Network Theory 1 Analoge Netzwerke

Transcription

1 etwork Theory Analoge etzwerke Prof. Dr.-Ing. Ingolf Willms and Prof. Dr.-Ing. Peter Laws Prof. Dr.-Ing. I. Willms etwork Theory S.

2 Chapter Introduction and Basics Prof. Dr.-Ing. I. Willms etwork Theory S.

3 . Preliminary remarks Components like resistor, coil etc. are network elements E K Two-poles have pins accessible Two-ports have 4 pins Two main tasks of network theory: etwork analysis: gives mathematical description of network properties etwork synthesis: gives structure and values of components due to given task Prof. Dr.-Ing. I. Willms etwork Theory S. 3

4 . Preliminary remarks Three solutions steps of network synthesis:. Solution step : etwork characterization by a given characterizing etwork function (specification). Solution step : Approximation of characterizing etwork function by a realizable etwork function 3. Solution step 3: Realization of the found realizable network function by a selected circuit (design) Prof. Dr.-Ing. I. Willms etwork Theory S. 4

5 Solution step :. Preliminary remarks Characterizing network functions can be: Impedance function Z ( c j ) of a LTI two-pole network (one port network) A transfer function Hc ( ) of a LTI two-port network (or its system function) The phase angle ( ) or a phase delay the transfer function ( ) g or the magnitude of Prof. Dr.-Ing. I. Willms etwork Theory S. 5

6 . Preliminary remarks Example of a characterizing network function and its approximation Frequency domain 0 @ I M I A tolerance scheme for H ( ) a M I of a low-pass M Prof. Dr.-Ing. I. Willms etwork Theory S. 6

7 . Preliminary remarks D? J Time domain )! ) ) ) J Characterizing impulse response (Tolerance pattern for an impulse response) Prof. Dr.-Ing. I. Willms etwork Theory S. 7

8 Solution step : Realization by:. Preliminary remarks finite number of linear elements time invariant elements (LTI elements) passive elements Additional possibilities: active LTI elements controlled voltage + current sources Danger: Poles in right p-plane Instability ( cases) In best case useful only for signal generators Prof. Dr.-Ing. I. Willms etwork Theory S. 8

9 To observe:. Preliminary remarks Rational real fractional network function cannot realize in general a) Arbitrary phase function b) Arbitrary magnitude c) Thus also no abitrary H ( ) c h () c t ( ) H ( ) Thus differences are always to be expected between desired functions (index c) and realisable functions (index a). Differences are measured by Hj ( ) in a certain range e H ( e j ) will depend on parameters of Ha ( j) H ( j) H ( ) H ( ) e a c c C Prof. Dr.-Ing. I. Willms etwork Theory S. 9

10 . Preliminary remarks The typical error function h () e t in time domain: he() t ha() t hc() t with approximations intervall t t t Approximation criteria:. CHEBYSHEF criterion or regular approximation: Limits max. deviation, example see S.6. Criterion of the smallest mean square error: min H ( ) e d or t min e ( ) t h t dt Solution of the approximation tasks: - Simple if approx. function linearly depends on approx. parameters - Leads to solution of linear system of equations and can be extended by weighting functions min H ( ) Q( ) d e or t min e ( ) ( ) t h t q t dt Prof. Dr.-Ing. I. Willms etwork Theory S. 0

11 Approximation criteria:. Preliminary remarks 3. Criterion of maximum smoothing: The error function and its derivatives should show for orders as high as possible the value of zero at a prescribed location within the approximation interval. 4. Interpolation criterion: Adjusting the approximation parameters in such a way that the error function at given discrete points within the approximation interval is zero. The number of these discrete interpolation points corresponds to the number of approximation parameters. Disadvantage: Large deviations at many other locations! Prof. Dr.-Ing. I. Willms etwork Theory S.

12 . Preliminary remarks Solution step 3: Circuit realization - Several realization possibilities exist depending on certain defaults: a passive or active network a pure reactance network an active or passive RC network a network with passive symmetrical bridges arrangements of pieces of line (for MW frequencies) - etworks composed with a minimum of components should also be preferred Prof. Dr.-Ing. I. Willms etwork Theory S.

13 Chapter Introduction. LTI Concentrated etwork Elements Prof. Dr.-Ing. I. Willms etwork Theory S. 3

14 . LTI concentrated network elements.. The Ohm s resistance: The symbol and the defining reference directions: EJ EJ EJ K J 0 I? D A H 9 E@ A HI 4 K J 5 F K A K A I = J H + ut () R it () LAPLACE-Transform: U( p) R I( p) Prof. Dr.-Ing. I. Willms etwork Theory S. 4

15 . LTI concentrated network elements.. The capacity C The relationship between the current i(t) and voltage u(t) of an electric capacitor is described as: du() t it () C dt as well as t 0 C t ut () ut ( ) i( ) d 0 Laplace transform: Admittance: Y( p) I( p) p C U( p) pc Impedance: Z(p) = / Y(p) Prof. Dr.-Ing. I. Willms etwork Theory S. 5

16 . LTI concentrated network elements..3 The inductivity L di() t ut () L and dt t 0 L t it () it ( ) u( ) d 0 Laplace transform: U( p) pli( p) Prof. Dr.-Ing. I. Willms etwork Theory S. 6

17 . LTI concentrated network elements..4 The ideal transformer: The transmission characteristics of a lossless transformer with leakage field in the time domain: di di u() t L M dt dt di di u () t M L dt dt with L L : primary inductivity : secondary inductivity Laplace transform with zero initial condition gives: U( p) pli ( p) pm I( p) U( p) pm I( p) pli( p) Prof. Dr.-Ing. I. Willms etwork Theory S. 7

