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Ultra low power circuits require robust and reliable operation despite the unavoidable use of low currents and the weak inversion transistor operation region. For analogue domain filtering doubly terminated LC ladder based filter topologies are thus highly desirable as they have very low sensitivities to component values: non-exact component values have a minimal effect on the realised transfer function. However, not all transfer functions are suitable for implementation via a LC ladder prototype, and even when the transfer function is suitable the synthesis procedure is not trivial. The modern circuit designer can thus benefit from an updated treatment of this synthesis procedure. This paper presents a methodology for the design of doubly terminated LC ladder structures making use of the symbolic maths engines in programs such as M

Despite the rise of digital filters, analogue domain filters are still an essential part of many electronic systems; for example in high frequency circuits where high frequency and high resolution ADCs are ostly [

At the circuit level, many different analogue filter topologies are available, with cascades of second order sections [

Filters based upon LC ladder prototypes, where the filter transfer function is implemented by lossless inductor and capacitor elements connected between terminating resistances (see

Unfortunately, not all transfer functions can be implemented as doubly terminated LC ladder topologies. Even when these topologies are possible, the synthesis procedure—moving from the wanted

This paper re-visits and reviews the synthesis procedure for the generation of doubly terminated LC ladders. In particular we focus on the use of symbolic maths engines in programs such as M

Inevitably it is not possible to consider all possible design cases and issues in one article; instead a refreshed overview of the topic is provided. To keep the description here clear and concise the main body of the paper presents the detailed design of a non-trivial, high order, real-life LC ladder filter transfer function. The example used here takes the previously reported transfer function from [_{m}C

The remainder of this paper is structured as follows. Section 2 presents some prerequisites of circuit theory necessary for the synthesis procedure. This is to ensure that there is no confusion in terminology used in future sections. Section 3 then presents a design procedure for a singly terminated LC ladder network as the starting point to obtain a doubly terminated network. The generation of a doubly terminated topology is then given in Section 4. The derivation of both of the singly and doubly terminated topologies are given for the minimum capacitor case, and the minimum inductor counterpart is considered in Section 5. The topologies are also derived assuming terminating resistors of 1 Ω. Section 6 presents the component scaling to give values that are more realistic for on-chip implementation and Section 7 presents the gyrator substitution for the inductors which cannot be implemented on-chip. This gives an end _{m}C

An amount of terminology is essential for the LC ladder synthesis procedure. This will already be familiar to most readers, in which case it may be skipped, but nevertheless the needed terminology is re-defined here to ensure that only consistent and clear terminology is used in the remainder of the paper.

An even function is one that satisfies the condition

In general an even polynomial is of the form

Given a polynomial

A Hurwitz polynomial is one that has all of its poles in the open left hand plane of the

Given a transfer function

For a passive network _{12} = _{21}. It is also possible to define

In general a _{ij}_{ij}

For a given circuit it is possible to find the _{ij}

In general all four _{11} and _{22}_{12}_{12}_{1} = 0 they have no effect on the circuit being analysed and so do not appear in _{12}_{1} = 0 for the determination of _{12}_{1} ≠ 0 the shunt components will affect the nodal equations and so they do appear in _{11}. Poles which are present in _{11} and/or _{22}_{12}

The core constraints on an arbitrary transfer function to be realisable as an LC ladder structure are that its numerator must be a purely even or odd function of

Based upon classical synthesis procedures [_{in}

For simplicity a design is considered for the case _{s}

In comparison, the wanted transfer function, _{o}_{e}

Comparing

From this, analytic forms for the required ^{−3}
^{−8}
^{−6}
^{−5}
^{−4}
^{−3}
_{12} and _{11}^{−2}
^{−6}
^{−5}
^{−2}

From _{22}

The design procedure now proceeds by examining the form of _{11} and _{12}. _{11} has a private pole (one that is not present in _{12}), given by _{12} has six transmission zeros: one at

The key to LC ladder synthesis is realising the correct positions of all of these zeros using suitable arrangements of components and then choosing the values of the components so that _{11} is numerically satisfied. The realised _{12} can then differ by at most a constant gain factor from the wanted _{12}, which corresponds to the LC ladder matching the wanted filter shape, but not necessarily the gain. However, this can be readily compensated for by the presence of an ideal transformer, amplifier, or simply compensating for the expected values in the next part of any end system.

