ACTIVE FILTER SYNTHESIS BASED ON MESH CURRENT
EMULATION OF LC LADDER STRUCTURES
Andrei Câmpeanu
Universitatea "Politehnica" Timişoara, Departamentul de Comunicaţii, Bv. V. Parvan Nr.2, Romania,
email: andrei.campeanu@etc.utt.ro
ABSTRACT
In a series of papers, [1]-[3], A. Câmpeanu and M. Naforniţă proposed a new active RC filter synthesis
method based on mesh current emulation of LC ladder filter networks. Initially, the method was
successfully applied to highpass and lowpass polynomial filters. This paper extends the method to the
very important classes of lowpass and highpass finite-zeros (elliptic) filters.
Based on the mesh current method, simple and systematic design rules are proposed, which permits a
direct active RC filter implementation of the passive network. It is also outlined a method of reducing
the number of amplifiers required. Some results obtained using circuit simulation show the efficiency
of the method and the importance of the results.
1.
INTRODUCTION
Because of their low-sensitivity properties, the double terminated lossless ladder filters are very
suitable to be implemented as RC-active filters. The most used approach to the design of active filters
based on LC ladder simulations is based on simulating the current voltage relationships existing in the
LC-ladder prototype, [4]-[6]. This paper is specifically concerned with this kind of approach.
A. Campeanu and M. Naforniţă [1]-[3], have given a procedure to derive an RC-active filter from the
passive ladder network using the mesh currents description of the passive circuit. The proposed
method have used as building blocks, modified biquad resonator filter cells with multiple inputs to
implement lowpass and highpass RC polynomial active filters. Here we propose an extension of the
mesh currents simulation method to lowpass and highpass finite-zeros filters. For this very important
case, most proposed RC-active filter synthesis methods [6]-[7] for emulating high-order LC ladder
filters are subject to either complicated design procedures or extra numbers of active elements. Mesh
currents emulation active filters have the advantages that we can realize every loop in the original
ladder prototype by a specific multiple-inputs multiple-outputs RC-active cell. Furthermore,
systematic design rules and equivalencies that simplify the design procedures are established. The
multiple-inputs multiple-outputs RC-active cells are implemented using conventional summation and
integration devices, that consist only of op-amps with grounded inputs. These circuits are insensitive
to parasitic input capacitances of op-amps and are convenient for electronic control of parameters
2.
CIRCUIT TOPOLOGIES
The procedure for the mesh currents emulation method starts by taking an LC-ladder prototype
network and slicing it into individual loops. We describe the work of the passive network presented in
Fig. 1 using mesh currents that flows in the loops of the network: I1 , I 2 , …, I n . Kirchoff’s theorem
applied in any of the network loops gives:
V
Z
Z
(1)
I k = kl + 2 kl−2 I k −1 + 2lk I k +1
Zk
Zk
Zk
Fig. 1 Using mesh currents to describe the work of a passive ladder filter
where Z kl is the mesh impedance of the k-th loop. Vk represents the value of voltage sources in the
loop. Only for k = 0, Vk = E ≠ 0 and I k −1 = 0 . For the last mesh of the network, I k +1 = 0 .
According to (1), the ladder filter network can be implemented using RC active circuits cells with
multiple inputs. Moreover, as the denominators of all the terms on the right side of (1) are identical,
Z kl , a unique active circuit structure with multiple inputs can be used to implement the desired
functions. As will be shown in the next Section, linked to the position of the loop in the ladder, the
implementations are first or second order RC active circuits having multiple inputs and outputs.
Fig. 2 π-shaped passive filter structure: a. lowpass filter, b. highpass filter.
Previous papers, [1]-[3], used T-shaped LC ladder structures to synthesize RC-active filters. This
approach gives the advantage of using a unique active cell repeatedly for each loop of the passive
structure. Extending the method to the case of finite-zeros filters, the previous approach becomes
unsatisfactory because in a single circuit loop there are as far as two arms with LC resonant series
circuits. A better solution is to use π-shaped filter structures (Fig. 2), because in this case each circuit
loop has no more than a single LC resonant parallel circuit. This second approach is used in this paper
unless it gives different transfer functions and different active filter cell structure for almost every loop
of the original ladder prototype. Furthermore, in the next section, systematic design rules are
established to simplify design procedures.
3.
MESH CURRENTS SIMULATION OF LC LADDER LOOPS
The examination of π-shaped double terminated passive filter structure has identified 2 distinct basic
types of circuit loops. Based on the mesh currents description of a certain loop, the structures of
