UNIVERSITY OF LAGOS

ELECTRICAL AND ELECTRONICS ENGINEERING

DEPARTMENT

EEG 407/416 ACTIVE NETWORKS

Dr. M. A. K. Adelabu
 FILTER CIRCUITS
• Filters are generally transmission circuits defined
with transfer function. H(s)
– Attenuation function

– Amplification (Gain) function

–

• H(s) is a complex function characterized with

magnitude and phase functions.

Active Networks ......Dr. M. A. K. Adelabu 2

 FILTER DESIGN
This requires determining the (amplitude or phase
characteristic) function of the desired filter type.
Usually accomplished with one of the hitherto developed and
documented approximation functions. The inputs for these
functions are:
• Maximum allowed attenuation(loss) of the function in the
filter’s pass band
• Minimum required attenuation (loss) of the function in the
filter’s stop band.
• Characteristic passband and stopband edge frequencies.
Approximation functions include:
- Amplitude functions: Bode plot, Butterworth and
Chebyshev polynomial functions; and Cauer real function.
- Phase functions: Bessel real function and delay equalizers.
Active Networks ......Dr. M. A. K. Adelabu 3
 NETWORK FUNCTIONS

• For synthesis, the network (real) function is often

represented in its factored form:

• m > n, then m is the order of filter

• Magnitude function is

• Phase function is

Active Networks ......Dr. M. A. K. Adelabu 4

 SPECIFIC EXAMPLES

• m ≤ 1 or n ≤ 1, m = n = 1; a first order filter

H(s) is a bilinear function

• m ≤ 2, n ≤ 2 or m = n = 2; a second order
filter, H(s) is a biquadratic function.

Active Networks ......Dr. M. A. K. Adelabu 5

 APPROXIMATION FUNCTION
• This produces a polynomial function to realise
normalised low pass filter functions. The
characteristic parameters are;
• ω = ωp pass band frequency
• ω = ωs stop band frequency
• ε - deviation error between desired and
approximated function
• n – order of function required to achieve desired
filter.

Active Networks ......Dr. M. A. K. Adelabu 6

 BUTTERWORTH APPROXIMATION FUNCTION
• The LPF function assumes the form

Where and

H(s) is a loss function. At frequencies ω  0

The normalized loss function is obtained as

where normalized frequency is

Filter polynomial is expressed as

And its coefficients are tabulated for the normalized low pass filter.
Buttherworth filters have maximally flat response at dc which zero(minimal) attenuation.
The attenuation increases steadily until it reaches infinite (theoretically) value.

Active Networks ......Dr. M. A. K. Adelabu 7

 CHEBYSHEV APPROXIMATION FUNCTION
• This also applies polynomial function to approximate
the required filter function. In this case the set of
Chebyshev polynomials defined as follows are used
to generate the coefficients of the function.

and normalized frequency is

The low-pass approximation function admits the
function

Higher order Chebyshev polynomials are generated

with recursive equations.
Active Networks ......Dr. M. A. K. Adelabu 8
CHEBYSHEV APPROXIMATION FUNCTION

Active Networks ......Dr. M. A. K. Adelabu 9

 CAUER (ELLIPTIC) APPROXIMATION

• This is the third of the magnitude approximation

functions. Unlike the first two, to obtain the
denominator of H(s), a rational function R(s) is
used in place of a polynomial P(s). The most
applied functions are Chebyshev rational
functions Rn(ω), that are realized with the use of
functions satisfying elliptical integrals.
• These function have similar equi-ripple
characteristics as Chebyshev filters, in this case, in
both pass and stop bands.

