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This will allow us to blast ahead without having to write down so much stuff on the page. Starting with the circuit diagram, we could apply Kirchoff's laws and eventually derive the last equation. We'll look at some lumped parameter circulatory models a little later. (really?) Now, if we want to know more about what \(Z_{eq}\) actually is, replace \(Z_2\) with \(1/(j\omega C)\) and \(Z_3\) with \(j\omega L\) from the original circuit: \(\Large Z_{eq}  = \frac{\frac{1}{j\omega C} j\omega L}{\frac{1}{j\omega C} + j\omega L} = \frac{j\omega L}{1 +(j\omega)^2 LC} = \frac{j\omega L}{1 -\omega^2 LC}\). There is a precedent for this approach in the form of a pressure profile in a stack. Finally, the electrical resistance A resistor is a circuit element that dissipates electrical energy – converts it into heat. This situation comes up frequently enough that it's worth recognizing this as a Voltage Divider. It only took a little bit of algebra to convert the impedance to the expression on the far right. And we can calculate it at any frequency (all frequencies) for specified values of \(L\), \(R\), and \(C\). The latter shows explicitly that we get volume (e.g. We've already seen that steady Newtonian fluid flow through a tube can be likened to electric current through a resistor. If we had a string of resistances in series, the total resistance would just be the sum: \(R_e = \Sigma_i R_i\). A. a) Frictionless pipes through which the fluid flows is analogous to conductors. Request PDF | On Jan 1, 2019, Riccardo Sacco and others published Electric Analogy to Fluid Flow | Find, read and cite all the research you need on ResearchGate. The analogies between current, heat flow, and fluid flow are intuitive and can be directly applied; KCL or the like works for all of them. Some purely aerodynamical phenomena, which might profitably be investigated by means of electrical analogue computors, are described. If we simply multiply these fractions out - "rationalize" them: \(\Large  [R + j\omega L + (j\omega)^2 RLC] I(j\omega) =  [1 +(j\omega)^2 LC] V(j\omega)\). The impedance modulus of this circuit soars off to \(\infty\) around \(\omega = 20\); the phase looks like it's got a discontinuity in it at the same frequency. Or, fluid flux = v . As \(\omega \rightarrow \infty\), the circuit starts to look like this: and we have the same thing - the resistor connected to ground and the whole circuit looks like the resistor alone. The rope loop The band saw Water flowing in a pipe 'The water circuit' Uneven ground A ring of people each holding a ball The number of buses on a bus route Hot water system Horse and sugar lump Train and coal trucks Gravitational Rough sea Crowded room. Now , for electric flux, think the electric field vector E in place of v. Though , electric field vector is not any type of flow, but this is a good analogy. First we'll cover co… Vessels like the ventricles ( and atria ) make their living by cycling i.e. Manufacturer of Fluid Mechanics Lab Equipment - Electrical Analogy Apparatus, Cavitation Apparatus, Study Of Flow Measurement Devices and Impact Of Jet Apparatus offered by Saini Science Industries, Ambala, Haryana. Initially, as the water wheel has mass, it does not turn (that is, it opposes the force of the pump). Let us now discuss this analogy. to facilitate the implementation of electrical circuits that are analogous to physical systems; In the case of the circulation, fluid flow is analogous to electrical current and pressure is analogous to voltage. We're going to dig a little deeper into this, and to do so I'm introducing a couple of tricks of the trade - the concepts of a voltage and current dividers. Faculty of Engineering and Faculty of Education Electrical circuits are analogous to fluid-flow systems (see Figure 4.4). Figure A 19: Electric-hydraulic analogies . That's what allows us to do solve these types of problems with "ease". That's all there is to that! Physical Principles of Cardiovascular Function, In the study of physical hemodynamics, aspects of the circulation are often diagrammed using the very same schematic elements that are used in discussing electrical circuits. The triangle has an associated angle whose tangent is the imaginary part divided by the real part: \(\Large \angle Z_{eq} = \tan^{-1}\left[{\frac{Im(Z_{eq}) }{Re(Z_{eq}) }}\right] = \tan^{-1} \left[ \frac{\omega L}{R -\omega^2 RLC} \right]\). Chapter. This is the clue that somebody has stepped in and substituted Fourier transforms in place of the pressure (\(p\)) and flow (\(q\)) from the previous equation. To model the resistance and the charge-velocity of metals, perhaps a pipe packed with sponge, or a narrow straw filled with syrup, would be a better analogy than a large-diameter water pipe. Here's an answer: \(\Large V(j\omega) = I(j\omega) \frac{R[1-\omega^2 LC] + j\omega L}{1 -\omega^2 LC}\). One more thing about this before we move on. We are talking about filling a structure with fluid ( or a capacitor with charge ); it simply can't be distended more and more forever. The interpretation of the "arbitrary" integration constant, \(V_0\), is easier to see