equivalent RC-active cell with op amps were established. Instead of the original currents I k , the RCactive cell work is described by voltages Vk having the same subscripts and superscripts as the
original currents.
In the case of network loops having on series arms LC parallel circuits, the need of keeping the
number of op amps in the synthesized cell to a minimum, imposes the separation of the loop current,
I k in two terms, the first with a highpass behaviour, I k′ , the second one, I k′′ behaving like a lowpass
filter:
I k = α k I k′ + I k′′
(2)
where α k is a constant for a given loop.
According to (2), the outputs of the corresponding RC-active cell will satisfy a similar equation:
Vk = α kVk′ + Vk′′
(3)
Actually, the transposition of (3) in the active circuit cell means the use of two separate outputs for the
same cell: a highpass output, Vk′ , and a lowpass output, Vk′′ . Finally, the multiplication with α k and
the addition of the two terms in (3) is made in the input stage of the adjacent cells which use as input
signal the result of the equation. Consequently, the op amp needed to implement (3) is spared.
Next, the first and second order RC-active cells used to emulate the passive circuit counterparts are
presented.
A.
First order transfer function ladder network loops synthesis
The input and output terminations of π-shaped passive ladder network lowpass and highpass filters
present a first order transfer function between the mesh current of the loop and the voltage source from
the loop and the mesh current of the adjacent cell. For the output terminations, this is true only in the
case of odd-order filters. Fig. 3 presents the four different cases that lead to first order transfer
functions.
Fig. 3 First order transfer function passive filter ladder loops: a. and b. input and output terminations
of a lowpass filter, c. and d. input and output terminations of a highpass filter
As example, the mesh current I1 on the input termination of a π-shaped passive highpass filter (Fig.
3.c.) has the expression
E r1
s l2 r1
(4)
I1 =
+
I2
s l2 r1 + 1 s l2 r1 + 1
There are two distinct behaviours in the same expression: a lowpass transfer function given the voltage
E and a highpass transfer function relative to the mesh current of the adjacent cell, I 2 . A similar case
occurs for the lowpass circuit in Fig. 3. a. The circuits in Fig. 3. b. and d. have simpler first order
behaviours: only lowpass for the first of them, respectively only highpass for the second one.
Fig. 4 a. The active cell used to emulate first order transfer function passive filter ladder loops, b.
Symbol of the active filter cell
The active filter cell that implements equivalents first order transfer functions is presented in Fig. 4. a.
It uses only simple op-amp building blocks with grounded inputs: two adders and an integrator. To
enable the addition of the two output voltages given by a neighbouring active filter cell, the circuit has
two lowpass and two highpass inputs. The symbol used to represent the active cell is shown in Fig.
′
4.b. As in the next block symbols presented, the notations ×α LP and ×α HP , used on the inputs −VLP
′ , indicate that the output −V2′ of the adjacent cell will be multiplied at one of these inputs by
and −VHP
the suitable coefficient.
Returning to the input termination of a π-shaped passive highpass filter in Fig. 3. c., the design
relationships for the parts in the equivalent active circuit are very easy to establish. Choosing the value
of the resistor R , the other parts in the schematic are:
l
′′ = R, RHP
′ = R α 2 , RHP
′′ = R
(5)
C = 2 , RLP
r1 R
′ remains unused and the coefficient α 2 is established when the adjacent active cell is
The input −VLP
computed.
B.
Second order transfer function ladder network loops synthesis
For the π-shaped passive lossless filters, there are four different topologies for the interior network
loops that leads to a second order dependency between the mesh current of the loop and the mesh
currents of the adjacent loops (Fig. 5).
Fig. 5 Second order transfer function passive filter ladder loops: a. and b. interior loops of lowpass
and highpass polynomial filters, c. and d. interior loops of lowpass and highpass finite-zeros filters
The circuits represented in Fig. 5. a. and c. are used in polynomial lowpass and highpass filter
networks. They have a single reactive element on each loop of the mesh. The mesh current I k is
expressed in these cases by a lowpass, respectively a highpass undamped second order transfer
function of adjacent loops currents, I k −1 and I k +1 . The interior loops of finite zeros low-pass and
highpass filters include on the series arms LC parallel circuits as it show Fig. 5. b. and d. Their second
order transfer functions are undamped and exhibit two imaginary finite zeros in addition to a pair of
imaginary poles. In the case of lowpass finite-zeros filter loop (Fig. 5. b), the mesh current I k has an
expression which allow a very convenient decomposition in two terms like in (2): the first, I k′ , with a
highpass behaviour, the second one, I k′′ , with a lowpass behaviour.
Ik =


