passband of a high-pass filter is
all frequencies above the critical frequency.
• A simple RC circuit consisting of a single
resistor and capacitor can be configured as a
high-pass filter by taking the output across
the resistor as shown in Fig (b).
• The critical frequency of a high-pass RC
filter occurs when Zc = R, where fc = 1/2πRC

7. High Pass Filter Response

8. Band-Pass Filter & Response
• A band-pass filter passes all signals lying
within a band between a lower-frequency
limit and an upper-frequency limit and
essentially rejects all other frequencies that
are outside this specified band.
• A generalized band-pass response curve is
shown in Fig.
• The bandwidth BW is defined as the
difference between the upper critical
frequency fc2 and lower critical frequency fc1.
• The frequency about which the passband is
centered is called the center frequency, fo,
defined as the geometric mean of the critical
frequencies.
• Quality Factor (Q) of a band-pass filter is the
ratio of the center frequency to the
bandwidth.
– The value of Q is an indication of the
selectivity of a band-pass filter. The
higher the value of Q, the narrower the
bandwidth and the better the selectivity
for a given value of fo.
– Band-pass filters are sometimes
classified as narrow-band (Q > 10) or
wide-band (Q < 10).

9. Band-Stop Filter & Response
• Band-stop filter, also known as notch,
band-reject, or band-elimination filter.
• An operation as opposite to that of the
band- pass filter because frequencies
within a certain bandwidth are rejected,
and frequencies outside the bandwidth
are passed.
• A general response curve for a band-
stop filter is shown in Fig.
• Notice that the bandwidth is the band of
frequencies between the 3 dB points.
just as in the case of the band-pass filter
response.

10. Filter Response Characteristics (a)
• Butterworth, Chebyshev, or Bessel response
characteristics can be realized with most active
filter circuit configurations by proper selection of
certain component values.
• A general comparison of the three response
characteristics for a low-pass filter response
curve is shown in Fig.
– High-pass and band-pass filters can also be
designed to have anyone of the
characteristics.
• The Butterworth Characteristics
– The Butterworth characteristic provides a
very flat amplitude response in the
passband and a roll-off rate of -20
dB/decade/pole.
– The Butterworth response is often referred
to as a maximally flat response.
• The Chebyshev Characteristic
– Filters with the Chebyshev response
characteristic are useful when a rapid roll-
off is required because it provides a roll-off
rate greater than -20 dB/decade/pole.
• The Bessel Characteristic The Bessel response
exhibits a linear phase characteristic, meaning
that the phase shift increases linearly with
frequency. The result is almost no overshoot on
the output with a pulse input.

11. Filter Response Characteristics (b)
• An active filter can be designed to have either
a Butterworth, Chebyshev, or Bessel response
characteristic regardless of whether it is a
low-pass, high-pass, band-pass, or band-stop
type.
• The damping factor (DF) of an active filter
circuit determines which response
characteristic the filter exhibits.
• To explain the basic concept, a generalized
active filter is shown in Fig.
– It includes an amplifier, a negative
feedback circuit, and a filter section.
The amplifier and feedback are
connected in a non inverting
configuration.
– The damping factor is determined by
the negative feedback circuit.

12. Filter Response Characteristics (c)

13. Filter Response Characteristics (d)

14. Filter Response Characteristics (e)

15. Low-Pass Filters

16. Low-Pass Filters
• Example: A filter transfer function is given by
Calculate (a) the closed-loop gain A=|H(jω)| (b) A in dB when
ω=0, (c) A in dB when ω= 2 rad/sec and (d) A in dB when ω=
10rad/sec

17. Low-Pass Filters

18. ButterWorth Filters
• The simplest all-pole low-pass filter is the
Butterworth filter, whose magnitude in
the nth-order case
• Where ALP is the gain and ωc is the cut-off
frequency.
– as the order increases, the Butterworth magnitude
more nearly approaches the ideal case.
• Example
Calculate the H(s) for a first-order
low-pass Butterworth filter with ωc = 1
rad/sec and a gain of ALP

19. ButterWorth Filters
• Higher-order transfer Functions
– Second-order factors form
• where a and b are normalized coefficients,
or coefficients when the cut-off frequency is
normalized to ωc = 1 rad/sec
– If the order is higher than two, then
H is a second-order factor of the
transfer function and is thus the
transfer function of a second order
stage of the filter.
– If the filter is odd, then it will have
one first-order stage with a transfer
function of the form
– If the order is 3, 5, 7 and so on, then
the transfer function will have one
factor like first order and one or more
like second-order.
– In every case K is the stage gain and
the filter gain ALP is the product of
the stage gains.
– The normalized coefficients a and b
are given in table.

20. ButterWorth Filters
1. Example: Calculate the transfer function of a third order and a
fourth order low-pass Butterworth Filter with a gain of 4 and a
cut-off frequency of 2000 Hz.
2. Example: Calculate the magnitude in dB of a second-order
low-pass Butterworth Filter with a gain of 4 and a cutoff
frequency of 2000 rad/s for (a) the general case, (b) ω= 0 rad/s,
(c) ω= 2000 rad/s and (d) ω= 20,000 rad/s
3. Example: Calculate the transfer function of a second-order
low-pass Butterworth filter with a gain of 10 and a cutoff
frequency of 100 Hz.

21. First-Order Filters (K>1)
• A first-order low-pass filter, or a first-
order stage of a higher-order low-pass
filter, has a transfer function
– Where K is the stage gain, ωc is the cutoff
frequency and b is the normalized low-pass
coefficient given in Table 5.1
– If the filter is first order then b=1
• Filter Circuit
– The circuit that achieves the above transfer
function is given figure 5.10
– The Op-Amp and the two resistors R2 and R3
constitute a voltage-controlled voltage source
(VCVS) with gain µ

22. First-Order Filters (K=1)
• If the gain is K=1, then replace the VCVS
by a voltage follower
• Again select C, obtain b as before from
table 5-1 and find R1.
• Example : Show a complete design of a
first-order low-pass filter with a gain of 4
and cutoff frequency of 1000 Hz.
• Example: Show a complete design of
first-order low-pass filter with a gain of 1
and a cutoff frequency of 2000 Hz. Use a
capacitance of 5nF.
• Example: Show a complete design of
first-order low-pass filter with a gain of 2
and a cutoff frequency of 100 Hz. Use a
capacitance of 0.1µF.