Passive Network Synthesis Revisited Malcolm C. Smith Department of Engineering University of Cambridge U.K. Mathematical Theory of Networks and Systems (MTNS) 17th International Symposium Kyoto International Conference Hall Kyoto, Japan, July 24-28, 2006 Semi-Plenary Lecture 1
Passive Network Synthesis Revisited Malcolm C. Smith Outline of Talk 1. Motivating example (vehicle suspension). 2. A new mechanical element. 3. Positive-real functions and Brune synthesis. 4. Bott-Duffin method. 5. Darlington synthesis. 6. Minimum reactance synthesis. 7. Synthesis of resistive n -ports. 8. Vehicle suspension. 9. Synthesis with restricted complexity. 10. Motorcycle steering instabilities. MTNS 2006, Kyoto, 24 July, 2006 2
Passive Network Synthesis Revisited Malcolm C. Smith Motivating Example – Vehicle Suspension Performance objectives 1. Control vehicle body in the face of variable loads. 2. Insulate effect of road undulations (ride). 3. Minimise roll, pitch under braking, acceleration and cornering (handling). MTNS 2006, Kyoto, 24 July, 2006 3
Passive Network Synthesis Revisited Malcolm C. Smith Quarter-car Vehicle Model (conventional suspension) load disturbances m s spring damper m u k t tyre road disturbances MTNS 2006, Kyoto, 24 July, 2006 4
Passive Network Synthesis Revisited Malcolm C. Smith The Most General Passive Vehicle Suspension m s Replace the spring and damper with a Z ( s ) general positive-real impedance Z ( s ). m u But is Z ( s ) physically realisable? k t MTNS 2006, Kyoto, 24 July, 2006 5
Passive Network Synthesis Revisited Malcolm C. Smith Electrical-Mechanical Analogies 1. Force-Voltage Analogy . voltage ↔ force current ↔ velocity Oldest analogy historically, cf. electromotive force. 2. Force-Current Analogy . current ↔ force voltage ↔ velocity electrical ground ↔ mechanical ground Independently proposed by: Darrieus (1929), H¨ ahnle (1932), Firestone (1933). Respects circuit “topology”, e.g. terminals, through- and across-variables. MTNS 2006, Kyoto, 24 July, 2006 6
Passive Network Synthesis Revisited Malcolm C. Smith Standard Element Correspondences (Force-Current Analogy) v = Ri resistor ↔ damper cv = F v = L di dF inductor ↔ spring kv = dt dt C dv m dv = i capacitor ↔ mass = F dt dt i i v 2 v 1 Electrical F F Mechanical v 2 v 1 What are the terminals of the mass element? MTNS 2006, Kyoto, 24 July, 2006 7
Passive Network Synthesis Revisited Malcolm C. Smith The Exceptional Nature of the Mass Element Newton’s Second Law gives the following network interpretation of the mass element: • One terminal is the centre of mass, • Other terminal is a fixed point in the inertial frame. Hence, the mass element is analogous to a grounded capacitor. F Standard network symbol for the mass element: v 2 v 1 = 0 MTNS 2006, Kyoto, 24 July, 2006 8
Passive Network Synthesis Revisited Malcolm C. Smith Table of usual correspondences Electrical Mechanical F F i i v 2 v 1 v 2 v 1 spring inductor F i i • v 2 v 1 v 1 = 0 v 2 capacitor mass F F i i v 2 v 1 v 2 v 1 damper resistor MTNS 2006, Kyoto, 24 July, 2006 9
Passive Network Synthesis Revisited Malcolm C. Smith Consequences for network synthesis Two major problems with the use of the mass element for synthesis of “black-box” mechanical impedances: • An electrical circuit with ungrounded capacitors will not have a direct mechanical analogue, • Possibility of unreasonably large masses being required. Question Is it possible to construct a physical device such that the relative acceleration between its endpoints is pro- portional to the applied force? MTNS 2006, Kyoto, 24 July, 2006 10
Passive Network Synthesis Revisited Malcolm C. Smith One method of realisation rack pinions terminal 2 flywheel terminal 1 gear Suppose the flywheel of mass m rotates by α radians per meter of relative displacement between the terminals. Then: F = ( m α 2 ) ( ˙ v 2 − ˙ v 1 ) (Assumes mass of gears, housing etc is negligible.) MTNS 2006, Kyoto, 24 July, 2006 11
Passive Network Synthesis Revisited Malcolm C. Smith The Ideal Inerter We define the Ideal Inerter to be a mechanical one-port device such that the equal and opposite force applied at the nodes is proportional to the relative acceleration between the nodes, i.e. F = b ( ˙ v 2 − ˙ v 1 ) . We call the constant b the inertance and its units are kilograms. The ideal inerter can be approximated in the same sense that real springs, dampers, inductors, etc approximate their mathematical ideals. We can assume its mass is small. M.C. Smith, Synthesis of Mechanical Networks: The Inerter, IEEE Trans. on Automat. Contr. , 47 (2002), pp. 1648–1662. MTNS 2006, Kyoto, 24 July, 2006 12
