Second Order Active RC Blocks _Lecture#2 manuscript2

1. 1
Manuscript 2
Second Order Active RC Blocks
Part 3
Second order filters (biquads) are useful building blocks due to their simplicity, mathematical
properties (readily obtainable), and physical characteristics (measurable and adjustable). They are
considered as basic building blocks in the design of large class of high order active filters using cascade
connections of biquads.
General second order function is biquadratic function (ratio of two second order polynomials in S).
Second order active RC block has transfer function in the s-plane of the following form:
𝑇(𝑆) =
𝑉
𝑉
=
𝑁(𝑆)
𝐷(𝑆)
= ±𝐾 ∙
(𝑆 + 𝑧 ) ∙ (𝑆 + 𝑧∗)
(𝑆 + 𝑝 ) ∙ (𝑆 + 𝑝∗)
𝑇(𝑆) =
𝑉
𝑉
=
𝑁(𝑆)
𝐷(𝑆)
= ±𝐾 ∙
𝑎 𝑆 + 𝑎 𝑆 + 𝑎
𝑏 𝑆 + 𝑏 𝑆 + 𝑏
After the poles are properly placed, N(S) can be adjusted to make the magnitude response possess
the following particularly conventional characteristics:
(i) Low pass filter  a2 = a1 = 0, b2 = 1
(ii) High pass filter  a1 = ao = 0, b2 = 1
(iii) Band pass filter  a2 = ao = 0, b2 = 1
(iv) Band reject (Band elimination = Notch) filter  a1 = 0 , b2 = 1
(v) All pass filter (phase equalizer): a2 = b2 = 1 , a1 = - b1 , ao = bo
(vi) Gain equalizer: a2 = b2 = 1, ao = bo , a1 ≠ b1
These designations refer to the magnitude |T(jω)| (magnitude response), and indicate how the
amplitude of the sine input signals with different frequencies are affected in the steady-state as they
are processed by a circuit having T(S) as a transfer function. Also, they indicate how the phase (angle)
of the input signals are shifted over range of frequencies. The numerator coefficients of the T(S)
determine the transmission zeros and therefore determines the type of the biquad filter.
The biquadratic transfer function has two-pairs of the zeros and poles occur as complex conjugate;
the poles are located in the Left-Half of the s-plane (stable system), while the zeros may be located
anywhere in the s-plane (except on the positive real axis; under BPF it is one on the origin; under HPF
it is dual on the origin). The pair of complex conjugate zeros are characterized by two parameters;
namely ωz and Qz , while the two parameters for the pair of complex conjugate poles (natural modes)
are ωp (or ωo) and Qp (or Q).
Ideally;
ωo is the pole frequency and it is defined as the approximated frequency at which |T(jω)| attains its
maximum.
Q is the pole quality factor (pole Q). It is a (FOM) measure of the sharpness of the maximum of
|T(jω)|(sharpness of a bump at ωo). Also, it is a measure of the selectivity of the filter; i.e., the higher
Q (the closer the pole to the jω-axis) the more selective the filter response becomes. It determines
the distance of the poles from the jω-axis. An infinite value of Q locates the poles on the jω-axis and
can yield sustained oscillation in the circuit realization. A negative value of Q implies that the poles
are in the RH plan, which certainly produces oscillations.

2. 2
ωz is the zero frequency and it is defined as the approximated frequency at which |T(jω)| attains its
minimum.
Qz is the zero quality factor. It is a (FOM) measure of the sharpness of the minimum of |T(jω)|.
Thus, the biquadratic transfer function can be expressed in term of ωz, ωp, Qz, and Qp as
𝑇(𝑆) =
𝑉
𝑉
=
𝑁(𝑆)
𝐷(𝑆)
= ±𝐾 ∙
𝑘 𝑆 + 𝑘
𝜔
𝑄 𝑆 + 𝑘 𝜔
𝑆 +
𝜔
𝑄 𝑆 + 𝜔
Constants k1, k2, and k3 are -1, 1 or 0. There are six possible cases as listed below
(i) Low pass filter  k2 = k3 = 0, k3 = 1
(ii) High pass filter  k2 = k3 = 0, k1 = 1
(iii) Band pass filter  k1 = k3 = 0, k2 = 1
(iv) Band reject (Band elimination = Notch) filter  k2 = 0 , k1 = k3 = 1
(v) All pass filter (phase equalizer): k1 = k3 = 1 , k2 = - 1 , = , 𝜔 = 𝜔
(vi) Gain equalizer: k1 = k2 = k3 = 1 , 𝑄 < 𝑄 (dip), 𝑄 > 𝑄 (gain boost), 𝜔 = 𝜔
In general, the Dc gain magnitude is 20 log 𝐾 and the infinite frequency magnitude is 20 log|𝐾|.
Typical zero-pole pattern is shown below. ωz and ωp are related to the frequencies at which the loss
peaks and transmission poles occur. Similarly, Qz and Qp are the corresponding nominal selectivities.
The complex poles have the form p1, p1* = – α ± jβ, where α and β can be obtained in terms of ωo and
Q. The poles are located at
𝑝 , 𝑝∗
= −
𝜔
2𝑄
1 ± 1 − 4𝑄
Thus, for Q ≤ 0.5, the poles are real (passive RC filters ) and for Q > 0.5, the poles are complex conjugate
(active RC filters) and may be expressed as
𝑝 , 𝑝∗
= −𝛼 ± 𝑗𝛽
where
𝛼 =
𝜔
2𝑄
and 𝛽 = 𝜔 1 −
1
4𝑄
(𝜔 ) = 𝛼 + 𝛽 𝑎𝑛𝑑 𝑄 =
𝜔
2𝛼