Active Networks ......Dr. M. A. K. Adelabu 10

 CAUER (ELLIPTIC) APPROXIMATION
• The transfer function of elliptic filters takes the form

• Where

• Values of the coefficients are tabulated as for Butterworth and

Chebyshev filters.
• Items required to characterize an elliptic filter function are:
i. Order of filter n
ii. Passband ripple constant Kp (e.g 1.0dB )
iii. Stopband tolerance constant Ks
iv. Stopband frequency ωs
• Any two of items (ii) to (iv) will uniquely define the filter of any
order.
Active Networks ......Dr. M. A. K. Adelabu 11
 COMPARISON OF MAGNITUDE FUNCTIONS
• Characteristic:
– Buttherworth: maximally flat at dc and in the pass band and
stop band; gets poorer as ω approaches ωp; has least
attenuation in stop band amongst the three.
– Chebyshev: has uniformly distributed (equi-ripple) attenuation
in passband and flat response in stop band; has greater
attenuation than Butterworth in stop band.
– Cauer: has uniformly distributed (equi-ripple) in both pass and
stop bands; has greatest attenuation in stop band of the three.
• Function:
– Butterworth and Chebyshev defined with polynomials, while
Cauer with real functions.
• Coefficients: Normalized low pass filter values tabulated for
the three of them. Real values are obtained after
denormalisation.

Active Networks ......Dr. M. A. K. Adelabu 12

 FREQUENCY TRANSFORMATION

• To obtain network functions for other filter

types, frequency transformation is carried out.
This is equivalent to mapping the filter
characteristic from the desired form to low
pass equivalent, obtain the network function
and carry out reverse transformation.

HP, BP, BS filter LP Normalized HHP (s)

requirements Equivalents LP HBP (s)
H(s) HBS(s)

Active Networks ......Dr. M. A. K. Adelabu 13

 FREQUENCY TRANSFORMATION

High Pass Filter HP(ω) Low Pass Filter (Normalized) LP(Ω)

• HP LP LPN HP

• dc ꝏ
• ꝏ dc

Active Networks ......Dr. M. A. K. Adelabu 14

 FREQUENCY TRANSFORMATION
Band Pass Filter HBP(s) Low Pass Filter HLP(S)

Active Networks ......Dr. M. A. K. Adelabu 15

 FREQUENCY TRANSFORMATION

Band Reject Filter HBR (s) Low Pass Filter HLP(S)

• This is inverse of the case LP equivalents
of band pass filter.

ω0 ꝏ
ω1 –1

ω2 +1

ω3

ω4

Active Networks ......Dr. M. A. K. Adelabu 16

 PHASE APPROXIMATION FUNCTIONS

• Phase characteristics are associated with

transmission of signals where there is need to
conserve the shape of the waveform, ex. pulse
shape. This is often associated with the time
delay experienced by such signal in passing
through a network. One type of filters that
meet this criterion has function that admits
representation with the Bessel recursive
function, hence they are called Bessel filters.

Active Networks ......Dr. M. A. K. Adelabu 17

BESSEL FILTERS

Active Networks ......Dr. M. A. K. Adelabu 18

CLASS WORK/ASSIGNMENT

Active Networks ......Dr. M. A. K. Adelabu 19

 UNIVERSITY OF LAGOS
ELECTRICAL AND ELECTRONICS ENGINEERING
DEPARTMENT

EEG 407/416 ACTIVE NETWORKS

Dr. M. A. K. Adelabu
 FILTER SYNTHESIS
Process of realizing circuit configuration from given
network function. Filters can be realized as
– Passive - use of lumped or distributed passive
elements; R, C, L (M), transmission line fragments.
– Active – use of passive elements with amplifying
devices.
• Passive with inductance simulation/emulation
• R,C with amplifying devices.
– Switched /Digital – use of switching (gating) elements,
amplifying devices and some passive elements
– Digital/Software – software routines with processing
and memory devices.
2
 METHODS OF PASSIVE NETWORKS’ SYNTHESIS

• Inspection
• Driving Point Synthesis
– Partial fraction expansion (Foster method)
– Continued fraction expansion (Cauer method)
• Transfer Function Synthesis
– Singly terminated ladder networks
– Zero shifting technique
– Doubly terminated networks

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 INDUCTANCE SIMULATION TECHNIQUES
• Network elements used to simulate
inductances
– Generalized Immitance Converter (GIC)
– Generalized Immitance Inverter (GIV)
– Gyrator
– Frequency Dependent Negative Resistance (FDNR)

4
 DIRECT SYNTHESIS OF ACTIVE NETWORKS
• Use of RC elements with amplifying devices.
– Transistorized amplifiers
– Operational amplifiers
• Switched capacitor ampli-filters.