Passive Network Synthesis Revisited Malcolm C. Smith A new correspondence for network synthesis Electrical Mechanical F F i i 1 Y ( s ) = k Y ( s ) = s v 2 v 1 Ls v 2 v 1 dt = 1 di dF spring L ( v 2 − v 1 ) dt = k ( v 2 − v 1 ) inductor i i F F Y ( s ) = bs Y ( s ) = Cs • • v 2 v 1 v 2 v 1 F = b d ( v 2 − v 1 ) i = C d ( v 2 − v 1 ) capacitor inerter dt dt F F i i Y ( s ) = 1 Y ( s ) = c v 2 v 1 R v 2 v 1 i = 1 F = c ( v 2 − v 1 ) damper R ( v 2 − v 1 ) resistor 1 Y ( s ) = admittance = impedance MTNS 2006, Kyoto, 24 July, 2006 13
Passive Network Synthesis Revisited Malcolm C. Smith Rack and pinion inerter made at Cambridge University Engineering Department mass ≈ 3.5 kg inertance ≈ 725 kg stroke ≈ 80 mm MTNS 2006, Kyoto, 24 July, 2006 14
Passive Network Synthesis Revisited Malcolm C. Smith Damper-inerter series arrangement with centring springs MTNS 2006, Kyoto, 24 July, 2006 15
Passive Network Synthesis Revisited Malcolm C. Smith Alternative Realisation of the Inerter screw nut flywheel MTNS 2006, Kyoto, 24 July, 2006 16
Passive Network Synthesis Revisited Malcolm C. Smith Ballscrew inerter made at Cambridge University Engineering Department Mass ≈ 1 kg, Inertance (adjustable) = 60–180 kg MTNS 2006, Kyoto, 24 July, 2006 17
Passive Network Synthesis Revisited Malcolm C. Smith Electrical equivalent of quarter car model F s m s k − 1 t Y ( s ) + z r ˙ − Y ( s ) m u F s m u m s k t z r Force Current Y ( s ) = Admittance = Velocity = Voltage MTNS 2006, Kyoto, 24 July, 2006 18
Passive Network Synthesis Revisited Malcolm C. Smith Positive-real functions Definition . A function Z ( s ) is defined to be positive-real if one of the following two equivalent conditions is satisfied: 1. Z ( s ) is analytic and Z ( s ) + Z ( s ) ∗ ≥ 0 in Re(s) > 0. 2. Z ( s ) is analytic in Re(s) > 0, Z ( jω ) + Z ( jω ) ∗ ≥ 0 for all ω at which Z ( jω ) is finite, and any poles of Z ( s ) on the imaginary axis or at infinity are simple and have a positive residue. MTNS 2006, Kyoto, 24 July, 2006 19
Passive Network Synthesis Revisited Malcolm C. Smith Passivity Defined Definition . A network is passive if for all admissible v , i which are square integrable on ( −∞ , T ], � T v ( t ) i ( t ) dt ≥ 0 . −∞ Proposition . Consider a one-port electrical network for which the impedance Z ( s ) exists and is real-rational. The network is passive if and only if Z ( s ) is positive-real. R.W. Newcomb, Linear Multiport Synthesis, McGraw-Hill, 1966. B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis, Prentice-Hall, 1973. MTNS 2006, Kyoto, 24 July, 2006 20
Passive Network Synthesis Revisited Malcolm C. Smith O. Brune showed that any (ratio- nal) positive-real function could be realised as the impedance or admittance of a network comprising resistors, capacitors, inductors transformers . and (1931) MTNS 2006, Kyoto, 24 July, 2006 21
Passive Network Synthesis Revisited Malcolm C. Smith Minimum functions A minimum function Z ( s ) is a positive-real function with no poles or zeros on j R ∪ {∞} and with the real part of Z ( jω ) equal to 0 at one or more frequencies. Re Z ( jω ) ω 1 ω 2 ω 0 MTNS 2006, Kyoto, 24 July, 2006 22
Passive Network Synthesis Revisited Malcolm C. Smith Foster preamble for a positive-real Z ( s ) Removal of poles/zeros on j R ∪ {∞} . e.g. s 2 + s + 1 1 = s + s + 1 s + 1 ↑ lossless ↓ s 2 + 1 � − 1 � 2 s = s 2 + 1 + 1 s 2 + 2 s + 1 Can always reduce a positive-real Z ( s ) to a minimum function. MTNS 2006, Kyoto, 24 July, 2006 23
Passive Network Synthesis Revisited Malcolm C. Smith The Brune cycle Let Z ( s ) be a minimum function with Z ( jω 1 ) = jX 1 ( ω 1 > 0). Write L 1 = X 1 /ω 1 and Z 1 ( s ) = Z ( s ) − L 1 s . Case 1. ( L 1 < 0) L 1 < 0 Z ( s ) Z 1 ( s ) ( negative inductor! ) Z 1 ( s ) is positive-real. Let Y 1 ( s ) = 1 /Z 1 ( s ). Therefore, we can write 2 K 1 s Y 2 ( s ) = Y 1 ( s ) − s 2 + ω 2 1 for some K 1 > 0. MTNS 2006, Kyoto, 24 July, 2006 24
Passive Network Synthesis Revisited Malcolm C. Smith The Brune cycle (cont.) Then: L 1 < 0 L 2 > 0 Z ( s ) Z 2 ( s ) C 2 > 0 where L 2 = 1 / 2 K 1 , C 2 = 2 K 1 /ω 2 1 and Z 2 = 1 /Y 2 . Straightforward calculation shows that Z 2 ( s ) = sL 3 + Z 3 ( s ) ↑ proper where L 3 = − L 1 / (1 + 2 K 1 L 1 ). Since Z 2 ( s ) is positive-real, L 3 > 0 and Z 3 ( s ) is positive-real. MTNS 2006, Kyoto, 24 July, 2006 25
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