3. 3
Thus, the maximum value for the complex poles (maximum |T(jω)| when D(S) is minimum) occurs
approximately at the pole frequency ωo (under Q > 5)
|𝑝 , 𝑝∗| = [𝑅𝑒{𝑝 }] + [𝐼𝑚{𝑝 }] = 𝜔
∡𝑝 , 𝑝∗
= − tan 2𝑄 1 −
1
4𝑄
∡𝑝 , 𝑝∗
= − tan 4𝑄 − 1
Solving for Q,
4𝑄 − 1 = − tan 𝜓
4𝑄 = tan 𝜓 + 1
𝑄 =
sec 𝜓
4
𝑄 =
sec 𝜓
2
Also Q can be given as
𝑄 =
𝜔
𝐵𝑊
=
[𝑅𝑒{𝑝 }] + [𝐼𝑚{𝑝 }]
2𝑅𝑒{𝑝 }
In a similar manner, the complex zeros have the form z1, z1* = – γ ± jξ, where γ and ξ can be obtained
in terms of ωz and Qz. The zeros are located at
𝑧 , 𝑧∗
= −
𝜔
2𝑄
1 ± 1 − 4𝑄
Thus, the minimum value for the complex zeros (minimum |T(jω)| when N(S) is minimum) occurs
approximately at the zero frequency ωz (under Qz >> 1)
|𝑧 , 𝑧∗| = [𝑅𝑒{𝑧 }] + [𝐼𝑚{𝑧 }] = 𝜔
Very often the complex zeros are on the jω-axis in which case Qz = ∞.
Qz can be given as
𝑄 =
𝜔
𝐵𝑊
=
[𝑅𝑒{𝑧 }] + [𝐼𝑚{𝑧 }]
2𝑅𝑒{𝑧 }

4. 4
Some of the important counters for the biquadratic transfer function in the s-plane are depicted
below.

5. 5
It is noted that lines of constant Q are lines of constant angle ψ. Counters of constant ωo are circles of
radius ωo with their centers at the origin. That is, ωo is the distance of the pole locations from the
origin. Finally, lines of constant ωo/2Q are lines parallel to the imaginary axis. In circuit design, we will
ordinarily deal with Q values greater than 1. This has implications with respect to the pole positions.
Hence, we conclude that we will normally be interested in a small sector of the s-plane (shaded area).
For interesting values of Q, the poles move closer to the jω-axis so that small element errors can shift
the poles into the RH plane and may cause the circuit to oscillate. For Q greater than 5, β ≈ ωo with an
error less than 1% and the horizontal distance α is less than 0.1ωo. At this level, we need to relate Q
and ωo to the magnitude and the phase responses which we will in turn relate to biquad filter*
specifications.

6. 6
The classification of the magnitude and the phase responses of the biquad transfer function
After all poles are properly placed on the LH plane, the magnitude responses |T(jω)| are classified to
different responses according to the location of the zeros. These classifications are as follows:
 Biquadratic transfer function with low-pass magnitude response:
The biquadratic transfer function with low-pass magnitude characteristics is given as:
𝑇(𝑆) =
𝑉
𝑉
=
𝑁(𝑆)
𝐷(𝑆)
=
±𝐾𝜔
𝑆 +
𝜔
𝑄 𝑆 + 𝜔
where K is the DC gain; i.e. |T(jω=0)|. The pole-zero pattern of the low pass biquad transfer function
is shown below.

7. 7
The magnitude response, |T(jω)|, the phase response θ(ω) considering +K, and the loss (attenuation)
α(ω) in dB, are
|𝑇(𝑗𝜔)| =
𝐾𝜔
(𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
𝜃(𝜔) = − tan
𝜔𝜔
𝑄(𝜔 − 𝜔 )
𝛼(𝜔) = −20 log|𝑇(𝑗𝜔)| = −𝐴(𝜔) 𝑑𝐵
The biquad filter response behaves as low pass filter providing the following
|𝑇(𝑗𝜔)| =
𝐾 ; 𝜔 = 0
𝐾𝑄 ; 𝜔 = 𝜔
0 ; 𝜔 = ∞
𝜃(𝜔) =
0°
; 𝜔 = 0
−90°
; 𝜔 = 𝜔
−180°
; 𝜔 = ∞
𝛼(𝜔) =
−20log 𝐾 ; 𝜔 = 0
−20log 𝐾𝑄 ; 𝜔 = 𝜔
∞ ; 𝜔 = ∞
Plots |T(jω)|and α(ω) for the case when K = 1 (i.e. the DC gain of unity) are shown below.

8. 8
𝑇(𝑗𝜔)| =
𝜕|𝑇(𝑗𝜔)|
𝜕𝜔
=
𝜕
𝜕𝜔
⎝
⎛
𝜔
(𝜔 − 𝜔 ) +
𝜔𝜔
𝑄 ⎠
⎞ = 0
𝑇(𝑗𝜔)| =
𝜕
𝜕𝜔
⎝
⎜
⎜
⎛
𝜔
𝜔 + 𝜔 − 2𝜔 𝜔 +
𝜔 𝜔
𝑄 ⎠
⎟
⎟
⎞
= 0
𝑇(𝑗𝜔)| =
−0.5𝜔 4𝜔 − 4𝜔 𝜔 + 2
𝜔𝜔
𝑄
𝜔 + 𝜔 − 2𝜔 𝜔 +
𝜔 𝜔
𝑄
/
= 0

9. 9
𝑇(𝑗𝜔)| = −0.5𝜔 4𝜔 − 4𝜔 𝜔 + 2
𝜔𝜔
𝑄
= 0
𝜔 4𝜔 − 4𝜔 𝜔 + 2
𝜔𝜔
𝑄
= 0
𝜔 4𝜔 𝜔 − 4𝜔 +
2𝜔
𝑄
= 0
One solution 𝜔 = 0
4𝜔 𝜔 − 4𝜔 +
2𝜔
𝑄
= 0
𝜔 − 𝜔 +
𝜔
2𝑄
= 0
𝜔 = 𝜔 −
𝜔
2𝑄
𝜔 = ± 𝜔 −
𝜔
2𝑄
= ±𝜔 1 −
1
2𝑄
Considering the positive frequency, second solution is 𝜔 = 𝜔 1 −
𝜔 =
⎩
⎪⎪
⎨
⎪⎪
⎧0 (𝑛𝑜 𝑝𝑒𝑎𝑘, 𝑓𝑙𝑎𝑡 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒);
1
2
< 𝑄 ≤
1
√2
; 𝑐𝑎𝑠𝑒𝐼
𝜔 1 −
1
2𝑄
;
1
√2
< 𝑄 ≤ 5 ; 𝑐𝑎𝑠𝑒𝐼𝐼
𝜔 ; 𝑄 > 5 ; 𝑐𝑎𝑠𝑒𝐼𝐼𝐼
𝑇 𝑗𝜔 ; 𝑐𝑎𝑠𝑒𝐼𝐼 =
𝜔
𝜔 − 𝜔 1 −
1
2𝑄
+
⎝
⎛
𝜔 1 −
1
2𝑄
𝑄
⎠
⎞
𝑇 𝑗𝜔 =
𝜔
𝜔
4𝑄
+
𝜔 1 −
1
2𝑄
𝑄
=
𝜔
𝜔
4𝑄
+
𝜔
𝑄
−
𝜔
2𝑄
𝑇 𝑗𝜔 =
𝜔
𝜔
𝑄
−
𝜔
4𝑄
=
𝑄
1 −
1
4𝑄