A transmission zero at the origin can be realised by one of the arrangements in ^{−1/2}. Also, it is possible to shift the position of zeros through the use of a partial pole removals [

To realise the network it is simply a matter of arranging the potential elements from

Ideally combinations of the networks illustrated in

For the Cauer 1 realisation the component values are given by a continued fraction expansion around infinity

Unfortunately, due to the arrangements of zeros needed a Cauer based network does not give a satisfactory result (see Section 3.4). We utalise an alternative method based upon the solution of non-linear simultaneous equations to provide the wanted network.

It is clearly seen that the network to be synthesised needs two elements from _{11} and so must be realised at port one of the network, not in the middle. A network that satisfies these requirements without any elements cancelling is given in

From these it can be seen that _{11} and _{12} are indeed of the correct form (have the same structure in terms of the number of poles and zeros). It is noted that due to the series connection of an inductor and capacitor to port two of the network, for the derived

From _{3} = _{4} to allow the equations to be written in a more compact form, the coefficients of the implemented

Comparing the coefficients of

Note that there are only seven component parameters, but eight equations. An exact solution is thus not possible. In general, the first equation,

The multiplicative simultaneous equations are readily solved using the M

If a starting guess that gives only positive component values cannot be found it is possible instead to view the solution of the equations as a constrained minimisation problem which can be solved via the

Alternative cost functions, such as using _{1} and Huber norms, rather than the _{2} norm, are also possible and could be used if desired, as could any desired optimisation technique. Ideally

Solving these equations for the singly terminated LC ladder gives the required component values. The best approximation found is given by
_{1}
=
24.1 mF
_{2}
=
60.6 mF
_{3}
=
48.6 mF
_{1}
=
364.5 mH
_{2}
=
123.4 mH
_{3}
=
18.3 mH
_{4}
=
400 mH

At this point it is noted that the alternative network illustrated in

The Cauer network from _{11} the private pole that it requires, but the series capacitor realising the second transmission zero at

All of these networks (with the series capacitor in series with each different inductor) will give the correct _{1}
=
133.3 mF
_{2}
=
68.5 mF
_{3}
=
504.1 mF
_{1}
=
400 mH
_{2}
=
18.3 mH
_{3}
=
48.3 mH
_{4}
=
13.4 mH

The resulting Bode magnitude plot is also shown in

To achieve the lowest sensitivity LC filter network a doubly terminated ladder, as illustrated in _{s}_{l}

Realising the correct transfer function thus now depends on realising the correct _{22} in addition to _{11} and _{12}.

It is possible to derive the required form of _{22} using the Feldtkeller relation [_{11} and _{22} can differ by at most one, due to the private pole at port 1. Also, to achieve the 7th order transfer function where all transmission zeros are at either zero or infinity the realisation of each

Thus there can be at most one component to be added to port 2 of the network to realise the required transfer function. Given the form of the singly terminated network (

A series inductor;

A shunt inductor;

A shunt capacitor;

No extra component required.

A series capacitor would simply alter the effective value of _{1} and so is not suitable. Given these four possibilities, by finding the resulting transfer function of each case, simple elimination shows that in the case considered here no extra components are required. The required form of _{22} is already satisfied, it is simply a matter of realising it numerically.

All four _{l}_{s}_{1}
=
15.3 mF
_{2}
=
60.9 mF
_{3}
=
50.0 mF
_{1}
=
500.7 mH
_{2}
=
135.6 mH
_{3}
=
19.1 mH
_{4}
=
384.7 mH

The resulting doubly terminated network is illustrated in

The network of

This is done by repeating the network synthesis procedure, but now in terms of the _{3} could be placed in parallel with any of the shunt capacitors and the same transfer function will be derived, subject to the satisfactory solution of the simultaneous equations. It is placed at the end of the network here to provide a direct dual to the minimum capacitance network of

Analysing the network of _{1}
=
384.7 mF
_{2}
=
19.1 mF
_{3}
=
135.6 mF
_{4}
=
500.7 mF
_{1}
=
50.4 mH
_{2}
=
60.9 mH
_{3}
=
15.3 mH

Note that in on-chip implementations inductors generally cannot be realised directly Instead they are simulated using gyrators (see Section 7). These are the only active elements in an otherwise passive network. The number of inductors is thus ideally minimised to reduce the power consumption, and so only the the minimum inductor topology of

The component values for the network of _{s}_{l}

To make the maximum capacitance value equal to 30 pF, readily achievable on-chip, the value of _{1}
=
23 pF
_{2}
=
1.14 pF
_{3}
=
8.1 pF
_{4}
=
30 pF
_{1}
=
841.2 MH
_{2}
=
1.016 GH
_{3}
=
255.4 MH

However, these inductor values are still not suitable for on-chip implementation, and have to be simulated with gyrators.

