TELLEGEN S THEOREM APPLIED TO MECHANICAL, FLUID AND THERMAL SYSTEMS

Size: px

Start display at page:

Download "TELLEGEN S THEOREM APPLIED TO MECHANICAL, FLUID AND THERMAL SYSTEMS"

Lester Strickland
5 years ago
Views:

1 Session 2793 TELLEGEN S THEOEM APPLIED TO MECHANICAL, LUID AND THEMAL SYSTEMS avi P. amachandran and V. amachandran 2. Department of Electrical and Computer Engineering, owan University, Glassboro, New Jersey Department of Electrical and Computer Engineering, Concordia University, Montreal, Canada H3G M8 Introduction Tellegen s theorem [][2] has been applied to several electrical systems that are linear or nonlinear, reciprocal or nonreciprocal, time-variant or time-invariant, and so forth. Tellegen s theorem states that the total power delivered to all the components of an electrical network is zero [][2]. This result has profound consequences in that the sum of the products of voltages and currents should be zero. If the voltage and current variables of a circuit are interchanged, we get an equivalent circuit with the same total delivered power. The two circuits are equivalent in terms of Kirchoff s Laws and topological equivalence [][2]. However, there is no dependence on the components of the circuit. or example, one may be a nonlinear resistive circuit and the other may be a linear circuit. The significant implication of Tellegen s theorem comes from the fact that two or more circuits with the same power constraints can be configured with no restrictions on linearity, time-invariance and method of analysis. The practical implication comes in the design process where students can exmine several equivalent networks and use for example, the simpler one in part of a design. So far, the discussion has centered about analog electrical networks only. Tellegen s theorem has been extended to discrete systems used in discrete signal processing [3]. This provides the derivation of different structures and a desirable one is chosen to suit the other properties required for a design. In this paper, we show that Tellegen s theorem can be extended to other types of systems. We consider mechanical, fluid and thermal systems. Across and Through Variables It is known that these four systems (electrical, mechanical, fluid and thermal) can be given a unified treatment by the use of across-variabes {av(t)} and through-variables {tv(t)} [4]. Proceedings of the 200 American Society for Engineering Education Annual Conference & Exposition

2 It is to be noted that for each system these variables are different and Table gives these variables, along with their units. System Across-variable av(t) Through-variable tv(t) Electrical Voltage (volts) v(t) Current (amperes) i(t) Mechanical Velocity v(t) orce f(t) (Translational) (meters/second) (Newtons) Mechanical (otational) Angular Velocity ω(t) (radians/second) Torque T(t) (Newtons)(meters) luid Pressure difference p(t) low rate q(t) (Pascals or (meter) 3 /second Newtons/meter 2 ) Thermal Degree Difference θ o C Heat low ϕ(t) ( o Celsius) (watts) Table Across and Through Variables for Different Types of Systems In each system, there will be mathematical relationships between av(t) and tv(t) depending on the property or physical element. These are: av(t) k tv(t) av(t) k 2 [ tv(t) ] () d (b) dt av(t) k 3 tv(t) dt (c) The relationship () corresponds to that of a resistor or equivalent; the relationship (b) corresponds to that of an inductor or equivalent; and the relationship (c) corresponds to that of a capacitor or equivalent. These relationships are not required in the statement of the extended Tellegen s theorem that we propose. However, they are required in the analysis. The analysis may be carried out either in the time-domain or in the Laplace transform domain. The Laplace transform domain equations corresponding to Eqs. (), (b) and (c) are given below: Av k. Tv () Av (sk 2 ). Tv () k Av 3 Tv (2c) s These equations are given assuming that the initial conditions are zero. If initial conditions are present, they are added as suitable sources. Proceedings of the 200 American Society for 2Engineering Education Annual Conference & Exposition