10. 10
𝑇 𝑗𝜔 =
⎩
⎪⎪
⎨
⎪⎪
⎧1 (𝑛𝑜 𝑝𝑒𝑎𝑘, 𝑓𝑙𝑎𝑡 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒) ;
1
2
< 𝑄 ≤
1
√2
; 𝑐𝑎𝑠𝑒𝐼
𝑄
1 −
1
4𝑄
;
1
√2
< 𝑄 ≤ 5 ; 𝑐𝑎𝑠𝑒𝐼𝐼
𝑄 ; 𝑄 > 5 ; 𝑐𝑎𝑠𝑒𝐼𝐼𝐼
Note that for Q > 5 ( ≪ 1), |T(jωpeak)| = |Tmax| = Q, and ωpeak = ωo. The phase response will have
same three fixed points for all values of Q (0°, – 90°, and – 180° asymptote). As Q increases, the
behavior of the phase response approaches the ideal behavior.
For 0.5 < Q ≤ 0.707; half-power frequencies is
0.707 =
𝜔
(𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
0.5 =
𝜔
(𝜔 − 𝜔 ) +
𝜔𝜔
0.707
0.5𝜔 + 0.5𝜔 − 𝜔 𝜔 + 0.5
𝜔 𝜔
0.5
= 𝜔
0.5𝜔 = 0.5𝜔
𝜔 = 𝜔
-----------------------------------------------------------------------------------------------------------------------
For 0.707 ˂ Q ≤ 5; half-power frequencies are obtained under
𝑇
√2
=
𝑄
2 −
1
2𝑄
=
𝜔
(𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
For 0.707 < Q ≤ 5; ω1 and ω2 are the lower and upper half-power frequencies, they are obtained as
follows:
|𝑇(𝑗𝜔 )| = |𝑇(𝑗𝜔 )| =
𝑇
√2
=
𝑄
2 −
1
2𝑄
By using zero-pole constellation, p1 for ω1 and ω2 are
𝑝 | ≈ −
𝜔
2𝑄
+ 𝑗𝜔 1 −
1
2𝑄
𝑝 | ≈ −
𝜔
2𝑄
+ 𝑗𝜔 1 +
1
2𝑄
This yields to

11. 11
𝜔 ≈ 𝜔 1 −
1
2𝑄
, 𝜔 ≈ 𝜔 (1 +
1
2𝑄
)
In terms of the poles, the transfer function can be written as
𝑇(𝑆) =
1
(𝑆 + 𝑝 )(𝑆 + 𝑝∗)
The denominator can be written as two factors; jω+p1 = m2 ϕ2 and jω+p2 = m1 ϕ1 . Thus,
|𝑇(𝑗𝜔)| =
1
𝑚 𝑚
𝜃(𝜔) = −(𝜑 + 𝜑 )
The phasor plots (zero-pole constellation) are shown for ω1, ωo and ω2 (values of frequencies one
below ωo, one at ωo, and one above ωo) visualizing the behavior of the low pass filter. It is observed
that m2 has shortest length near ωo (phasor with frequency at ωo), the magnitude response reaches a
peak (ωpeak) near ωo.
Note for Q > 5, ω2 – ω1 = ωo/Q ≈ -3dB bandwidth (BW) and for Q ≤ 0.707, ωo = ω1 = ω2.
Under the assumption of ωo = 1rad/sec, we got the following plots. The asymptotic magnitude Bode
plot decreases at the rate of – 12dB/octave (– 40dB/decade), and this is sometimes described as two-
pole roll-off.

12. 12
Now, the low pass filter specifications will be the half-power frequency (–3dB bandwidth) and the
maximum magnitude value (|T(jωpeak)| = |Tmax|). The term half-power comes analogously from the
equation for power P= I2
R, when P is reduced by , then it is necessary that current be reduced by
√
.
Here, we are dealing with frequency for |T(jω)|, and the –3dB bandwidth ω2 - ω1 ≈ ω2 = ω-3dB
corresponds to the value of
| ( )|
√
≈ 0.707 ∙ |T(jω)|. Since |T(jωpeak)| is approximately equal Q (for
Q > 5), the filter specification of a relatively flat response in the passband implies a low value of Q (Q
≤ 0.707).