18 . LTI concentrated network elements The relation between the ideal transformer and its transmission characteristics: u () t üu () t i() t i() t ü where ü is transformer constant in Laplace transforms: U( p) üu( p) I( p) I( p) ü ü w w L L Prof. Dr.-Ing. I. Willms etwork Theory S. 8

19 . LTI concentrated network elements..5 The ideal Gyrator Changes resistance into conductance and vice versa or capacitors to coils Voltage-current relationship of ideal gyrators: i () t gu () t i () t gu () t Laplace transform I ( p) g U ( p) I ( p) g U ( p) (g is gyrator s value) Prof. Dr.-Ing. I. Willms etwork Theory S. 9

20 . LTI concentrated network elements Symbol of an ideal gyrator with the reference directions E J E J K J K J I ( p) g U ( p) I ( p) gu ( p) Z e ( p) with ( ) g Za p Z C U ( p) I ( p)/( g) I ( p) I ( p) gu ( p) g U ( p) Z a e U( p) ( p) I ( p) U ( p) I ( p) ( p) the input impedance the load impedance Prof. Dr.-Ing. I. Willms etwork Theory S. 0

21 . LTI concentrated network elements Examples A and B with g ms C F R Rg k R g A: Z ( p) R Z ( p) 0 a 3 e 6 g Za ( p) 0 S k B: Za( p) / pc Ze( p) ( ) g Za p 6 p0 F = ph S /( p0 F) 0 S Prof. Dr.-Ing. I. Willms etwork Theory S.

22 . LTI concentrated network elements..6 Independent sources: There are two different kinds of independent sources:. The independent (ideal) voltage supply: u(t) q U(p) q independent of i(t) I(p). The independent (ideal) current supply: independent of (t) i q u(t) (p) I q U(p) Prof. Dr.-Ing. I. Willms etwork Theory S.

23 . LTI concentrated network elements Symbols of uncontrolled sources and reference directions F F E F 7 G F 7 G F 7 F! G F " G F F 7 F ; E F 7 F ) Independent (ideal) voltage supply ) Voltage supply with an internal resistance 3) Independent (ideal) current source 4) Current source with an internal resistance Prof. Dr.-Ing. I. Willms etwork Theory S. 3

24 . LTI concentrated network elements..7 Dependent sources: Symbols of dependent sources (controlled) and reference directions of the electricity F F F 7 F 7 F 7 F 7 F EJ7 F = 7 F Voltage-controlled voltage supply EJ7 F > F Current-controlled voltage supply! F F C " 7 F F F 7 F 7 F 7 F 7 F Voltage-controlled current source Current-controlled current source Prof. Dr.-Ing. I. Willms etwork Theory S. 4

25 . LTI concentrated network elements..8 etwork equations: Impedance matrix Z Admittance matrix Y Cascade matrix A U( p) Z( p) Z( p) I( p) U ( p) Z ( p) Z ( p) I ( p) I( p) Y( p) Y( p) U( p) I ( p) Y ( p) Y ( p) U ( p) U( p) A( p) A( p) U( p) I ( p) A ( p) A ( p) I ( p) For conversion formulas see the matrix table slides. Prof. Dr.-Ing. I. Willms etwork Theory S. 5

26 Chapter Introduction.3 etwork Topology Prof. Dr.-Ing. I. Willms etwork Theory S. 6

27 .3 etwork topology The electrical characteristics of a network depend on: Properties of network elements used Topology or structure of the network elements Described by a graph representing the structure of the network Method of drawing the network topology: Example: on-directional graph of a two-pole as well as a four-pole network element!! " " Prof. Dr.-Ing. I. Willms etwork Theory S. 7

28 .3 etwork topology A given network and the nondirectional graph following from it 7 #! # 7 #!! # 7 7! $ = $ > % " 7 $ = $ % 7 $ > 7 % $ = $ > " " 7 " Prof. Dr.-Ing. I. Willms etwork Theory S. 8

29 .3 etwork topology One can cut a network in a manner that:. A part of the graph is completely separated from the remainder. This cut is not through a node 5? D EJJ! # $ > $ = % " 5? D EJJHE? D JK C Prof. Dr.-Ing. I. Willms etwork Theory S. 9

30 .3 etwork topology Applying cuts to all branches around a node leads (as charging up of a node cannot happen) to: Kirchhoff s current rule (summation on all branches cut): i () t 0 Laplace transform I ( p) 0 as well as I * ( p) 0 where I * ( p) is the complex conjugate of I ( p) Prof. Dr.-Ing. I. Willms etwork Theory S. 30

31 .3 etwork topology Kirchhoff s voltage rule (summation along one loop): u () t 0 t U Laplace transformation ( p) 0 as well as U * ( p) 0 TELLEGE s theory (summation covering all branches): i () t and u () t z are the branch current and branch voltage and it is valid: u () t i () t 0 Laplace trans. z U p I p * ( ) ( ) 0 and z * U p I p ( ) ( ) 0 Prof. Dr.-Ing. I. Willms etwork Theory S. 3

32 .3 etwork topology In a given network only a certain maximum number of branch currents arises, which. are independent. thereby specifying the remaining branch currents The independent branches of a network are branches in which the independent currents flow. The independent branches of a network are determined by first designing a complete tree. A complete tree of a connected graph is a partial graph which contains all of its nodes and some of its branches, but contains no loops. Prof. Dr.-Ing. I. Willms etwork Theory S. 3