3 Table 2 gives the equations in the Lapalce-transform domain (and hence the impedances) in the above-mentioned systems corresponding to Eqs. (), () and (2c). esistance {Eq.()} Inductance {Eq.()} Capacitance {Eq.(2c)} Electrical V I V (sl) I I V Mechanical k (Translational) V s b V (b damper constant) (k spring constant) T B ω K T ω (B damper s constant) (K spring constant) luid P f Q P (sl f ) Q Mechanical (otational) Thermal ϕ Thermal θ (Does not exist) (sm) V (m mass in kg) T (sj) ω (J moment of inertia in (kg.meter 2 ) P Q Proceedings of the 200 American Society for 3Engineering Education Annual Conference & Exposition ϕ ( f ) θ Table 2 Laplace Transform elationships Between Across and Through Variables for Different Types of Systems rom Table 2, it is noted that a damper is the mechanical equivalent of a resistor and the spring is the mechanical equivalent of an inductor. The mechanical equivalent of a capacitor is either mass or moment of inertia. or fluid systems, the notions of resistance, inductance and capacitance exist. A tank is the physical realization of a capacitor. or thermal systems, only the concepts of resistance and capacitance exist. Application of Tellegen s Theorem To All our Systems Let N a be the equivalent electrical network of a system and let N b be another electrical equivalent network of the same system in that both N a and N b have the same topology. The extended Tellegen s theorem that we propose has four parts and they are given below. The relationships are written in the Laplace-transform domain. However, similar relationships can be written in the time-domain also. elationship (): elationship (2): B k Av ak. Tv ak 0 () f

4 B k Av bk.tv bk 0 () or both equations () and (), B is the number of branches in the network graph and is the same in both the systems. The subscript ak refers to the k th branch of Network N a. Similarly, the subscript bk refers to the k th branch in Network N b. The summation is taken over all the branches. Hence, in Network N a, the branches are numbered as a, a2, and so on. Similarly, in Network N b, the branches are numbered as b, b2, and so on. The relationships in equations () and () can readily be interpreted as the consequence of the Law of Conservation of Energy in a system. or example, in an electrical system, the sum of the products of voltage and current (which is the power) in each branch is equal to zero. elationship (3): elationship (4): B k Av ak. Tv bk 0 (3c) B k Av bk.tv ak 0 (3d) Equations (3c) and/or (3d) can be interpreted as the sum of the products of the av s (or tv s) in Network N a and the tv s (or av s) in Network N b will always be equal to zero. The following examples illustrate the above concept. Example Mechanical System igure (a) shows a mechanical system (called N a ) and igure (b) shows its equivalent circuit in the Laplace-transform domain. v v b v sa k m k m igure (a) A mechanical system designated as N a Proceedings of the 200 American Society for 4Engineering Education Annual Conference & Exposition

5 V V V sa + igure (b) The equivalent circuit of igure (a) in the Laplace transform domain The impedances are given by Analysis yields s s,, and (4) k sm k b + sm ( + + ) Vsa ( + + ) + ( + ) (5a) ( + ) Vsa ( + + ) + ( + ) (5b) ( ) Vsa ( + + ) + ( + ) (5c) V (5d) V (5e) We shall now consider another mechanical system (called N b ) shown in igure 2(a) and its Lapalce-transform domain equivalent circuit in igure 2(b). v b b b v sb v b k m b k b igure 2(a) The mechanical system called N b. Proceedings of the 200 American Society for 5Engineering Education Annual Conference & Exposition

6 b b V b V V sb + igure 2(b) The Laplace-transform equivalent circuit of igure 2(a) The impedances are given by Analysis yields b s s b,, and (6) k k b + sm b b b ( + + ) Vsb ( + + ) + ( + ) Proceedings of the 200 American Society for 6Engineering Education Annual Conference & Exposition (7a) b ( + ) Vsb ( + + ) + ( + ) (7b) b ( ) Vsb ( + + ) + ( + ) (7c) V b (7d) V (7e) igures (b) and 2(b) show that networks N a and N b have the same topology. It is readily verified that all the four relationships given in Equation (3) are satisfied. Specifically, Equation () becomes - V sa + (V sa V ) + V + (V V ) + V 0 (8a) Equation () becomes - V sb b + (V sb V b ) b + V b + (V b V ) + V 0 (8b) Equation (3c) becomes - V sa b + (V sa V ) b + V + (V V ) + V 0 (8c) Equation (3d) becomes - V sb + (V sb V b ) + V b + (V b V ) + V 0 (8d)