13. 13
The pole Q is a very important parameter in filter realization, since it indicates how close the pole is
to the imaginary axis and hence how selective the filter would. Since BW = ωo/Q, the higher Q the
smaller BW will be and the more selective the filter will be.
In passive circuit realization increasing Q requires that the elements be of better quality; inductors
and capacitors should be lossless or less lossy. In active circuits, increasing Q requires using more
active elements (op amp) in the realization, so that the network is not too sensitive. In analog filters
both the passive components (R and C) and active components (op amp open-loop gain ‘A’) affect Q,
but high Q, poles are very close to the jω-axis. Therefore, any small change in the components would
result in moving the poles to the RH plane and the circuit will no longer be stable. Thus, for high-Q
realization, one should use circuits with Q’s that have low sensitivities with respect to R, C, and A. It
has been shown that, the more the number of the op amps in a circuit, the less sensitive the Q of the
circuit (with respect to R, C, and A) would be. That is; Q of a second order circuit with three op amps
is less sensitive than the one with two op amps and they are in turn less sensitive than single op amp
circuit. A familiar application of the low pass filter is in the high-pitched tone control of some Hi-Fi
components (high-fidelity) amplifiers. The treble control varies the cut-off frequency of a low pass
filter and it is used to attenuate the high frequency record scratch-noise.
 Biquadratic transfer function with high-pass magnitude response:
The biquadratic transfer function with high-pass magnitude characteristics is given as:
𝑇(𝑆) =
𝑉
𝑉
=
𝑁(𝑆)
𝐷(𝑆)
=
±𝐾𝑆
𝑆 +
𝜔
𝑄 𝑆 + 𝜔
where K represents the high frequency gain (HFG); i.e. |T(jω=∞)|=K. The pole-zero pattern of the high
pass biquadratic transfer function is shown below:
The magnitude response, |T(jω)|, the phase response θ(ω) considering +K, and the loss (attenuation)
α(ω) in dB, for K = 1 are
|𝑇(𝑗𝜔)| =
−𝜔
(𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
𝜃(𝜔) = 180°
− tan
𝜔𝜔
𝑄(𝜔 − 𝜔 )
𝛼(𝜔) = −20 log|𝑇(𝑗𝜔)| = −𝐴(𝜔) 𝑑𝐵
The biquad filter response behaves as high pass filter providing the following

14. 14
|𝑇(𝑗𝜔)| =
0 ; 𝜔 = 0
𝐾𝑄 ; 𝜔 = 𝜔
𝐾 ; 𝜔 = ∞
𝜃(𝜔) =
180°
; 𝜔 = 0
90°
; 𝜔 = 𝜔
0°
; 𝜔 = ∞
𝛼(𝜔) =
∞ ; 𝜔 = 0
−20log 𝐾𝑄 ; 𝜔 = 𝜔
−20 log 𝐾 ; 𝜔 = ∞
Plots |T(jω)|and α(ω) for the case when K = 1 (i.e. unity HGF ), are shown below.
Note that the frequencies ωpeak, ω1, and ω2 can be obtained in a similar manner as in the low pass filter
case. Thus,
𝜔 ≈ 𝜔 1 −
1
2𝑄
, 𝜔 ≈ 𝜔 (1 +
1
2𝑄
)
𝜔 =
⎩
⎪⎪
⎨
⎪⎪
⎧ ∞;
1
2
< 𝑄 ≤
1
√2
; 𝑐𝑎𝑠𝑒𝐼
𝜔
1 −
1
2𝑄
;
1
√2
< 𝑄 ≤ 5 ; 𝑐𝑎𝑠𝑒𝐼𝐼
𝜔 ; 𝑄 > 5 ; 𝑐𝑎𝑠𝑒𝐼𝐼𝐼
And,

15. 15
𝑇 𝑗𝜔 =
⎩
⎪⎪
⎨
⎪⎪
⎧1 (𝑓𝑙𝑎𝑡 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒) ;
1
2
< 𝑄 ≤
1
√2
; 𝑐𝑎𝑠𝑒𝐼
𝑄
1 −
1
4𝑄
;
1
√2
< 𝑄 ≤ 5 ; 𝑐𝑎𝑠𝑒𝐼𝐼
𝑄 ; 𝑄 > 5 ; 𝑐𝑎𝑠𝑒𝐼𝐼𝐼
Note that for Q > 5, |T(jωpeak)| = |Tmax| = Q, and ωpeak = ωo.
 Biquadratic transfer function with band-pass magnitude response:
The biquadratic transfer function that has a band-pass magnitude characteristics is given as:
𝑇(𝑆) =
𝑉
𝑉
=
𝑁(𝑆)
𝐷(𝑆)
=
±𝐾𝑆
𝑆 +
𝜔
𝑄 𝑆 + 𝜔
where K represents the center frequency gain (CFG); i.e. |T(jω=ωo)|=K/(ωo/Q)., ωo is the center
frequency, and ωo/Q is -3dB bandwidth. The pole-zero pattern of the band pass biquadratic transfer
function is shown below:
The magnitude response, |T(jω)|, the phase response θ(ω) considering +K, and the loss (attenuation)
α(ω) in dB, are
|𝑇(𝑗𝜔)| =
𝐾𝜔
(𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
𝜃(𝜔) = 90°
− tan
𝜔𝜔
𝑄(𝜔 − 𝜔 )
𝛼(𝜔) = −20 log|𝑇(𝑗𝜔)| = −𝐴(𝜔) 𝑑𝐵
The biquad filter response behaves as band pass filter providing the following
|𝑇(𝑗𝜔)| =
0 ; 𝜔 = 0
𝐾𝑄 𝜔⁄ ; 𝜔 = 𝜔
0 ; 𝜔 = ∞

16. 16
𝜃(𝜔) =
90°
; 𝜔 = 0
0°
; 𝜔 = 𝜔
−90°
; 𝜔 = ∞
𝛼(𝜔) =
∞ ; 𝜔 = 0
−20log 𝐾𝑄 𝜔⁄ ; 𝜔 = 𝜔
∞ ; 𝜔 = ∞
Plots |T(jω)|and α(ω) for the case when K = ωo/Q (i.e. the center frequency gain of unity) are shown
below.
From the symmetry of the band pass response, ωo is the geometric mean (not arithmetic) of ω1 and
ω2 :
𝜔 = 𝜔 𝜔
And the bandwidth is given as
𝐵𝑊 = 𝜔 − 𝜔 =
𝜔
𝑄