33 .3 etwork topology umber z u of linearly independent loop equations for in total z branches and k nodes is: zu z k The figure shows branches of a complete tree (, 3, 5, 6a, 6b) and the independent branches (, 4, 7) of the graph as well as the appropriate loop current s. J Prof. Dr.-Ing. I. Willms etwork Theory S. 33

34 .3 etwork topology Method of branch current determination I( p) m J( p) where J ( p) are the independent loop currents Independant branches produce a loop when adding it to a complete tree. In each of these loops the corresponding loop current runs through an independant branch which is not contained in the other loops! This can be expressed more generally in matrix form: I( p) M J( p) Branch current vector - (z x ) Incidence matrix Loop current vector ( zu x ) Prof. Dr.-Ing. I. Willms etwork Theory S. 34

35 Incidence matrix coefficients:.3 etwork topology m 0 if branch v belongs to loop µ; branch and loop directions agree if branch v belongs to loop µ; branch and loop directions do not agree if branch v does not belong to loop µ Kirchhoff s voltage rule can then be formulated as z m U ( p) 0 with U ( p) as the voltage of the branch v Prof. Dr.-Ing. I. Willms etwork Theory S. 35

36 .3 etwork topology All branch voltages can be combined into the branch voltage vector: U ( ) p U( p) U ( p) Uz ( p) Kirchhoff s voltage rule can then be represented as follows: T M U ( p) 0 Replacing branch voltages with voltages at all components gives (and under the condition that the network does not contain uncontrolled power sources, controlled sources, transformers and gyrators): U( p) Z I( p) U ( p) BI Branch impedance matrix Prof. Dr.-Ing. I. Willms etwork Theory S. 36 s Voltage source vector

37 .3 etwork topology Branch impedance matrix: (purely diagonally with z x z elements) Z ( ) 0 0 p ZBI 0 Z ( p) Z z ( p) Prof. Dr.-Ing. I. Willms etwork Theory S. 37

38 .3 etwork topology ow the following equations will be combined: U( p) ZBI I( p) Us ( p) I( p) M J( p) U( p) ZBI M J( p) Us ( p) T Another relation results from using: M U ( p) 0 They give: T T T M U ( p) M Z M J ( p) M U ( p) 0 BI s Thus the loop impedance matrix Z(p) appears: ( z x z ) Z Z Zz u T Z Zz u Z( p) M ZBI M Zz u Z zuzu Prof. Dr.-Ing. I. Willms etwork Theory S. 38 u u

39 .3 etwork topology Thus the previous equation can be simplified to: T Z( p) J( p) M U ( p) It is a system of z u linearly independent equations! Thus the following procedure results based on: a) a given loop impedance matrix T b) a given vector M U ( p) s s ) Determination of the current loop vector J(p) ) From this the current branch vector is determined by I ( p) M J( p) 3) Finally the branch impedance vector is determined using U( p) Z I( p) U ( p) BI s Prof. Dr.-Ing. I. Willms etwork Theory S. 39

40 a.3 etwork topology For RLC-networks with independent voltage sources the loop impedance matrix can be set up as follows: Zmm Z for all branches, which belong to loop m Here Z is the sum of all impedances in a loop. Z mm a Z for all branches, which belong to loop and loop nm v m n m if the direction of loop and n at the common branch are equal m If the direction of loop and n at the common branch are not equal Prof. Dr.-Ing. I. Willms etwork Theory S. 40

41 Chapter Introduction.4 etwork Functions Prof. Dr.-Ing. I. Willms etwork Theory S. 4

42 .4 etwork functions The network function L (p) is defined as the ratio of the Laplace transform of the response signal and the Laplace transform of the input signal under the conditions that:. All network elements are linear and time invariant (in short LTI or LZI). All network elements are in the energyless initial condition (zero state). If the response signal and the input signal are at the same branch, the network function is called two-terminal function or impedance function or admittance function. Prof. Dr.-Ing. I. Willms etwork Theory S. 4

43 .4 etwork functions An example of an impedance function: I Lv ( p) ULv ( p) ULv ( p) Branch v Remainder network Two-terminal network Input I L ( p) and response signal UL ( p) of a two-port LTI system gives the network impedance function: Z L ( p) U I L L ( p) ( p) Prof. Dr.-Ing. I. Willms etwork Theory S. 43

44 .4 etwork functions If the excitation signal and the response are located at different branches of the network or at different ports, then one calls the appropriate network function an effective function H ( ) L p An example of an effective function: I ( ) Lv p I ( p) U ( ) Lv p U L ( p) Branch v LZI twoport (Remainder network) Input UL ( p) and the response signal U L ( p) of a two-port LTI network: UL ( p) HL( p) U ( p) L Branch Prof. Dr.-Ing. I. Willms etwork Theory S. 44

45 .4 etwork functions Modern network theory uses a description of the transmission characteristics of networks in a representation which follows partly that of classical network theory: R I I I A U 0 U LZI-two port network R U The classical representation is in time domain instead of p domain: ut () U cos ( t ) eff u and it () Ieff cos ( tu ) Prof. Dr.-Ing. I. Willms etwork Theory S. 45

46 .4 etwork functions For passive networks it is reasonable to look at both voltage transmission and current transmission through the network. This is done for a typical case with impedance matching at the input. Voltage amplification then is: Current amplification then is: A A U U with R U U R R L 0 / L U U I I I I A I R R L I U0 / R U0 /R An operation transmission factor then can be defined: U I U U / R U R A AA H ( p) B U I LB U0 / U0 / R ( U0 /) / R U0 / R The inverse of this factor leads then to the insertion loss function. Prof. Dr.-Ing. I. Willms etwork Theory S. 46