7 Example 2 luid System igure 3(a) depicts a fluid system which we can denote as N a and figure 3(b) shows the electrical circuit equivalent which has the same topology as the circuit in igure (b). Cf C f + (t) p sa L f (t) p L f (t) p f igure 3(a) A fluid system designated as N a Q Q P P P sa + Q Q igure 3(b) The equivalent circuit of igure 3(a) in the Laplace transform domain The impedances are given by sl f, f, sl f, igure 4(a) depicts a fluid system which we can denote as N b and figure 4(b) shows the electrical circuit equivalent which has the same topology as the circuit in igure 2(b). The fluid systems of this example are equivalent to the mechanical systems of example. The circuit topologies of the mechanical and fluid systems are all identical. f f f + Proceedings of the 200 American Society for 7Engineering Education Annual Conference & Exposition

8 C bf bf p b (t) L bf f + (t) p sb (t) p f igure 4(a) The fluid system called N b. Q b b P b Q P P sb + Q Q igure 4(b) The equivalent circuit of igure 4(a) in the Laplace transform domain The impedances are given by b bf, f, sl bf, The analysis and verification of Tellegen s theorem proceeds in an identical manner as for Example. Summary and Conclusions In this paper, it has been shown that Tellegen s theorem, enunciated for electrical systems so far, can be applied to other types of systems like mechanical, fluid and thermal systems also. Though one example of a pair of mechanical systems and one of a pair of fluid systems are given (due to space considerations), similar examples can be given for other systems also. This paper makes a thrust at multidisciplinary engineering education by showing that a powerful theorem in circuit theory can be applied in other systems thereby further enforcing the isomorphism among various systems. bf f f + Proceedings of the 200 American Society for 8Engineering Education Annual Conference & Exposition

9 It is suggested that courses in Circuits, Statics, Dynamics and luids integrate Tellegen s theorem and our extension to provide students with a better insight across enginering disciplines. This will partially address the demand of industry for acquiring engineers with a broad set of skills and a comprehension of diverse prctical applications and who can move across rather artificial program boundaries with great ease. eferences. J. Choma, Electrical Networks, John Wiley and Sons, N. Balabanian and T. Bickart, Linear Network Theory: Analysis, Properties and Synthesis, Matrix Publishers Inc., A. V. Oppenheim and. W. Schafer, Digital Signal Processing, Prentice-Hall Inc., J.. Lindsay and V. amachandran, Modeling and Analysis of Linear Physical Systems, Weber Systems Inc., January 99. Biography avi P. amachandran is an Associate Professor in the Department of Electrical and Computer Engineering at owan University. He received his Ph.D. from McGill University in 990 and has worked at AT&T Bell Laboratories and utgers University prior to joining owan. V. amachandran is a Professor in the Department of Electrical and Computer Engineering at Concordia University. He received his Ph.D. from the Indian Institute of Science in 965 and was on the faculty at the Technical University of Nova Scotia prior to joining Concordia. Proceedings of the 200 American Society for 9Engineering Education Annual Conference & Exposition

THE USE OF KIRCHOFF S CURRENT LAW AND CUT-SET EQUATIONS IN THE ANALYSIS OF BRIDGES AND TRUSSES

THE USE OF KIRCHOFF S CURRENT LAW AND CUT-SET EQUATIONS IN THE ANALYSIS OF BRIDGES AND TRUSSES Session TH US O KIRCHO S CURRNT LAW AND CUT-ST QUATIONS IN TH ANALYSIS O BRIDGS AND TRUSSS Ravi P. Ramachanran an V. Ramachanran. Department of lectrical an Computer ngineering, Rowan University, Glassboro,