17. 17
Use the above two equations to get expression of the lower and upper cut-off frequencies (half-power
frequencies)
𝜔 −
𝜔
𝜔
=
𝜔
𝑄
𝜔 − 𝜔 =
𝜔 𝜔
𝑄
𝜔 −
𝜔 𝜔
𝑄
− 𝜔 = 0
Solving quadratic equation a = 1 , b = – ωo / Q , c = – ωo
2
, Δ = b2
– 4 ac = (ωo
2
/ Q2
) – 4 (– ωo
2
) = (ω2
/
Q2
) + 4 (ωo
2
)
𝜔 =
𝜔
2𝑄
±
𝜔
𝑄 + 4𝜔
2
= 𝜔
1
2𝑄
± 1 +
1
4𝑄
Similarly for ω1
𝜔
𝜔
− 𝜔 =
𝜔
𝑄
−𝜔 + 𝜔 =
𝜔 𝜔
𝑄
𝜔 +
𝜔 𝜔
𝑄
− 𝜔 = 0
Solving quadratic equation a = 1 , b = ωo / Q , c = – ωo
2
, Δ = b2
– 4 ac = (ωo
2
/ Q2
) – 4 (– ωo
2
) = (ω2
/ Q2
)
+ 4 (ωo
2
)
𝜔 =
−𝜔
2𝑄
±
𝜔
𝑄 + 4𝜔
2
= −𝜔
1
2𝑄
± 1 +
1
4𝑄
= 𝜔
−1
2𝑄
∓ 1 +
1
4𝑄
Considering the positive frequencies,
𝜔 = 𝜔
−1
2𝑄
+ 1 +
1
4𝑄
𝜔 = 𝜔
1
2𝑄
+ 1 +
1
4𝑄
There are two types of band pass filters which are classified as per quality factor. Wide band pass filter
is considered for Q < 10, while narrow band pass filter is considered for Q >10.

18. 18
An interesting example that illustrates the application LPF, HPF, and BPF is in the detection of the
signals generated by a telephone set with push buttons as in touch-toneTM
dialing (contrary to rotary
dialing=pulse dialing). The international standard for telephone signaling utilizes dual-tone multi-
frequency (DTMF) signaling. Pressing a push button from the 10 decimal digits 0 to 9 (and an additional
six extra buttons, used for special purposes ) generates a pair of tones, one from the low-band (697Hz-
941Hz) and one from the high-band (1209Hz-1633Hz); that is, each button is identified by a unique
pair of signal frequencies as illustrated in the figure below.
As the telephone number is dialed a set of signals is transmitted to the telephone office. There, these
tones are identified and converted to a suitable set of DC signals that are used by the switching system
to connect the caller to the party being called. After amplification, the two tones are separated into
their respective groups by the LPF and HPF. The separated tones are then converted to square waves
of fixed amplitude using limiters. The next step in the detection scheme is to identify the individual
tones in the respective groups. This is accomplished by the 8 band pass filters shown the figure below.
Each of these band pass filters passes one tone, rejecting all the neighboring tones. The band pass
filters are followed by detectors that are energized when their input voltage exceeds a certain
threshold voltage, and the output of each detector provides the required DC switching signal.
 Biquadratic transfer function with band-reject magnitude response:
The biquadratic transfer function that has a band-reject magnitude characteristics is given as:
𝑇(𝑆) =
𝑉
𝑉
=
𝑁(𝑆)
𝐷(𝑆)
=
±𝐾(𝑆 + 𝜔 )
𝑆 +
𝜔
𝑄 𝑆 + 𝜔
where K pass-band gain; i.e. |T(jω=0,∞)|=K(ωz
2
/ωo
2
)., ωz is the zero frequency, and ωo/Q is -3dB
bandwidth. There are two types of band reject filters which are classified as per quality factor. Wide

19. 19
band reject filter is considered for Q < 10, while narrow band reject filter is considered for Q >10. The
latter filter is called the notch filter.
Further, depending the value of ωo as compared to ωz the following three classes of biquadratic band
reject transfer function are generated:
(i) Biquadratic transfer function with a standard (symmetrical) notch response if ωz = ωo
(ii) Biquadratic transfer function with a high-pass notch response (2nd
order HPNF) if ωz < ωo
(iii) Biquadratic transfer function with a low-pass notch response (2nd
order LPNF) if ωz > ωo
(i) Second order standard (symmetrical) notch filter:
In this case the transfer function of the filter is
𝑇(𝑆) = ±𝐾
(𝑆 + 𝜔 )
𝑆 +
𝜔
𝑄 𝑆 + 𝜔
The pole-zero pattern of the standard notch biquadratic transfer function is shown below:
The magnitude response, |T(jω)|, the phase response θ(ω) considering +K, and the loss (attenuation)
α(ω) in dB, are
|𝑇(𝑗𝜔)| =
𝐾(𝜔 − 𝜔 )
(𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
𝜃(𝜔) = − tan
𝜔𝜔
𝑄(𝜔 − 𝜔 )
𝛼(𝜔) = −20 log|𝑇(𝑗𝜔)| = −𝐴(𝜔) 𝑑𝐵
The biquad filter response behaves as standard notch filter providing the following
|𝑇(𝑗𝜔)| =
𝐾 ; 𝜔 = 0
0 ; 𝜔 = 𝜔
𝐾 ; 𝜔 = ∞
𝜃(𝜔) =
0°
; 𝜔 = 0
−90°
; 𝜔 = 𝜔
0°
; 𝜔 = ∞
𝛼(𝜔) =
−20log 𝐾 ; 𝜔 = 0
∞ ; 𝜔 = 𝜔
−20 log 𝐾 ; 𝜔 = ∞
Plots |T(jω)|and α(ω) for the case when K = 1 (i.e. the pass-band gain of unity) are shown below.