s 2 c2 k −1l2 k −1 + 1
c2 k
c2 k − 2
I k −1 +
I k +1  = α k I k′ + I k′′

c
+
c
c
+
c

c c 

2k − 2
2k
s 2  c2 k −1 + 2 k − 2 2 k  l2 k −1 + 1  2 k − 2 2 k
+
c
c
2k − 2
2k 

(6)
where

c c 
s 2  c2 k −1 + 2 k − 2 2 k  l2 k −1
c


c2 k
c2 k − 2
c2 k −1
2 k − 2 + c2 k 

,
I k′ =
I k −1 +
I k +1  , α k =

c2 k − 2 c2 k
+
+
c
c
c
c


c
c
2
2k −2
2k
2k −2
2k


2 k −2 2k
+
c
s  c2 k −1 +
2 k −1
 l2 k −1 + 1
c2 k − 2 + c2 k
c2 k − 2 + c2 k 

I k′′ =
1

c c 
s 2  c2 k −1 + 2 k − 2 2 k  l2 k −1
c2 k − 2 + c2 k 

(7)


c2 k
c2 k − 2
I k −1 +
I k +1 

c
+c
c2 k − 2 + c2 k

+ 1  2k −2 2 k
It is very simple to prove that the output voltages Vo′ and Vo′′ of the RC-active circuit in Fig. 6. a.
perform identical behaviours with regard to the input voltages as the passive circuit:

s 2 R 2C 2  R
R
R
R
V0′ = 2 2 2
 − ( −Vi′) − ( −Vi′′) − ( −V j′ ) − ( −V j′′) ,
s R C + 1  Ri′
Ri′′
R ′j
R′′j

(8)


R
R
R
R
1
V0′′ = 2 2 2
 − ( −Vi′) − ( −Vi′′) − ( −V j′ ) − ( −V j′′)
s R C + 1  Ri′
Ri′′
R ′j
R′′j

Since the complete synthesis of the function in (6) needs a supplementary amplifier to multiply Vo′ by
α k and then to add the result to Vo′′ , the solution, which keeps at a minimum the number of op-amps,
consists to transfer this operation to the input stages of the adjacent filter cells.
Fig. 6 a. The active cell used to emulate second order transfer function of passive filter ladder loops, b.
Symbol of the active filter cell
Finally, the emulation of the passive filter loop in Fig. 5. b. by the active filter cell in Fig. 6. a. uses the
following relationships:
Ri′ =
c2 k −2 + c2 k
R,
c2 kα k −1

c c
l2 k −1  c2 k −1 + 2 k −2 2 k
c2 k − 2 + c2 k
+c
c

Ri′′ = 2 k − 2 2 k R, C =
c2 k
R



,
(9)
+c
c2 k − 2 + c2 k
c
R, R ′′j = 2 k − 2 2 k R
c2 k − 2α k +1
c2 k − 2
In (9), we suppose the value of R being chosen.
The second order active filter cell in Fig. 6. a. is used successfully to implement all the passive filter
loops in Fig. 5. If the passive filter is polynomial, only one of two active filter cell outputs is used: the
lowpass output, Vo′′ , for the circuit in Fig. 5. a. and the highpass output, Vo′ , for the circuit in Fig. 5. c.
Finally, the finite-zeros highpass ladder network filter in Fig. 5. d. reveals the same transfer function
as its lowpass counterpart in Fig. 5. b.
R ′j =
4.
DESIGN PROCEDURE AND SIMULATION RESULTS
The design procedure of an RC-active filter based on the mesh currents emulation method is
straightforward. First, divide the original ladder prototype in distinct loops and choose appropriate
filter cells among the cells introduced in the previous Section. Next, connect neighbouring cells with
appropiate interconnections. Finally, determine the values of resistors and capacitances via the derived
design relationships.
To demonstrate the flexibility of the proposed active filters synthesis, a fifth-order LC elliptic lowpass
filter is realised firstly. Its prototype is shown in Fig. 7.a., having 0.1dB ripple in the passband and
minimum 43dB attenuation in the stopband. The bandwidth of the active filter was 50 kHz.
Fig. 7. b. shows the implementation of the LC ladder network filter at the active filter cell level. The
synthesis employs a total number of 12 op amps. The simulation of the active circuit employed
LT1252 (LINEAR TECHNOLOGY) op amps and gave the result shown in Fig. 7. c.
Fig. 7 a. Fifth-order LC lowpass filter prototype, b. The corresponding RC-active filter derived
from the prototype on the basis of mesh currents emulation method, c. Amplitude response of the
active lowpass filter
The second design case refers to a third-order elliptic highpass filter. The LC π-shaped passive
prototype is shown in Fig. 8. a. It has 0.5 dB ripple in the passband and at least 35 dB attenuation in
the stopband. The bandwidth of the active filter is 10 kHz. The active filter implementation, shown in
Fig. 8. b. employs also LT1252 op-amps.
Fig. 8 a. Third-order LC highpass filter prototype, b. The corresponding RC-active filter derived
from the prototype on the basis of mesh currents emulation method, c. Amplitude response of the
active highpass filter
5.
CONCLUSIONS
A new passive filter implementation based on mesh currents emulation of LC ladder networks was
introduced for finite-zeros lowpass and highpass active filters. The active filter is obtained by the
adequate interconnection of 2 different types active cells. Design principles and procedures jointly
with the results of circuit simulations are also presented. In conclusion, the mesh currents emulation of
LC ladder networks represents a very powerfull method for realising active filters.
6.
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