47 .4 etwork functions Definition of the insertion loss function: U 0 U 4R R H ( ) BD p j AB U R U R This definition follows the idea that the magnitude of the insertion loss function can be expressed using effective powers (related to the input and to the load): P 0max. U 0 U P 4 R R Here voltages represent effective values 0 Therefore here the magnitude of the insertion loss function is considered and can be rewritten using the two effective powers. Prof. Dr.-Ing. I. Willms etwork Theory S. 47

48 .4 etwork functions Thus we obtain: H BD ( ) P U o max R P R U 0 Here the two effective powers are represented by: P0max. The maximum effective power passed to an external load resistance attached to a voltage supply (or the input power) (The maximum is given in case of RL R ) P The effective power at the load R This definition of the insertion loss function is due to the fact that in early days of communications technology the transmission of telephone/telegraph signals over lossy long lines was quite an important problem. R L Prof. Dr.-Ing. I. Willms etwork Theory S. 48

49 .4 etwork functions Long lines (compared to wavelength) have to be operated with suitable loads for avoiding reflections at input and output ports. At all interfaces loads have to be equal to the wave impedance of the line, e.g.: Zw 600R R The same is true for two-ports for RF signals! Definition of the effective transmission factor: g ( ) a ( ) jb ( ) B B B Attenuation constant (Phase) Wavelength constant Relation with insertion loss function: Using the natural logarithm it holds: jhbd H ( ) H ( ) e BD jh ( ) BD HBD HBD e ln ( ) ln ( ) ln H ( ) jh ( ) a ( ) jb ( ) BD BD B e BD ( ) g ( ) a ( ) jb ( ) e B B B B Prof. Dr.-Ing. I. Willms etwork Theory S. 49

50 Prof. Dr.-Ing. I. Willms etwork Theory S etwork functions 0 ( ) ln ( ) ln B BD U R a H R U 0 0 Im ( ) arctan Re B U U b U U Thus we arrive at the following relation with the insertion loss function due to: 0 BD ( ) U R H R U If impedances (instead of resistors) are used, also these values go into the determination of the angle!

51 .4 etwork functions owadays the usual method of the description of the transmission characteristics of two-port networks is the LAPLACE transform of the voltage signals u(t) or of the current signals i(t). Four operational cases conc. the effective functions are distinguished. The first operational case Two-port network without input&output load : ) A two-port network fed by voltage source U ( p) 0 and negligible internal resistance, two-port load R (zero-load): R 0 Voltage effective funktion (expressed by two-port matrix elements): U( p) Z( p) Y( p) U ( p) Z ( p) Y ( p) A ( p) 0 Prof. Dr.-Ing. I. Willms etwork Theory S. 5

52 .4 etwork functions ) A two-port network fed by voltage supply source U ( p) 0 and internal resistance R, two-port load R 0 and U ( p) 0 (short-circuit at output) 0 Transmission admittance function: I ( p) Z ( p) U ( p) A ( p) A Y( p) 0 det Z 3) A two-port network fed by power supply source with I ( p) 0 and internal conductance Y (/ R) 0 with load R (open circuit). Transmission impedance function: U ( p) Y ( p) Z ( p) I ( p) A ( p) 0 det Y Prof. Dr.-Ing. I. Willms etwork Theory S. 5

53 .4 etwork functions 4) A two-port network fed by power supply source I ( p) 0 and internal conductance Y (/ R) 0 and R 0 (closed circuit at output). Current effective funktion: IA( p) Z( p) Y( p) I ( p) Z ( p) Y ( p) A ( p) 0 Prof. Dr.-Ing. I. Willms etwork Theory S. 53

54 H Leu.4 etwork functions The second operational case Two-port with resistance at input" ) A two-port fed by voltage supply source U ( p) and internal 0 resistance with two-port load R (zero-load). ( p) U( p) U( p) Z( p) U ( p) R I ( p) Z ( p) R 0 0 Y( p) Y ( p) R det Y A ( p) A ( p) R ) A two-port fed by power supply source I ( p) 0 and internal conductance Y (/ R) 0 with load R 0. H Lei ( p) IA( p) R IA( p) Z( p) R I ( p) U ( p) det Z Z ( p) R Y( p) R R Y ( p) R A ( p) A ( p) R 0 0 Prof. Dr.-Ing. I. Willms etwork Theory S. 54

55 .4 etwork functions The third operational case "two-port with output load" (load resistance is finite and nonzero). A two-port fed by voltage supply source U ( p) and internal 0 resistance R with two-port termination gives voltage effective function: H Lau ( p) 0 U( p) R IA( p) Z( p) R U ( p) U ( p) det Z Z ( p) R Y ( p) R R R 0 0 A Y( p) R A ( p) A( p) R with I ( p) I ( p). A two-port fed by power supply source with I ( p) 0 and internal conductance Y (/ R) 0 and load R gives current effective function: H Lai IA( p) U( p) Z( p) ( p) I0( p) RI0( p) Z( p) R Y( p) Y ( p) R dety A ( p) A ( p) R Prof. Dr.-Ing. I. Willms etwork Theory S. 55

56 .4 etwork functions The fourth operational case "two-port with input and output load" (characterized by finite resistance values) with Voltage effective function: U( p) Z( p) HLu ( p) AU det Z R Current effective function: U 0( p) Z( p) Z( p) R R R A ( p) A ( p) R A ( p) R A ( p) R R I ( p ) H p A R H p A Li ( ) I Lu ( ) I 0( p) R I p I p ( ) A ( ) Prof. Dr.-Ing. I. Willms etwork Theory S. 56

