Passive Network Synthesis without Transformers

Passive Network Synthesis without
Transformers
In honour of Yutaka Yamamoto’s 60th birthday
Malcolm C. Smith
with
Jason Zheng Jiang
Cambridge University Engineering Department
YY Fest
29 March 2010
Kyoto

Passive Network Synthesis without Transformers M.C. Smith
The three linear, passive two-terminal electrical
elements
resistor capacitor inductor
Why three?
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Passive Network Synthesis without Transformers M.C. Smith
forbidden
v
i
i
Passive
Network
O. Brune (1931): the drivingpoint
impedance of a linear
passive two-terminal network is
positive-real. Conversely:
Any (rational) positive-real function
can be realised using resistors,
capacitors, inductors and
transformers.
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Passive Network Synthesis without Transformers M.C. Smith
Bott-Duffin Synthesis
R. Bott and R.J. Duffin showed
that transformers were unnecessary
in the synthesis of positive-real
functions. (1949)
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Passive Network Synthesis without Transformers M.C. Smith
Foster preamble for a positive-real Z(s)
Removal of poles on jR ∪ {∞}
Z = sL+Z1, (Z1 proper)
Removal of zeros on jR ∪ {∞}
As Z =
s2 +ω2 −1 + Y1
Subtract minimum real part
Z = R+Z2
Not necessarily a unique process
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L
L
R
Y1
Z1
C
Z2

Passive Network Synthesis without Transformers M.C. Smith
Minimum functions
A minimum function Z(s) is a positive-real function with no poles
or zeros on jR ∪ {∞} and with the real part of Z(jω) equal to 0 at one
or more frequencies.
ReZ(jω)
0 ω1
ω2
ω
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Passive Network Synthesis without Transformers M.C. Smith
Biquadratic minimum function
Z(s) = As2 + Bs + C
Ds 2 + Es + F
with A,B,...,F > 0 and BE = ( √ AF − √ CD) 2 > 0 is a minimum
function. Bott-Duffin realisation:
Z(s)
3 capacitors, 3 inductors and 2 resistors!!
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Passive Network Synthesis without Transformers M.C. Smith
Some of the literature on RLC synthesis
E. L. Ladenheim, A Synthesis of Biquadratic
Impedances, Master’s thesis, Polytechnic
Inst. of Brooklyn, 1948.
R. H. Pantell, A new method of driving
point impedance synthesis, Proc. IRE, 861,
1954.
F. M. Reza, Conversion of a Brune Cycle,
Trans. I. R. E., March, PGCT-1, 1954.
F. M. Reza, A bridge equivalent for a Brune
cycle terminated in a resistor, Proc. I. R. E.,
March, 42(8):1321, 1954.
J. E. Storer, Relationship between the
Bott-Duffin and Pantell Impedance
Synthesis, Proc. IRE March, Sep., 42:1451,
1954.
F. M. Reza, A supplement to Brune
synthesis, Commun. and Electronics.,
March, 85–90, 1955.
E. A. Guillemin, Synthesis of Passive
Networks, John Wiley & Sons, 1957.
S. Seshu, Minimal Realizations of the
Biquadratic Minimum Function, IRE
Transactions on Circuit Theory, Dec.,
345–350, 1959.
R. M. Foster, Academic and Theoretical
Aspects of Circuit Theory, Proc. IRE,
866–871, 1962.
R. M. Foster, Biquadratic Impedances
Realisable by a Generalization of the
Five-Element Minimum-Resistance
Bridges, IEEE Trans. on Circuit Theory,
363–367, 1963.
R. M. Foster, Minimum Biquadratic
Impedances, IEEE Trans. on Circuit
Theory, 527, 1963.
S. Seshu , Author’s Reply, IRE Transactions
on Circuit Theory, Dec., 527, 1963.
R. M. Foster and E. L. Ladenheim, A Class
of Biquadratic Impedances, IEEE Trans. on
Circuit Theory, 262–265, 1963.
E. L. Ladenheim, Three-Reactive
Five-Element Biquadratic Structures, IEEE
Trans. on Circuit Theory, 455–456, 1963.
E. L. Ladenheim, A Special Biquadratic
Structure, IEEE Trans. on Circuit Theory,
88–97, 1964.
P. M. Lin, A theorem on equivalent
one-port networks, IEEE Trans. on Circuit
Theory, 619–621, 1965.
M. Reichert, Die Kanonisch und
Übertragerfrei realisierbaren
Zweipolfunktionen zweiten Grades
(Transformerless and canonic realisation of
biquadratic immittance functions), Arch.
Elek. Übertragung, Apr., 23:201–208,
1969.
C. G. Vasiliu, Three-Reactive Five-Element
Structures, IEEE Trans. Circuit Theory,
Feb., CT-16, 99, 1969.
C. G. Vasiliu, Series-Parallel Six-Element
Synthesis of the Biquadratic Impedances,
IEEE Trans. on Circuit Theory, 115–121,
1970.
C. G. Vasiliu, Series-Parallel Six-Element
Synthesis of the Biquadratic Impedances,
IEEE Trans. on Circuit Theory, 115–121,
1970.
C. G. Vasiliu, Four-Reactive Six-Element
Biquadratic Structures, IEEE Trans. on
Circuit Theory, Sep., 530–531, 1970.
C. G. Vasiliu, Correction to ‘Series-Parallel
Six-Element Synthesis of the Biquadratic
Impedances’, IEEE Trans. on Circuit
Theory, Nov., 207, 1970.
G. Dittmer, Zur Realisierung von
RLC-Brückenzweipolen mit zwei
Reaktanzen und mehr als drei
Widerständen (On the realisation of RLC
two-terminal bridge networks with two
reactive and more than three resistive
elements), Nachrichtentechnische
Zeitschrift, 225-230, 1970.
P. M. Lin, On Biquadratic Impedances with
Two Reactive Elements, IEEE Trans. on
Circuit Theory, 277, 1971.
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Passive Network Synthesis without Transformers M.C. Smith
Ladenheim’s master’s thesis (1948)
Ladenheim considered all networks
with at most five elements and at most
two reactive elements, and reduced
the whole set to 108 networks (1948).
Questions not answered:
◮ What is the totality of
biquadratics which may be
realised?
◮ How many different networks
are needed?
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Passive Network Synthesis without Transformers M.C. Smith
The Concept of Regular Positive Real Functions
Definition
A positive-real function Z(s) is defined to be regular if the smallest
value of Re(Z(jω)) or Re Z −1 (jω) occurs at ω = 0 or ω = ∞.
Example 1. Z1(s) =
Imaginary Axis
2
1
0
−1
2 s+2
s+1
−2
0 1 2 3 4
Real Axis
Smallest value of Re(Z1(jω)) occurs
at ω = ∞, hence Z1(s) is regular.
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Passive Network Synthesis without Transformers M.C. Smith
The Concept of Regular Positive Real Functions
Example 2. Z2(s) =
Imaginary Axis
15
10
5
0
−5
−10
Z2(s)
−15
0 5 10 15 20 25
Real Axis
2 s+5
s+1 , which is non-regular:
Imaginary Axis
1
0.5
0
−0.5
Y2(s) = 1/Z2(s)
−1
0 0.2 0.4 0.6 0.8 1
Real Axis
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Passive Network Synthesis without Transformers M.C. Smith
Network Quartets
◮ Exchange of capacitors and
inductors
◮ Duality (series ⇆ parallel,
capacitor ⇆ inductor)
Lemma 1.
Na is regular =⇒ Nb,Nc,Nd are
all regular.
A network quartet which has
only one network:
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IV