5
 NETWORK FUNCTIONS
• As stated earlier, the basic network (transfer)
functions for realization are in bilinear or
biquadratic forms.
• Bilinear function.

• Biquadratic function.

6
 DIRECT SYNTHESIS OF ACTIVE NETWORKS
• Cascaded topology
• Coupled topology
• Advantages of cascaded topology
– Individual biquads/bilinear are isolated from each
other, such that change in one circuit does not affect
its neighbour(s). This allows for easy tuning.
– Higher order function is reduced to a product of
biquads/bilinears for realisation.
• Coupled topology are usually complex without
isolation between blocks, thus, making tuning
more difficult. Advantage is its usually lower
sensitivity.

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 TOPOLOGIES

T1, H1(s) T2, H2(s) TN, HN(s) VOUT

VIN

Cascaded network

VIN Σ Σ VOUT

Hi(s)
Coupled network

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 BILINEAR REALISATION
• Inverting Op-Amp • Non -inverting Op-Amp

Z2
+

_
VIN
Z1 _
VOUT
Z2
VIN +
Z1
VOUT

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 BIQUAD REALISATION

VS RC VIN VOUT

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 BIQUAD REALISATION
POSITIVE FEED BACK TOPOLOGY NEGATIVE FEEDBACK TOPOLOGY

Zeros of the feedback network

determine the poles of the
• Zeros of transfer function transfer function.
are the zeros of the Zeros of the feedforward
feedforward RC network, network determine the zeros
which can be complex. of the transfer function.
• Poles of the transfer Poles and zeros can be complex.
function can be located However, for a stable
network, poles cannot lie in
anywhere in the left half of the right –half of s plane.
s plane, being determined Poles of RC network do not
by the poles of RC network contribute to the transfer
and the factor k function.

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 REALISATION OF NETWORK ELEMENTS

• Coefficient matching
• Adjustment of gain constant
• Impedance scaling
• Frequency scaling

12
EXERCISES

13
UNIVERSITY OF LAGOS
ELECTRICAL AND ELECTRONICS
ENGINEERING DEPARTMENT

EEG 407/416 ACTIVE NETWORKS

Dr. M. A. K. Adelabu
madelabu@unilag.edu.ng; 0808 –717—1607
INTRODUCTION
Electronic network;
 required to transfer or transmit signals from point to
point.
 classified as passive or active and more recently as
analogue and digital
 Can be source, load or transmission network
 source ;- no signal input port, only an output port
 Load;- only an input port, no output port
 Transmission network – both input and output ports

2
INTRODUCTION
Passive networks - consist mainly of passive components
R, C, L/M
Active networks – consist of both passive and active
networks
Switched (Digital) networks – operate on discrete signals
and principles
Network Analysis - used to determine circuit’s or
network’s characteristics.
Network Synthesis – used based on the knowledge of
their characteristics to realise circuits or networks for
desired applications.
3
NETWORK FUNCTION
Network function
𝑎𝑛 𝑠 𝑛 + 𝑎𝑛−1 𝑠 𝑛−1 + ⋯ … + 𝑎1 𝑠 + 𝑎0
𝐻 𝑠 = 𝑚 𝑚 −1
; 𝑎𝑛 ≠ 0; 𝑏𝑚
𝑏𝑚 𝑠 + 𝑏𝑚 −1 𝑠 + ⋯ . +𝑏1 𝑠 + 𝑏0
≠0
ai (i = 1,2,……n) and bj (j = 1,2,……m) are real coefficients.
In factored form;
𝑎𝑛 𝑠 − 𝑧1 𝑠 − 𝑧2 … … … … (𝑠 − 𝑧𝑛 )
𝐻 𝑠 =
𝑏𝑚 𝑠 − 𝑝1 𝑠 − 𝑝2 … … … (𝑠 − 𝑝𝑚 )
Where zi are zeros since 𝐻(𝑠)𝑠= 𝑧 𝑖 = 0 and pi are poles
4
where 𝐻(𝑠)𝑠=𝑝 𝑗 = ∞
NETWORK FUNCTION
PROPERTIES
Poles and zeros often plotted on the s-plane where 𝑠 = 𝜎 + 𝑗𝜔
jω
x o x – pole o - zero
x o
x o σ
x o
x o