20. 20
For stopband symmetry we have 𝜔 = 𝜔 𝜔 = 𝜔 𝜔 . Sometimes the specifications of the
standard notch filter are given in terms of the depth of the notch and the band of the frequency to be
eliminated. Let the required loss 𝛼 in dB over a bandwidth 𝐵𝑊 = 𝜔 − 𝜔 .Then, it can be shown
that BWx and the -3dB bandwidth (BW) are related by
𝐵𝑊 =
𝜔
𝑄
= 𝐵𝑊 10 . − 1
To show this we proceed as follows:
|𝑇(𝑗𝜔)| =
(𝜔 − 𝜔 )
(𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
And
𝛼 = 𝛼(𝜔 ) = −20 log|𝑇(𝑗𝜔 )| = 20 log
⎝
⎜
⎛ (𝜔 − 𝜔 ) +
𝜔 𝜔
𝑄
(𝜔 − 𝜔 )
⎠
⎟
⎞
𝛼 = 10 log 1 +
𝜔 𝜔
𝑄 (𝜔 𝜔 − 𝜔 )
= 10 log 1 +
𝜔
𝑄 (𝜔 − 𝜔 )
𝛼 = 10 log 1 +
𝜔
𝑄 (𝐵𝑊 )
Solving for 𝐵𝑊 =
𝐵𝑊 =
𝜔
𝑄
= 𝐵𝑊 10 . − 1
The -3dB bandwidth (BW) can also be obtained as
𝐵𝑊 = 𝜔 − 𝜔
The magnitude response at these two frequencies is 0.707 considering K = 1.
|𝑇(𝑗𝜔)|
√2
=
(𝜔 − 𝜔 )
(𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
=
1
√2

21. 21
√2(𝜔 − 𝜔 ) = (𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
2(𝜔 − 𝜔 ) = (𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
(𝜔 − 𝜔 ) −
𝜔𝜔
𝑄
= 0
𝜔 = 𝜔
−1
2𝑄
+ 1 +
1
4𝑄
𝜔 = 𝜔
1
2𝑄
+ 1 +
1
4𝑄
In general, let us assume the loss at ω1 and ω2 is K/q (K is not unity), then
𝐾(𝜔 − 𝜔 )
(𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
=
𝐾
𝑞
(𝜔 − 𝜔 )𝑞 = (𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
(𝜔 − 𝜔 ) (𝑞 − 1) −
𝜔𝜔
𝑄
= 0
Solving for ω1 and ω2 and considering the positive frequencies,
𝜔 =
𝜔 2𝑞 𝑄 + 4𝑄 (𝑞 − 1) + 1 − 2𝑄 + 1
(𝑞 − 1)𝑄
√2
𝜔 = −
𝜔 2𝑞 𝑄 + 4𝑄 (𝑞 − 1) + 1 − 2𝑄 + 1
(𝑞 − 1)𝑄
√2
And their difference can be shown to be
𝐵𝑊 = 𝜔 − 𝜔 =
𝜔
𝑄 𝑞 − 1
Under the assumption 𝑞 = √2, then 𝐵𝑊 = .

22. 22
The value of the Q to meet specific loss 1/q over a band Δf:
𝑄 =
𝑓
∆𝑓 𝑞 − 1
It is also useful to note the symmetry relationships for the magnitude and the phase
|𝑇(𝑗𝜔 )| = |𝑇(𝑗𝜔 )|
𝜃(𝜔 ) = −𝜃(𝜔 )
These relations hold for all frequencies that satisfy
𝜔 = 𝜔 𝜔
This kind of filter is useful in applications where a specific frequency must be eliminated. For example,
instrumentation system required that the powerline frequency interference of 50/60Hz be eliminated.
Another example is in the system used for billing of long distance telephone calls. In a normal long
distance call, a single frequency tone is transmitted from the caller to the telephone office until the
end of the dialing of the number. As soon as the called party answers, the signal tone ceases and the
billing begins. The billing continues as long as the signal tone is absent. An exception to this system
needs to be made for long distance calls that are toll-free, such as calls to the operator for information.
To prevent these calls from being billed, the signal tone is transmitted to the telephone office through
the entire period of the call. However, the signal tone is usually within the voice frequency band, it
must be removed from the voice signal before being transmitted from the telephone office to the
listener. A second order (or higher) standard notch filter might be used to remove the signal tone.
(ii) Second order high-pass notch filter:
In this case, the transfer function of the second order high-pass notch filter is
𝑇(𝑆) = ±𝐾
(𝑆 + 𝜔 )
𝑆 +
𝜔
𝑄 𝑆 + 𝜔
where ωz < ωo, and K is the pass band gain. The magnitude response, |T(jω)|, the phase response θ(ω)
considering +K, and the loss (attenuation) α(ω) in dB, are
|𝑇(𝑗𝜔)| =
𝐾(𝜔 − 𝜔 )
(𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
𝜃(𝜔) = − tan
𝜔𝜔
𝑄(𝜔 − 𝜔 )

23. 23
𝛼(𝜔) = −20 log|𝑇(𝑗𝜔)| = −𝐴(𝜔) 𝑑𝐵
The biquad filter response behaves as standard notch filter providing the following considering +K
|𝑇(𝑗𝜔)| =
⎩
⎪
⎨
⎪
⎧
𝐾(𝜔 𝜔⁄ ) ; 𝜔 = 0
0 ; 𝜔 = 𝜔
𝐾(𝜔 − 𝜔 )
𝜔 𝑄⁄
; 𝜔 = 𝜔
𝐾 ; 𝜔 = ∞
𝜃(𝜔) =
0°
; 𝜔 = 0
−90°
; 𝜔 = 𝜔
0°
; 𝜔 = ∞
𝛼(𝜔) =
⎩
⎪
⎨
⎪
⎧
−20log 𝐾(𝜔 𝜔⁄ ) ; 𝜔 = 0
∞ ; 𝜔 = 𝜔
−20log
𝐾(𝜔 − 𝜔 )
𝜔 𝑄⁄
; 𝜔 = 𝜔
−20 log 𝐾 ; 𝜔 = ∞
The pole-zero pattern of the high pass notch biquadratic transfer function, its magnitude response
|T(jω)|and the loss α(ω) in dB for the case when K = 1 (i.e. the pass-band gain of unity) are
(iii) Second order low-pass notch filter:
In this case, the transfer function of the second order low-pass notch filter is
𝑇(𝑆) = ±𝐾
(𝑆 + 𝜔 )
𝑆 +
𝜔
𝑄 𝑆 + 𝜔
where ωz > ωo, and K is the pass band gain. The magnitude response, |T(jω)|, the phase response θ(ω)
considering +K, and the loss (attenuation) α(ω) in dB, are same as second order high-pass notch filter.
The pole-zero pattern of the low-pass notch biquadratic transfer function, its magnitude response
|T(jω)|and the loss α(ω) in dB for the case when K = 1 (i.e. the pass-band gain of unity) are