Passive Network Synthesis without Transformers M.C. Smith
Properties of Regular Positive Real Functions
Lemma 2
The following networks are always regular:
Regular
Network
Regular
Network
Lemma 3
A network that has all reactive elements of the same kind can only
realise regular immittances.
Lemma 4
The following networks are always regular:
Passive
Network
Passive
Network
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Passive Network Synthesis without Transformers M.C. Smith
Five-Element Series-Parallel Networks with Two
Reactive Elements—Elimination Process
Assume structure 11 can be
non-regular:
Contradiction.
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Passive Network Synthesis without Transformers M.C. Smith
Five-Element Series-Parallel Networks with Two
Reactive Elements
IV
Theorem 1
A biquadratic impedance can be realised by series-parallel
five-element networks with two reactive elements if and only if it is
regular. Moreover, the following two network quartets cover all cases.
dual
s↔s −1 s↔s −1
dual
dual
s↔s −1 s↔s −1
Jason Z. Jiang and Malcolm C. Smith, Regular Positive-Real Functions and Passive Networks Comprising Two Reactive Elements, 10th
ECC, Pages 219–224, August 2009.
IV
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dual

Passive Network Synthesis without Transformers M.C. Smith
Five-Element Bridge Networks with Two Reactive
Elements
(1)
(2)
(3)
IV
IV
IV
dual
dual
s↔s −1
Theorem 2
Bridge networks with two
reactive and three resistive
elements can only realise
regular immittances except
for the third network
quartet.
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Passive Network Synthesis without Transformers M.C. Smith
A canonical form and the regular region
Extraction of a constant multiplier
and frequency scaling gives a
canonical form for biquadratics:
Zc(s) = s2 + 2U √ Ws + W
s 2 + (2V/ √ W)s + 1/W ,
where U, V, W > 0.
Positive-real ⇔ σc ≥ 0
λ † λ c = 0 † c = 0
Regular ⇔ λc,λ † c ≥ 0 σc σc < 0
V
2
1
λ † λ c > 0 † c > 0
W=0.5
Kc Kc < 0
λc λc > 0
Kc Kc = 0
λc λc = 0
0
0 1
U
σc σc = 0
2
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Passive Network Synthesis without Transformers M.C. Smith
The impedances that can be realised by the third bridge
network quartet with W ∈ (1/3,1).
V
γ2 = 0
2
1
λ † c = 0
γ1 = 0
W=0.6
λc = 0
0
0 γ2 = 0
1
U
σc = 0
2
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Passive Network Synthesis without Transformers M.C. Smith
Mechanical Network Synthesis
Theorem
It is possible to build a passive mechanism
of small mass whose impedance
(velocity/force) is any rational postive-real
function.
Proof
Bott-Duffin + ideal inerter: F = b(¨x1 − ¨x2),
where physical embodiments must satisfy:
◮ Inertance b (kg) is independent of mass;
◮ Inertance is independent of travel.
ms
Passive
Mechanism
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mu

Passive Network Synthesis without Transformers M.C. Smith
Ballscrew inerter made in Cambridge University
Engineering Department (2003)
Mass ≈ 1 kg, Inertance (adjustable) = 60–180 kg
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Passive Network Synthesis without Transformers M.C. Smith
Yutaka’s Birthday Puzzle
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