Network functions: analysis and synthesis
The same impedance can be built with series LC tank circuits or with a ladder of inductors and capacitors. The two circuits look nothing alike and are indistinguishable from their terminals.
01Network functions
A network function is the ratio of two transformed quantities of a circuit with zero initial conditions. If both are taken at the same pair of terminals, it is a driving-point function: an impedance \( Z(s) \) or an admittance \( Y(s) \). If they are taken at different points, it is a transfer function: a voltage gain, a transimpedance. For circuits of lumped components, any of them is a ratio of polynomials in \( s \) with real coefficients, and is determined, up to a constant, by its poles and zeros.
Analysis starts from the circuit and obtains the function; synthesis takes the reverse path: given the desired function, it finds a circuit that realizes it. It is the basis of filter design.
02What functions a passive circuit can have
Not every ratio of polynomials is the impedance of a circuit of resistors, capacitors and inductors. It has to be a positive real function: real for real \( s \), and with a nonnegative real part throughout the right half-plane. That has consequences that can be checked at a glance:
- The poles and zeros are in the left half-plane or on the imaginary axis.
- Those on the imaginary axis are simple and have real, positive residues.
- The degrees of the numerator and the denominator differ by at most one.
- On the \( j\omega \) axis, the real part (the resistance) is never negative: a passive circuit does not deliver energy.
For a transfer function the requirement is milder: it is enough for it to be stable, with the poles in the left half-plane; the zeros can be anywhere.
03LC networks: Foster’s reactance theorem
A network of only inductors and capacitors does not dissipate energy, so its impedance is purely imaginary on the \( j\omega \) axis: \( Z(j\omega) = jX(\omega) \). Foster proved that the reactance of any LC network always increases with frequency, and that therefore its poles and zeros alternate along the imaginary axis, starting and ending at zero or at infinity:
04Four canonical forms
The same LC impedance can be realized in four ways with the minimum number of elements:
| Form | Method | Circuit |
|---|---|---|
| Foster I | Partial fractions of \( Z(s) \) | Series connection of an inductor, a capacitor and parallel LC tank circuits, one for each finite pole |
| Foster II | Partial fractions of \( Y(s) \) | Parallel connection of an inductor, a capacitor and series LC branches |
| Cauer I | Continued fraction removing poles at infinity | Ladder of series inductors and shunt capacitors |
| Cauer II | Continued fraction removing poles at the origin | Ladder of series capacitors and shunt inductors |
The Cauer continued fraction is obtained by dividing: the quotient is a series inductor; the remainder is inverted, divided again, and the new quotient is a shunt capacitor; and so on until no remainder is left:
You choose an LC impedance (with normalized values: henry, farad and rad/s). The chart shows its reactance on the \( j\omega \) axis, with the zeros (dots) and the poles (dashed lines); below, the elements of the Foster I and Cauer I forms, which give the same impedance.
05Beyond LC networks
RC networks have analogous properties, with poles and zeros alternating along the negative real axis, and are synthesized with the same methods. Transfer functions are realized with doubly terminated ladders, between the generator resistance and the load resistance, which is exactly what the passive filters of analog filters do. Today synthesis is done by design software, but its results can only be understood with these methods.
06In the lab
With an impedance analyzer, measure the reactance of a four-element LC network between 1 kHz and 100 kHz and locate its poles and zeros. Verify that they alternate.
Build the Foster I and Cauer I forms of the same impedance, denormalized to 10 kHz and 1 kΩ, and compare their measurements.
Synthesize in Cauer II form the impedance \( (s^2 + 1)(s^2 + 3)/(s(s^2 + 2)) \) and verify the result.
07Common mistakes
- Trying to synthesize a function that is not positive real. The passive circuit does not exist.
- Confusing Cauer I with Cauer II. The first removes poles at infinity; the second, at the origin.
- Forgetting to denormalize. The synthesis values are in henries and farads for 1 rad/s and 1 Ω.
- Expecting the reactance of an LC network to decrease somewhere. By Foster, it always increases.
08Self-assessment
Can the function (s² + 2)/(s(s² + 1)) be the impedance of an LC network?
No: the zero is at √2 and the pole at 1, and with the pole at the origin the sequence would be pole, pole, zero. They do not alternate.
What are the Foster I elements of Z(s) = s + 1/s?
A 1 H inductor in series with a 1 F capacitor.
How is a 1 H inductor denormalized for 1 kΩ and 10 kHz?
\( L = 1 \cdot 1000 / (2\pi \cdot 10^4) \approx 15.9 \) mH.
How many elements does the minimum realization of a degree-4 LC impedance have?
Four, equal to the higher degree of the numerator and denominator.
09Further reading
- Mac E. Van Valkenburg. Introduction to Modern Network Synthesis. Wiley, 1960. Positive real functions and Foster and Cauer synthesis, clearly explained.
- Gabor C. Temes and Jack W. LaPatra. Introduction to Circuit Synthesis and Design. McGraw-Hill, 1977. Synthesis applied to the design of passive and active filters.
- Ernst A. Guillemin. Synthesis of Passive Networks. Wiley, 1957. The classic treatise on network synthesis.