57 .4 etwork functions Combination of corresponding current and voltage effective functions give the operation effective function: H ( p) A H ( p) H ( p) LB B Lu Li det Z R R Z ( p) R R Z ( p) Z ( p) R R R R R A ( p) R A ( p) R R A ( p) A ( p) R R R R The determination of the formula makes use of the the definition of A B, the relations for voltages at input and output and the two-port cascade matrix equations. Prof. Dr.-Ing. I. Willms etwork Theory S. 57

58 H L ( p).4 etwork functions Each of these effective functions follows the general definition: L Zerostate Response Signal L Input Signal This is often also called system function or system-driving function. An alternative representation is made using rational real fractions in p: with p M M m am p ( p p0 i ) P( p) m0 i L( ) L( ) Q( p) n bn p ( p p k ) n0 k H p A H p e 0i i.. M zeros of the numerator P(p) jh L ( p) p k k.. zeros of the denominator Q(p) Prof. Dr.-Ing. I. Willms etwork Theory S. 58

59 One can write.4 etwork functions i ( p p ) p p e j ( p) j ( ) 0 i 0 i k p and ( p p k) p p k e so that H ( p) A L p p p p p p p p p p p p M and ( p) H ( p) L ( p) ( p) ( p) ( p) ( p) ( p) ( p) A M with ( p A ) ( A ) Prof. Dr.-Ing. I. Willms etwork Theory S. 59

60 .4 etwork functions Example of a pole-zero diagram Prof. Dr.-Ing. I. Willms etwork Theory S. 60

61 .4 etwork functions In practice, one often looks at the corresponding transfer function: H ( ) H ( p j) F Condition: Poles of must lie in left half p plane H ( ) L p L In this case the following the equation holds: ( j p0) ( j p0) ( j p0 M ) HF( ) HL( j) A ( j p ) ( j p ) ( j p ) Fourier transform gives the impulse response h(t) of the system! H ( ) H ( j) F L h(t) with p k Re 0 ote: This also is possible from the system function using inverse Laplace transform! Re 0 p k Prof. Dr.-Ing. I. Willms etwork Theory S. 6

62 Stability of the two-port network.4 etwork functions M H (p) 0 for p L is a condition for subsequent considerations! (with M being the order of the numerator, being the order of the denominator) The partial fraction method gives for H L ( p) A ( p p ) with the pole factors: d A H p p p ( )! v -fold poles at location : p v ( ) ( ) ( ) ( ) L dp p p Prof. Dr.-Ing. I. Willms etwork Theory S. 6

63 With the correspondence.4 etwork functions ( p p ) t e ( )! 0 p t für t 0 sonst the impulse response h(t) thus can be determined as: H ( ) L p h(t) 0 A t e ( )! p t für t 0 sonst t t h t A e A e e ( )! ( )! pt t jt () Prof. Dr.-Ing. I. Willms etwork Theory S. 63

64 .4 etwork functions Thus the impulse response h(t) will satisfy the following condition ht () W for all t > 0 (W is a finite value) only under the condition that: Re 0 for p max.re max. 0 p for and ) Amplitude delimitation of h(t) corresponds to the stability definition for two-port systems. ) In this case a special stability situation is given characterized by constant oszillations! Prof. Dr.-Ing. I. Willms etwork Theory S. 64

65 .4 etwork functions Stability of the system with the HURWITZ criterion: Hurwitz rational polynomial: All coefficients are real numbers All zeros are located in the left half of the complex p plane The stability criterion of a two-port system thus can be expressed: A two-port with the characteristic function or system function H ( ) ( )/ ( ) L p P p Q p is stable if the denominator polynomial Q(p) is a Hurwitz - polynomial. Prof. Dr.-Ing. I. Willms etwork Theory S. 65

66 Modified Hurwitz polynomial:.4 etwork functions All coefficients are real numbers There may be simple or multiple zeros in the left half p-plane There may be simple poles on the jω-axis but no multiple ones! A two-port network with the system function H ( ) ( )/ ( ) L p P p Q p is stable if the denominator polynomial Q(p) is a modified Hurwitz - polynomial. Prof. Dr.-Ing. I. Willms etwork Theory S. 66

67 Chapter Introduction.5 Two-terminal networks with special effective functions (System functions) Prof. Dr.-Ing. I. Willms etwork Theory S. 67

68 .5. The all-pass network The all-pass behaviour is defined by the system function: Q( p) with A being real and Q(p) HL ( p) A Q ( p ) being a Hurwitz - polynomial Then in addition to conjugated complex poles and zeros it holds: p 0i p k for corresponding i and k and the system function will be: b0 b p b p b p 0 Q( p) HL( p) A A Q( p) b b p b p b p ' ( p p ) ( p p) ( p p ) A ( p p ) ( p p ) ( p p ) gives square-symmetrical pole zero configuration typical for an all-pass network Prof. Dr.-Ing. I. Willms etwork Theory S. 68

69 For p j.5. The all-pass network the corresponding transfer function gives: ( j p ) ( j p ) ( j p ) ' ' H L( j) A A ( j p ) ( j p) ( j p ) jh F ( ) HL j HF e Constant with ( ) ( ) and for complex conjugated poles: j p j j and j p j j so that here p has the same distance to j as p = p! *, 0,, A pole with the index k contributes with: k k ( j pk) ( jk jk) arctan( ) arctan( ) k k A corresponding zero contributes with: k ( j pok ) ( jk jk ) arctan( ) k In total it results: k H F( ) arctan, k 0 k k Prof. Dr.-Ing. I. Willms etwork Theory S. 69