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NETWORK FUNCTION PROPERTIES
 GENERAL – for passive and all stable active
networks
 H(s) must be a rational function of s with
real coefficients.
 H(s) may not have poles on the right half of
s-plane
 H(s) may not have multiple poles along the
jω axis.

6
NETWORK FUNCTION
PROPERTIES

 Driving Point (DP) functions

 May not have poles or zeros in the right half
of s-plane
 May not have multiple poles or zeros on the
jω axis
 Degree of numerator and denominator
polynomials can differ by not more than 1;
that is n - m ≤ 1 or m – 1 ≤ 1
7
NETWORK FUNCTION PROPERTIES

 Passive RLC DP
 All properties of DP functions
 Must have a non-negative real part for H(s)
along s = jω
 Must have positive and real residues for
poles along the jω axis

8
NETWORK FUNCTION PROPERTIES
 Passive RC DP impedance functions
satisfy all conditions of RLC function and
 Their poles and zeros lie on the negative real
(σ) axis and alternate.
 At DC, the function is either a positive
constant or has a pole.
 At infinity, the function is either a positive
constant or zero.
 Residues of ZRC(s) are real and positive
 Slope of is negative
9
NETWORK FUNCTION PROPERTIES
 PassiveRC DP admittance functions satisfy all
conditions of RLC and
 Their poles and zeros lie on the negative real (σ)
axis and alternate.
 At DC, the function is either a positive constant or
has a zero.
 At infinity, the function is either a positive constant
or pole.
 Residues of YRC(s) are real and negative
 Slope of YRC(s)/s is real and positive
 Slope of is positive.
10
NETWORK FUNCTION
PROPERTIES
 Passive
LC DP functions satisfy properties of
RLC functions and
 Poles and zeros lie on jω axis, are simple and
alternate
 Function has a pole or zero at dc
 Function has a pole or zero at infinity
 Function must be odd/even or even/odd
 Slope of is positive.

11
NETWORK FUNCTION
PROPERTIES
 TransferTX functions – RLC passive or
RC active satisfy the general properties
and
 Complex poles and zeros occur in conjugate
pairs
 Gain TX functions

 Have no poles in the right half s-plane

 Poles along jω axis are simple.
 Zeros can be anywhere in the s-plane.

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NETWORK TYPES
 Filters – passive are characterized by attenuation as
the output signal is usually of less value than input.
Active filters can have unity or higher gain
characteristic.
 Filters are mainly used to achieve required frequency
characteristics.
 Classified as
 Low pass (LPF); f ϵ { 0 , fmax}
 High pass (HPF); f ϵ { fmin, ꝏ}
 Band pass (BPF); f ϵ { 0 , [fmin, fmax ], ꝏ}
 Band stop/reject (BSF or BRF); f ϵ {[0, fmin], [fmax , ꝏ]}
 Notch (APF); f ϵ {[ 0, fmin], [fmax , ꝏ]}; fmin ≡ fmax
13
PASSIVE –ACTIVE COMPARISON
 Advantages of Active Filters
 Reduction in size and weight
 Increased reliability
 Reduced cost due to ever decreasing cost of ICs and other
active components.
 Improved performance with high quality components.
 Reduction in parasitic influences
 Simpler design process
 Realisation of wider class of functions
 Possibility of gains or lossless transmission.
 Disadvantages
 Finite bandwidths
 Require power supply
 Higher sensitivity to variations in element values. 14
CLASS WORK

15