A final _{m}C

Many alternative gyrator circuits could potentially be used to replace the inductors in the circuit. For example, a floating inductor can be made with four single ended transconductors [

Here the three transconductor variation is used and is illustrated in

Unfortunately, the network form of _{s}_{l}_{1}
=
31.6 mF
_{2}
=
24.8 mF
_{3}
=
148 mF
_{4}
=
39.4 mF
_{1}
=
187.8 mH
_{2}
=
260.4 mH
_{3}
=
104.8 mH

This topology provides both the required transfer function (see _{2} and _{2}. The high

Substituting the identities into the network of _{m}C_{1}
=
3.16 pF
_{2}
=
2.48 pF
_{3}
=
14.8 pF
_{4}
=
3.94 pF
_{L}_{1}
=
18.8 pF
_{L}_{2}
=
26.0 pF
_{L}_{3}
=
10.5 pF

The Bode magnitude response of the _{m}C_{m}C

Filters based upon ladders of inductors and capacitors show inherently low sensitivity to non-exact component values; an essential design parameter for low power devices where robust operation is mandatory. For example, battery powered physiological monitors must be reliable, and must also operate at the very low frequencies associated with bodily phenomena whilst consuming very little power. Low frequency operation results in the presence of significant parasitic components and design in weak inversion for low power results in increased levels of device current mismatch, both making low sensitivity circuit design key.

Through the use of a detailed, high order, design example this paper has reviewed the procedure of LC ladder synthesis, making use of symbolic maths engines that can solve the non-linear simultaneous equations. Inevitably, in the space available it is not possible to consider and address all possible design issues that would potentially be of interest, but a clear overview has been provided. To conclude, the procedure considered in this article is summarised in

A generalised doubly terminated LC ladder filter. Doubly terminated LC ladder filters consist of inductors (_{s}_{l}

Two singly terminated two port networks with the output (

Realising transmission zeros. (^{−1/2}.

In principle two cascaded capacitors realise two transmission zeros but do not in practice.

Foster and Cauer network realisations. These allow simple determination of the required component values by continued and partial fraction expansions.

A prototype network with the desired

The final singly terminated LC ladder filter with component values.

Bode magnitude responses for the singly terminated LC ladder networks. The proposed network (

(

A doubly terminated two port network with voltage mode transfer function.

A minimum capacitor doubly terminated LC ladder realising the desired transfer function.

Bode magnitude responses for the doubly terminated LC ladder networks. All the proposed networks match the wanted response well: the minimum capacitor network (_{m}C

A minimum inductor 5th order Cauer low pass network.

A minimum inductor doubly terminated LC ladder realising the desired transfer function.

Gyrators simulating inductors [

The final doubly terminated LC ladder implementing

_{m}C

Summary of the core procedure used in this article to aid in the design of other transfer function cases.

1 | 3.1 | From the wanted transfer function, _{o}_{e} |

2 | 3.1 | Hence find the required forms of _{11} and _{12}. For even |

3 | 3.2 | From _{11} identify any poles private to port 1 of the required network. These are poles present in _{11} but not in _{12}. |

4 | 3.2 | From _{12} identify the transmission zeros required in the network. The positions of these transmission zeros are given by the zeros of _{12}. In addition, if the denominator of _{12} is order |

5 | 3.2 | Select suitable elements from |

6 | 3.2 | Arrange these elements to form a ladder structure. Any private poles from _{11} must be placed at port 1 of the network. Otherwise the elements can be placed in any order provided there are no two capacitors directly in series or inductors in parallel. Note that each alternative network may have different properties, for example in terms of the component values required. It may be desirable to investigate the properties of more than one potential network in the following steps. |

7 | 3.3 | For the ladder structure formed, perform nodal analysis and find the analytical expressions for the _{11} and _{12} parameters. Ensure that these have the same form (have the same structure in terms of the number of poles and zeros) as the parameters derived in step 2. |

8 | 4 | Find _{22} for the ladder structure formed. Assuming that _{s}_{g} |

9 | 3.3, 4 | Compare the form of the wanted |

10 | 4 | _{22}. Additional components from |

11 | 3.3, 4 | There are now two transfer functions: |

12 | 6 | Scale the terminating resistances (_{s}_{l} |

13 | 7 | The LC ladder has now been formed and the required component values found. This is then used to give the final doubly terminated network. If required, gyrators can be used to form an inductor-less circuit. Some example gyrators are given in |