70 .5. The all-pass network The corresponding all-pass group delay gives: db( ) g ( ) d( ) k k ( ) k arctan k k ( k) Examples of quadrant-symmetrical pole zero configuration for all-passes: k with b( ) H F ( ) All-pass first order network All-pass second order network Prof. Dr.-Ing. I. Willms etwork Theory S. 70

71 .5. The all-pass network Minimum phase two-port network: 0 F M 0 F M 0 ) F M I I I Two-port contains all-pass Minimum phase system Pure all-pass H ( p) H H L LM LA Two-ports containing all-passes can be divided into a minimum phase two-port and an all-pass two-port network. Prof. Dr.-Ing. I. Willms etwork Theory S. 7

72 .5. Bridge circuits A special two-port network with interesting properties is represented by: F = F F F! > F = F > F F 7 F 7 F? F 7 F 7 F? F " Bridge connection representation Cross connection representation Bridge network advantages: Easy synthesis relations (under conditions) Prof. Dr.-Ing. I. Willms etwork Theory S. 7

73 .5. Bridge circuits The Z-matrix of the asymmetrical bridge two-port of previous figure: Z Z Z Z Z ( Za Zd)( ZbZc) ZbZd ZaZc Z ZZ b d ZZ a c ( Za Zb)( Zc Zd) a ZbZc Z d Under the condition of Z Z and Z Z it results: Z Z Z a c b d ( Z Z ) ( Z Z ) a b b a Z Z ( Zb Za) ( Zb Za) Prof. Dr.-Ing. I. Willms etwork Theory S. 73

74 .5. Bridge circuits and the system becomes symmetric in structure: = F = F F F F F > F 7 F 7 F 7 F = F 7 F A symmetrical bridge Its equivalent circuit with an ideal transformer Two conditions sufficiently and necessary for the symmetry of a two-port (exchange of the ports has no effect) are fulfilled by this circuit: Z Z Z Z Prof. Dr.-Ing. I. Willms etwork Theory S. 74

75 .5. Bridge circuits The matrix Y of this symmetrical bridge two-port applies with and gives the result: Y Y a Y Z Y a a b b a Y Y ( Yb Ya) ( Yb Ya) Y b Z ( Y Y ) ( Y Y ) b Longitudinal branch impedance Z a and transverse branch impedance are always positive real functions of p for a passive symmetrical four-pole network! Z b Prof. Dr.-Ing. I. Willms etwork Theory S. 75

76 .5. Bridge circuits Condition of a two-port network with symmetrical structure:.. Z Z Z Z Z Z Z Z Z The properties of the two-port remains unchanged when exchanging the ports. Symmetric two-ports can be transformed into equivalent circuits using the symmetry rule of Bartlett Z Z Z Z.... Z 3 3 Z 4 Z 5 Z Symmetry line Z Prof. Dr.-Ing. I. Willms etwork Theory S. 76

77 .5. Bridge circuits Due to S. 73 a special operation condition of a two-port with the same currents at both ports can be used to determine the longitudinal impedance: I I gives U U due to symmetrical two-port and in addition to: U U ZIZI ZbI Zb I Due to the symmetrical two-port all node potentials φ are equal at corresponding nodes of both halfes of the two-port. Thus no current flows through branches crossing the symmetry line. Consequently, these branches could be extracted from the network without changes to currents or voltages. Considering the two-port with extracted branches gives an easier overview and corresponds to an open circuit operation of seperated halfes of the original network. So Z b can also be determined by looking at one separated half of the two-port in open circuit operation. Prof. Dr.-Ing. I. Willms etwork Theory S. 77

78 .5. Bridge circuits Again due to S. 73 a second special operation of a two-port with the opposite current values at port and port can be used to determine the needed transverse impedance Z a : I I gives U U due to symmetrical two-port and in addition to: U U ZIZI ZaI Za I Due to the symmetrical two-port and considering superposition of the cases when only the first and then only the second source is attached, all node potentials φ at the branches crossing the symmetry line now have the same value. Thus all of theses points can be connected to each other without changes to currents or voltages. Consequently, the two halfes of the symmetrical network can be separated without changing currents or voltages. This corresponds to a short circuit operation of the seperated halves of the original network. Considering the short circuited separated half of the two-port gives an easier overview and thus Z a can be determined by looking at this simplified network. Prof. Dr.-Ing. I. Willms etwork Theory S. 78

79 .5. Bridge circuits An example of the BARTLETT' symmetry rule: A J M A H 4 + A given two-port with symmetrical structure 5? D HEJJ Symmetrical allocation of the components 5? D HEJJ = > Determination of the longitudinal and transverse branch impedance - HC A > EI 4 A resulting symmetrical bridge two-port + Prof. Dr.-Ing. I. Willms etwork Theory S. 79

80 .5. Bridge circuits ote: ow a network with loads is considered for determination of relations with the operation effective function Method: Insertion of impedance matrix elements for a bridge network into formula for H LB (p) Details are shown in the next slide. Prof. Dr.-Ing. I. Willms etwork Theory S. 80

81 H ( p) H ( p) H ( p) = LB Lu Li.5. Bridge circuits det Z Z R R R Z det Z R Z R Z R R = (due to S.57) ( Za Zb ) Z Z ( Zb Za ) with Z Z Z ( Zb Za) ( Zb Za) and det Z (( Za Zb) ( Zb Za) 4ZaZb 4 4 R R R R Z Z R R R H LB ( p) RR Zb( p) Za( p) Z ( p) Z ( p) ( R R ) Z ( p) Z ( p) R R a b a b Prof. Dr.-Ing. I. Willms etwork Theory S. 8

