Equivalent Circuits. Henna Tahvanainen. November 4, ELEC-E5610 Acoustics and the Physics of Sound, Lecture 3
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1 Equivalent Circuits ELEC-E5610 Acoustics and the Physics of Sound, Lecture 3 Henna Tahvanainen Department of Signal Processing and Acoustics Aalto University School of Science and Technology November 4, 2015
2 1 General Principles Equivalent Circuits 2/22
3 1 Analogies Analogies Lumped Elements Applications Analogies are two or more systems that produce similar mathematical representations (apart from constant coefficients and units) e. g. lumped mechanical or acoustical systems and electric circuits In other words, acoustical and mechanical systems can be represented with electric circuits (especially when considering low frequencies). What is the advantage of this? There are well-known methods for solving electric circuits calculations are typically easier Equivalent Circuits 3/22
4 1 Lumped Elements Analogies Lumped Elements Applications If the dimensions of an object are small compared to the wavelength, the wave variable can be considered constant throughout the object. the state change can be considered simultaneous at every point on the object physical dimensions become irrelevant, the object may be treated as lumped (or point-like) e. g. inductors, capacitors, resistors (electronics) e. g. point masses, springs, dampers (mechanics) Remember: it depends on the frequency range of interest, whether the lumped representation is valid! Equivalent Circuits 4/22
5 1 Applications Analogies Lumped Elements Applications Where can the equivalent circuits be used in? electrical representation for a mechanical or acoustical system mechanical representation for an acoustical system linear representation for a rotational system unified representation of a multi-domain system Figure : Equivalent circuit for a SET power amp output chain. Equivalent Circuits 5/22
6 2 Analogies Between Domains Equivalent Circuits 6/22
7 2 Mass and Inductance Mass and Inductance Spring and Capacitance Resistance Other Components Type of Analogy Interconnection of Components The inertial components relate the potential variable to the time derivative of the flux variable: Mechanical domain F = m v Electrical domain U = Lİ Acoustical domain p = L a Q Analogy Table where: F - force v - velocity m - mass where: U - voltage I - current L - inductance where: p - pressure Q - vol. velocity L a - ac. ind. See the analogy? Equivalent Circuits 7/22
8 2 Spring and Capacitance Mass and Inductance Spring and Capacitance Resistance Other Components Type of Analogy Interconnection of Components Also, potential energy storage components relate the potential variable to the time integral of the flux variable: Mechanical domain F = K vdt Electrical domain U = 1 Idt C Acoustical domain p = 1 C a Qdt Analogy Table where: F - force v - velocity K - spring constant where: U - voltage I - current C - capacitance where: p - pressure Q - vol. velocity C a - acoust. capacitance. Equivalent Circuits 8/22
9 2 Resistance Mass and Inductance Spring and Capacitance Resistance Other Components Type of Analogy Interconnection of Components Finally, dissipative components relate the potential variable directly to the flux variable: Mechanical domain F = R m v Electrical domain U = RI Acoustical domain p = R a Q Analogy Table where: F - force v - velocity R m - mech. resistance where: U - voltage I - current R - resistance where: p - pressure Q - vol. velocity R a - acoust. resistance Equivalent Circuits 9/22
10 2 Other Components Mass and Inductance Spring and Capacitance Resistance Other Components Also other electric components (but not all) have mechanical or acoustical analogies e. g. an ideal transformer corresponds to a massless ideal lever (no acoustic counterpart, though!) Type of Analogy Interconnection of Components Analogy Table Equivalent Circuits 10/22
11 2 Type of Analogy Mass and Inductance Spring and Capacitance Resistance Other Components Type of Analogy Interconnection of Components Analogy Table In the previous discussion, the relation between the K-pair (impedance) in different domains was seen analogous. Thus, this type of analogy is called impedance analogy (or Maxwell s or direct analogy). However, alternative analogies exist, too! admittance (or mobility or Firestone or inverse) analogy treats the (I, F, p) as analogous m 1 C, 1 K L, R m 1 R, for other analogies and further info, see [?] The suitable analogy type depends on the system. Equivalent Circuits 11/22