24. 24
 Biquadratic transfer function with all pass magnitude response:
The biquadratic transfer function that has a all pass magnitude characteristics is given as:
𝑇(𝑆) =
𝑉
𝑉
=
𝑁(𝑆)
𝐷(𝑆)
= ±𝐾
𝑆 −
𝜔
𝑄 𝑆 + 𝜔
𝑆 +
𝜔
𝑄 𝑆 + 𝜔
The pole-zero pattern of the biquadratic all pass transfer function is shown below:
The complex poles and zeros of this function are symmetrical about the jω axis
The magnitude response, |T(jω)|, and the phase response θ(ω) considering +K are
|𝑇(𝑗𝜔)| =
𝐾 (𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
(𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
= 𝐾
𝜃(𝜔) = − tan
𝜔𝜔
𝑄(𝜔 − 𝜔 )
− tan
𝜔𝜔
𝑄(𝜔 − 𝜔 )
= −2 tan
𝜔𝜔
𝑄(𝜔 − 𝜔 )
The biquad filter response behaves as all pass filter providing the following considering +K
|𝑇(𝑗𝜔)| = 𝐾 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜔
𝜃(𝜔) =
0°
; 𝜔 = 0
−180°
; 𝜔 = 𝜔
−360°
; 𝜔 = ∞
For – K, the phase response will be
𝜃(𝜔) =
180°
; 𝜔 = 0
0°
; 𝜔 = 𝜔
−180°
; 𝜔 = ∞

25. 25
The magnitude response |T(jω)| of the biquadratic all pass transfer function is
Circuits with all pass transfer function are called all pass circuits. It should be noted that all pass circuits
do not discriminate any frequency. Actually all pass circuits are used for phase correction (delay
equalization). That is to equalize for a distorted delay characteristic of a given circuit, one connects it
in cascade with an all pass circuit such that the total delay (the sum of the delays of the given circuit
and the all pass circuit) is as desired.
Phase and delay of the second order all pass filter
(a) The phase:
The phase, 𝜃(𝜔), of a second order all pass filter is obtained as
𝜃(𝜔) = −2 tan
𝜔𝜔
𝑄(𝜔 − 𝜔 )
(b) The delay:
Thus far, we have discussed the gain (loss) characteristics of filters, but we have not paid any attention
to their phase characteristics. In many applications this omission is justifiable because the human ear
is insensitive to phase changes. Therefore, in the transmission of voice, we need not be concerned
with the phase characteristics of the filter function. However, in digital transmission systems, where
the information is transmitted as square wave time domain pulses, the phase distortions introduced
by the filter cause a variable delay and this cannot be ignored. Delay equalizers are used to
compensate for the delay distortions introduced by filters and other parts of the transmission system.
An ideal delay characteristic is flat for all frequencies, as depicted in figure below. A digital pulse
subjected to this flat delay characteristic will be translated on the time axis by To seconds, but will
otherwise be undistorted.

26. 26
Mathematically, the ideal delay characteristic is described by
𝑉 (𝑡) = 𝑉 (𝑡 − 𝑇 )
Taking Laplace transforms
𝑉 (𝑆) = 𝑉 (𝑆)𝑒
The gain function is thus
𝐻(𝑆) =
𝑉 (𝑆)
𝑉 (𝑆)
= 𝑒
𝐻(𝑗𝜔) =
𝑉 (𝑗𝜔)
𝑉 (𝑗𝜔)
= 𝑒
Thus, the magnitude and phase responses of this function are
|𝐻(𝑗𝜔)| = 𝑒 = 1
𝜃(𝜔) = −𝜔𝑇
This ideal delay characteristic has a constant amplitude, and the phase is a linear function of the
frequency. Observe that the delay To can be obtained by differentiating the phase function with
respect to ω. This, in fact, serves as a definition of delay.
The delay, 𝐷(𝜔), is related to the phase by (the change of the phase with the frequency also is known
as group delay)
𝐷(𝜔) = −
𝑑𝜃(𝜔)
𝑑𝜔
Therefore, the delay of the second order all pass section is obtained as
𝐷(𝜔) = 2
𝜔 𝑄(𝜔 − 𝜔 ) + 2𝜔 𝜔 𝑄
𝑄(𝜔 − 𝜔 )
1 +
𝜔𝜔
𝑄(𝜔 − 𝜔 )
= 2
𝜔 𝑄(𝜔 − 𝜔 ) + 2𝜔 𝜔 𝑄
𝑄(𝜔 − 𝜔 ) + (𝜔𝜔 )
𝐷(𝜔) = 2
𝜔 𝑄 + 𝜔 𝜔 𝑄
𝑄(𝜔 − 𝜔 ) + (𝜔𝜔 )
=
2𝜔 𝑄(𝜔 + 𝜔 )
𝑄(𝜔 − 𝜔 ) + (𝜔𝜔 )
In general the delay of filters will not be flat, and will therefore need to be corrected. This correction
is achieved by following the filter by a delay equalizer (all pass filter). The purpose of the delay
equalizer is to introduce the necessary delay shape to make the total delay (of the filter and equalizer)
as flat as possible. In addition, the delay equalizer must not perturb the loss characteristic of the filter;
in other words, the loss characteristic must be flat over the frequency band of interest. An example of
the application of delay equalizers is in the transmission of data on cables. The delay characteristic for
a typical cable is shown in figure below. A second (or higher) order delay equalizer compensates for
this distortion by introducing the complement of this delay shape as indicated in the figure.