82 .5. Bridge circuits Reciprocally wired symmetrical bridge two-port 4 F = F F 7 F 7 F > F 7 F 4 * F * F and the condition So for this circuit the effective transmission factor results as shown in last slide: H LB ( p) RR Zb( p) Za( p) Z ( p) Z ( p) ( R R ) Z ( p) Z ( p) R R a b a b Prof. Dr.-Ing. I. Willms etwork Theory S. 8

83 Z ( p) Z ( p) b a.5. Bridge circuits After solving for Z ( b p ) an easily usable relation results (for a given operation effective function and given loads): R R R R HLB ( p) Za ( p) RR Z ( p) R R H a LB ( ) RR RR The operation impedances (input and output impedances) can be determined as: For the input: For the output: Z Z B B ( p) ( p) U ( ) ( ) ( p) Za p Zb p R Za( p) Zb( p) I ( p) R Z ( p) Z ( p) U ( ) ( ) ( p) Za p Zb p R Za( p) Zb( p) I ( p) R Z ( p) Z ( p) U ( p) 0 0 a a b p b Prof. Dr.-Ing. I. Willms etwork Theory S. 83

84 .5. Bridge circuits A further simplification results with the condition of In this often wanted case it holds: H LB ( p) Z ( p) Z ( p) b R R b( ) a( ) b( ) a( ) R Z p Z p R Z p Z p a b a b a b R R ( Z ( p) Z ( p)) Z ( p) Z ( p) R Z ( p) R Z ( p) Z ( p) Z ( p) a B B U with ZB,( p) I R H p Za ( p) HLB ( p) R LB( ) Za ( p),, Z ( p) Z ( p) R Z ( p) Z ( p) ( p) ( p) a b a b R Z ( p) Z ( p) a b Prof. Dr.-Ing. I. Willms etwork Theory S. 84

85 .5. Bridge circuits Another simplification uses the additional condition Z ( p) Z ( p) R and gives: a b H LB ( p) R Za ( p) R Z ( p) a or H LB ( p) Zb( p) R Z ( p) R b dissolving Z ( p) R a H LB( p) H ( p) LB Z ( p) R b H LB ( p) H ( p) LB Z ( ) b p as well as Za ( p) depend only on the internal resistance and the Operation effective function! Prof. Dr.-Ing. I. Willms etwork Theory S. 85

86 To observe:.5. Bridge circuits ot all effective transmission factors lead to impedances realizable just by passive RLCÜ elements! The conditions of realizing rational two-poles just with passive RLCÜ elements are: Im Z( p) 0 for all p with p (DC-case due to 0) Z p for all p with p Re ( ) 0 Re 0 The last equation means that the two-pole cannot deliver energy (true for all passive two-poles)! (see also S. of chapter ) These conditions are now considered with respect to the two impedances of the symmetric bridge circuit. Prof. Dr.-Ing. I. Willms etwork Theory S. 86

87 .5. Bridge circuits Conditions for realizing the bridge circuit with the two impedances HLB ( p) Za ( p) R and H ( p) Z ( p) R a LB Hr( p) jhi( p) H ( p) jh ( p) r i Z ( p) R b HLB ( p) H ( p) The operation effective function is now split up into real and imaginary parts: H ( p) H ( p) jh ( p) This leads to: LB r i Z ( p) R b LB Hr( p) jhi( p) H ( p) jh ( p) r i Positive real parts of these impedances are given for (without proof ) H ( ) LB p for all p Re 0 This means just no amplification of currents/voltages! Prof. Dr.-Ing. I. Willms etwork Theory S. 87

88 .5. Bridge circuits It can also be shown that positive real parts of the bridge impedances relies on the additional conditions: a) The operation effective function has no poles in closed positive p plane Z ( p) Z ( p) R b) It holds B B The last relation is also a condition for being able to set up a chain of bridge circuits with in some sense effectless or non-reactive connections (i.e. no reflections, no effect of a second network on H LB (p) of a first network). Or: The load of one element in the chain thus has no negative effect on the transmission properties of the previous chain elements. Reason: The loads of all inputs and outputs are equal to: R R Prof. Dr.-Ing. I. Willms etwork Theory S. 88

89 .5. Bridge circuits Effectless chain network of two symmetrical bridge two-ports with dual branch impedances Zb( p) and Za ( p) For Z symmetric bridge circuits (connected in a chain) it holds: U ( p) H p H p Z Z LB ( ) ( ) LBv v U0( p) Prof. Dr.-Ing. I. Willms etwork Theory S. 89

90 .5. Bridge circuits Thus a given operation effective function can be realized using: A number of Z chained symmetric bridge circuits Symmetrical loads Identical operation input and output impedances For simplicity: Chain elements of the order of (in addition to one with the order of ) Prof. Dr.-Ing. I. Willms etwork Theory S. 90

91 Chapter Introduction.6 ormalization Procedures Prof. Dr.-Ing. I. Willms etwork Theory S. 9

92 .6. ormalized representation of network functions ormalization procedures help to keep the overview in a big network where usally it is hard to compare values. ormalisations are transforms of variables using reference constants. They can be applied to frequencies and component values. Typical normalization procedures applied to a network function (p) might be: An impedance-function Z(p) is normalized Its argument, the complex frequency is normalized p j An operation effective function is normalized Prof. Dr.-Ing. I. Willms etwork Theory S. 9

93 .6. ormalized representation of network functions The standardizing auxiliary variable used during the frequency normalization is the normalized complex frequency: p j j P (p) Frequency normalization nf ( P) ( P) ( P ) nf or p ( p) nf ( ) Prof. Dr.-Ing. I. Willms etwork Theory S. 93