12 2 Interconnection of Components Mass and Inductance Spring and Capacitance Resistance Other Components Type of Analogy Interconnection of Components Analogy Table The components connect to each other either in series or parallel. The type of connection depends on the chosen analogy. if there is the same force (or pressure) affecting the ends two components, the components are in parallel according to the impedance analogy series according to the admittance analogy if the ends of two components move at the same velocity (or there is a common volume flow through them), they are in series according to the impedance analogy parallel according to the admittance analogy Equivalent Circuits 12/22
13 2 Analogy Table Mass and Inductance Spring and Capacitance Resistance Other Components Type of Analogy Interconnection of Components Analogy Table Table : Component types and connections el. impedance el. admittance mechanical acoustical voltage U current I force F pressure p current I voltage U velocity v vol. velocity Q impedance Z admittance Y mech. imped. Z M ac. imped. Z A resistance R conductance G mech. res. R M ac. res. R A inductance L capacitance C mass m ac. ind. L A capacitance C inductance L compliance 1 ac. cap. C K A series conn. parallel conn. common velocity common vol. vel. parallel conn. series conn. common force common pressure Equivalent Circuits 13/22
14 3 How to Make Equivalent Circuits Equivalent Circuits 14/22
15 3 Outline of the Procedure Outline of the Procedure Impedance Analogy Admittance Analogy Star-to-Triangle Transform Bibliography In the following, we will learn how to make electrical equivalent circuits for mechanical systems using the impedance analogy admittance analogy and also how to transfer an impedance analogy into an admittance analogy. This is a rule of thumb -approach, meaning that the equivalent circuits can also be formed by analyzing the combined movement of objects (as done previously). Comparing these approaches would make a good brain exercise! Equivalent Circuits 15/22
16 3 Impedance Analogy Outline of the Procedure Impedance Analogy Admittance Analogy Star-to-Triangle Transform Bibliography The algorithm for going from mechanical to electrical representation is as follows: 1. make a circuit loop for each mass 2. make a circuit loop for each generator not connected to a mass 3. into each loop: add the electrical version of the directly involved components in series 4. connect different loops by combining their shared elements Let s make the equivalent circuit for a 2DOF mass-spring oscillator system... Equivalent Circuits 16/22
17 s 1 s 2 s 3 Outline of the Procedure Impedance Analogy Admittance Analogy Star-to-Triangle Transform Bibliography F m 1 m 2 R M1 R M2 R M3 Make loop for m 1, make loop for m 2, and connect the loops. m 1 s 1 R M1 R M3 s 3 m 2 F V s 2 s 2 R M2 R M2 Equivalent Circuits 17/22
18 3 Admittance Analogy Outline of the Procedure Impedance Analogy Admittance Analogy Star-to-Triangle Transform Bibliography Alternatively, the admittance analogy may be used. The algorithm for this is as follows: 1. for each mass, make circuit node with a grounded capacitor 2. make a grounded node for each generator not connected to a mass, and a node grounded for a rigid body 3. into each node: add the electrical version of the directly involved components in parallel 4. connect different node branches by combining their shared elements Let s re-make the equivalent circuit using the admittance analogy... Equivalent Circuits 18/22
19 s 1 s 2 s 3 Outline of the Procedure Impedance Analogy Admittance Analogy Star-to-Triangle Transform Bibliography F m 1 m 2 R M1 R M2 R M3 Make node for m 1, make node for m 2, make nodes for rigid bodies, and connect the nodes. F F s 1 s 1 s 2 s 2 s 3 s 3 1/R M1 1/R M1 1/R M2 m2 1/R M2 1/R M3 1/R M3 m 1 Equivalent Circuits 19/22
20 3 Star-to-Triangle Transform Outline of the Procedure Impedance Analogy Admittance Analogy Star-to-Triangle Transform Bibliography How to go from impedance analogy to admittance analogy? 1. inside each loop, insert a reference point 2. insert also a reference point outside the circuit 3. connect the points with lines 4. re-draw the connected pattern next to the circuit 5. for each line, change the intersecting series connection to parallel and switch the components into their dual versions Let s re-make the equivalent circuit using the admittance analogy... Equivalent Circuits 20/22
21 m 1 s 1 R M1 R M3 s 3 m 2 Outline of the Procedure F V s 2 R M2 Impedance Analogy Admittance Analogy Star-to-Triangle Transform Insert points in loops, and one outside, connect the points, re-draw the shape, insert branches... and ground. Bibliography F s 1 m 2 m1 1/R M1 1/R M3 s 3 s 2 1/R M2 Equivalent Circuits 21/22
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