27. 27
The sum of the delays of the cable and equalizer will then be flat, if the parameters ωo/Q and ωo are
chosen properly.
 Biquadratic transfer function of the gain equalizer (shelving filter (dip=bump=cut case),
peaking filter (gain boost=boost case)):
The biquadratic transfer function of a gain equalizer is given as:
𝑇(𝑆) =
𝑉
𝑉
=
𝑁(𝑆)
𝐷(𝑆)
= ±𝐾
𝑆 +
𝜔
𝑄 𝑆 + 𝜔
𝑆 +
𝜔
𝑄 𝑆 + 𝜔
where 𝑄 < 𝑄 provides a dip at 𝜔 while 𝑄 > 𝑄 provides a gain boost at 𝜔 .
The pole-zero pattern of the biquadratic gain equalizer transfer function is shown below:
The magnitude response, |T(jω)|, and the phase response θ(ω) considering +K are
|𝑇(𝑗𝜔)| =
𝐾 (𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
(𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
𝜃(𝜔) = tan
𝜔𝜔
𝑄 (𝜔 − 𝜔 )
− tan
𝜔𝜔
𝑄(𝜔 − 𝜔 )

28. 28
The second order gain equalizer filter may be used to obtain a bump or a dip at the pole frequency in
the gain specifications over a limited band of frequencies.
The magnitude transfer function at the bump or the dip is given by
𝑇 (𝑗𝜔) = |𝑝|
When |𝑝| < 1, it determines the level of the dip. When |𝑝| > 1, it determines the gain boost. let us
assume the loss (specified attenuation) at ω1 and ω2 is 1/q under K = 1 and 𝑄 = 𝑄 𝑝⁄ , then
|𝑇(𝑗𝜔)| =
(𝜔 − 𝜔 ) +
𝑝𝜔𝜔
𝑄
(𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
=
1
𝑞
(𝜔 − 𝜔 ) 𝑞 +
𝑝𝜔𝜔
𝑄
𝑞 = (𝜔 − 𝜔 ) +
𝜔𝜔
𝑄
(𝜔 − 𝜔 ) (𝑞 − 1) =
𝜔𝜔
𝑄
(1 − 𝑝 𝑞 )
where 𝐵𝑊 = 𝜔 − 𝜔 , solving the above equation for these frequencies, the BW expression will be
𝐵𝑊 = 𝜔 − 𝜔 =
𝜔
𝑄
1 − 𝑝 𝑞
𝑞 − 1
The required pole quality factor is
𝑄 =
𝑓
∆𝑓
1 − 𝑝 𝑞
𝑞 − 1
Under p = 0, the standard notch filter is obtained as mentioned previous.

29. 29
The gain equalizers are very much similar to the phase correcting circuits (delay equalizers) but they
deal with the magnitude rather than phase or delay. They may be used in cascade with a filter in order
to yield a desired magnitude at a desired frequency.
The concept of a variable equalizer was introduced by H. W. Bode in 1938. The variable equalizer is
capable of changing the amount of the equalization without changing the shape of the transfer
characteristics. Variable equalizers are used to compensate the variation of the loss-frequency
characteristic in communication cables. They are also applied in audio systems to provide the listener
with the most desirable sound by equalizing the appropriate frequency ranges.
The gain equalizer can provide a gain boost (bump) or a dip height at the pole frequency (ωo). Gain
equalizers are used to shape the gain versus frequency spectrum of a given signal. The shaping can
take the form of a bump or a dip, that is, an emphasis or deemphasis of a band of frequencies. Gain
equalizers differ from the filter types discussed thus far, In that the shapes they provide are not
characterized by a passband and a stopband. In fact, any gain versus frequency shape that does not
fall into the four standard categories (LP, HP, BP, BR) will be considered a gain equalizer. A familiar
application of gain equalizers occurs in the recording and reproduction of music on phonograph
records. High frequency background-hiss noise associated with the recording of sound is quite
annoying. One way of alleviating this problem is to increase the amplitude of the high frequency signal,
as shown in figure. This is known as pre-emphasis. Another problem associated with phonograph
recording is that, for normal levels of sound, the low frequencies require impractically wide excursions
in the record grooves. These excursions can be reduced by attenuating the low frequency band as
shown in the recording equalizer curve in the figure. In the playback system, which consists of a
turntable and an amplifier, the high frequencies must be de-emphasized and the low frequencies
boosted, as shown in the reproduction equalizer characteristic of figure below. After this equalization
the reproduced sound will have the same frequency spectrum as that of the original source generated
in the recording studio. To allow for different recording schemes, some high quality phonograph
amplifiers are equipped with variable shape equalizers, which are most conveniently designed using
active RC networks. (another application is graphic equalizer)

30. 30
Example:
Assume that it is desired to adjust the gain of a 2nd
order LP filter such that the Dc gain and the pole
frequency gain are equal to unity.
Solution:
Here we have
|𝑇(𝑗𝜔 = 0)| = |𝑇(𝑗𝜔 )| = 1
The transfer function of a cascade of the gain equalizer and the LP filter is
𝑇 (𝑆) =
𝑉 (𝑆)
𝑉 (𝑆)
= ±
𝐾
𝑆 +
𝜔
𝑄 𝑆 + 𝜔
The approach used in the synthesis is to arrange the given transfer function so that it leads itself to a
realization using simple first order circuits that have been studied in the previous lectures. The above
equation can be rewritten as (only the negative sign is used first):
𝑉 𝑆 𝑆 +
𝜔
𝑄
+ 𝜔 = −𝐾𝑉
𝑉 1 +
𝜔
𝑆 𝑆 +
𝜔
𝑄
= 𝑉
−𝐾
𝑆 𝑆 +
𝜔
𝑄
𝑉 = 𝑉
−𝐾 ℎ⁄
𝑆 +
𝜔
𝑄
+ 𝑉
− 𝜔 ℎ⁄
𝑆 +
𝜔
𝑄
−ℎ
𝑆
(−1)
The h parameter is introduced to add more flexibility in the realization. Flow diagram of this equation
is shown below:
𝑇(𝑆) = 𝑇 (𝑆)𝑇 (𝑆) =
±𝜔
𝑆 +
𝜔
𝑄 𝑆 + 𝜔
∙
𝑆 +
𝜔
𝑄 𝑆 + 𝜔
𝑆 +
𝜔
𝑄 𝑆 + 𝜔
Therefore,

31. 31
|𝑇(𝑗Ω)| =
(1 − Ω ) + (Ω 𝑄⁄ )
(1 − Ω ) +
Ω
𝑄
, 𝛺 =
𝜔
𝜔
The plot of the magnitude responses are shown below

32. 32

33. 33