94 .6. ormalized representation of network functions Example of a frequency normalization: etwork function (p) is given in the form of U(p), a voltage at a series connection of a resistance R, an inductance L and a capacity C. I(p) is the current of the series connection. U( p) R pl I( p) pc Frequency normalisation extends p as follows: p p U( p) R L I( ) p C Prof. Dr.-Ing. I. Willms etwork Theory S. 94

95 .6. ormalized representation of network functions Thus: Unf ( P) RP L Inf ( P) P C and p I( ) I( P ) Inf ( P) Example of the resistance normalization: This normalisation is applied to all impedance functions of the network function: U( p) R pl I( p) Z( p) I( p) pc Impedance functions Prof. Dr.-Ing. I. Willms etwork Theory S. 95

96 .6. ormalized representation of Thus the extension with network functions R starts with: R pl R U( p) R R I( p) R R pcr After extraction of the normalization resistance R from the parenthesis it results: U( p) R Z ( p) I( p) nw with Z nw Z( p) R pl I( p) R R R pcr unitless (normalized) impedance Prof. Dr.-Ing. I. Willms etwork Theory S. 96

97 .6. ormalized representation of network functions Use of auxiliary variables in network normalization Such variables can be applied both to normalized voltages/currents or for Laplace transforms. Example: U( p) ( ) ( ) I( p) Unu p Z p U U It is possible to normalize all other elements in formulas describing network functions/elements by applying such a normalization individually! This is called a complete normalization. Prof. Dr.-Ing. I. Willms etwork Theory S. 97

98 .6. ormalized representation of network functions An example of the complete normalization: U( p) R p LR R p U R I( ) U p R R R C From this arises:. U ( ) nu p U( p) U Voltage normalized concerning its value r R R RG. ormalized resistance Prof. Dr.-Ing. I. Willms etwork Theory S. 98

99 .6. ormalized representation of network functions l L 3. ormalized inductance R 4. c R C ormalized capacity p 5. P ormalized complex frequency p 6. Inf ( P) I I( P ) Frequency normalized current Prof. Dr.-Ing. I. Willms etwork Theory S. 99

100 .6. ormalized representation of network functions The equation R Unu ( p) rpl Inf ( P) Unuf ( P) U Pc using leads to: I U R Unuf ( P) rpl Pc I nf I ( P) ow I nf (P) is substituted: I nfi ( P) I nf I ( P) Prof. Dr.-Ing. I. Willms etwork Theory S. 00

101 .6. ormalized representation of network functions Then the completely normalized network function results: Unuf ( P) rpl Infi ( P) Pc Prof. Dr.-Ing. I. Willms etwork Theory S. 0

102 .6. ormalized representation of network functions Resistances R and conductances G appear in a network function after a complete standardization of (p) in the diagrams as pure numerical values. F 4 BE F H 4 7 F + M 7 7 K B F? Diagram and designation of the sizes before (left) and after the normalization (right) of the network function Prof. Dr.-Ing. I. Willms etwork Theory S. 0

103 .6. ormalization in the time and frequency domain The description of signals in the time domain has two common representation types:. The representation with units. The completely normalized representation The mathematical description of a signal x(t) in the time domain looks like this: x() t A s() t 0 A0 is a constant with a certain unit, for example Volt if x(t) is a voltage signal s(t) is a unitless function and t is its argument with the unit s. Prof. Dr.-Ing. I. Willms etwork Theory S. 03

104 .6. ormalization in the time and frequency domain The transform of the signal representation with units into the completely standardized representation of the signal happens with the help of two standardization operations. The first normalising operation leads to the dimensionless signal: x t xt () A s t 0 n () () A A The second normalising operation leads to dimensionless time: t T Prof. Dr.-Ing. I. Willms etwork Theory S. 04

105 .6. ormalization in the time and frequency domain x( T ) A0 Then one obtains: xn( T) s( T) A A By means of the relation a completely normalized signal results. A x T s s s 0 n( ) n( ) with n( ) ( ) An Application Under the condition of s () n 0, 0 the Laplace transform can be applied to a normalized signal as follows: P S ( P) s ( ) e d Ln 0 n Prof. Dr.-Ing. I. Willms etwork Theory S. 05

106 .6. ormalization in the time and frequency domain The inverse Laplace transformation is defined as: j 0 jp sn( ) lim SL n( P) e dp 0 j j0 In English literature usually instead of P the dimensionless variable s is used. Prof. Dr.-Ing. I. Willms etwork Theory S. 06

107 .6. ormalization in the time and frequency domain Using normalized variables the usual Laplace transform can be written as follows: p T L 0 S ( p) s( T ) e d( T ) The relations s () s( T ) Thus it follows: n L n 0 S ( p) T s ( ) e d! P L n Ln 0 S ( p) T s ( ) e d T S ( P) T p d( T ) T d give: Prof. Dr.-Ing. I. Willms etwork Theory S. 07

108 .6. ormalization in the time and frequency domain P Thus one obtains: S ( P) s ( ) e d Ln 0 n In a similar way by setting p = jω or P = jω the normalised version of the Fourier transform is obtained: ( ) p j SL p S ( ) S ( ) Ln P SFn( ) F Prof. Dr.-Ing. I. Willms etwork Theory S. 08

109 .6. ormalization in the time and frequency domain Thus it follows under certain restrictions: S ( ) S ( j) SFn( ) SLn( j) F L Here the following relationships hold: S ( ) T S ( ) with F Fn and of course: j S ( ) s ( ) e d Fn jt S ( ) s( t) e dt F n Prof. Dr.-Ing. I. Willms etwork Theory S. 09