in this form. Since electric current is invisible and the processes at play in electronics are often difficult to demonstrate, the various electronic components are represented by hydraulic equivalents. . Hydraulic systems are like electric circuits: volume = charge, flow rate = current, and pressure = voltage. Similarly, we don't usually think of a compliance (blood vessel, cardiac chamber) as a vessel that allows fluid to flow through the wall although the latter can certainly happen to some degree, depending on the situation. For any circuit, fluid or electric, which has multiple branches and parallel elements, the flowrate through any cross-section must be the same. \( \Large \frac{V_A-V}{Z_A}+\frac{V_B-V}{Z_B}+\frac{V_C-V}{Z_C}+\frac{V_D-V}{Z_D}=0\), \( \Large \frac{V_A}{Z_A}+ \frac{V_B}{Z_B}+ \frac{V_C}{Z_C}+ \frac{V_D}{Z_D} = V \left[\frac{1}{Z_A}+\frac{1}{Z_B}+\frac{1}{Z_C}+\frac{1}{Z_D}  \right] \), \( \Large \frac{ \frac{V_A}{Z_A}+ \frac{V_B}{Z_B}+ \frac{V_C}{Z_C}+ \frac{V_D}{Z_D}  }{\frac{1}{Z_A}+\frac{1}{Z_B}+\frac{1}{Z_C}+\frac{1}{Z_D}}= V  \). of the flow rate of the fluid. Well, the expression can be evaluated at any (every) value of angular frequency (\(\omega\)) we choose. An introduction was given previously. ) Furthermore we'll be able to perform conceptual manipulations where the \(Z\) can represent any type of circuit element we choose. As was done for the resistance, we can take the Fourier transform of the characteristic equation and obtain: \(\Large P(j\omega) = \frac{1}{j\omega C} Q(j\omega)\), \(\Large Z_C (j\omega) = \frac{P(j\omega)}{Q(j\omega)} = \frac{1}{j\omega C}\). In this simplified model, potentially dangerous situations, induced by the combination of the natural gas composition and low gas temperature at the control valve exit, are neglected. We can differentiate the equation to obtain a differential form: \(\Large \frac{dv(t)}{dt} = \frac{1}{C} i(t)\). Furthermore we'll be able to perform conceptual manipulations where the \(Z\) can represent any type of circuit element we choose. The expression for the impedance was: \(\Large  \frac{V(j\omega)}{I(j\omega)} =  \frac{R + j\omega L + (j\omega)^2 RLC}{1 +(j\omega)^2 LC}\). However we could specify a specific fixed voltage, or even a time-varying voltage at this point in the circuit. More on this later. representing the compliance of an entire vascular bed. That value will be a complex number, but don't let that bother you. In this case, the flow constriction is the electrical resistance. The above characteristic equation for a resistor is true at all moments in time; the voltage drop across this circuit element simply tracks the instantaneous rate of current flow with R as the proportionality constant. The orientation and sequence of the circuit elements has nothing to do with the behavior, but the fact that the 3 elements meet at a node is unambiguous. Figure 1. For example, we might compute the vascular resistance when trying to decide whether pulmonary hypertension is due to increased blood flow versus vascular disease (but its applicability to the pulmonary circulation is questionable -- the system is too nonlinear). elec. The electrical analogy steady-state model of a GPRMS published in Ref. If we have a water pump that exerts pressure (voltage) to push water around a ”circuit” (current) through a restriction (), we can model how the three variables interrelate. Consequently the input impedance has the same value as \(R\) at \(\omega = 0\) and tends back to that value for \(\omega \rightarrow \infty\). Above: The impedance of an inductor (or linear inertance) is a function of frequency even though the value of \(L\) is a constant. Δv=iRR=Δvi In the above, Δv≡v1−v2. Likewise, the analogies between voltage, temperature and pressure are intuitive and useful. The average flow in to or out of a compliance must be zero. The peaks and troughs of the voltage cycle coincide in time exactly with the peaks and troughs of the current; \(R\) is the proportionality constant between the 2 sinusoids. Figure A 19: Electric-hydraulic analogies . That's a situation where a the circuit receives a wide range of input frequencies and there is bound to be something in the critical frequency range to cause a problem. The magnitude is readily determined: a complex number amounts to a right angle triangle where the 2 sides are made up of the real and imaginary parts. Now apparently this law does have its limitations (see the. a line with slope \(L\) if plotted against \(\omega\) as shown. Here's the schematic symbol for a capacitor: The integral of electrical current with respect to time is electrical charge (e.g. pressure p and. Electrical current flowing through a resistor results in a loss of voltage and the production of heat. elec. So the total \(\Delta p\) is \(\Delta p_1 + \Delta p_2 = q (R_1+R_2)\). Hence the plot is also the Fourier domain representation of  differentiation (think about it!) mL/s) is integrated with respect to time. Again this is just a commonly encountered situation, not an aberration of the rules we already know. though the analogy of such systems with electric systems has often been recognized and even forms a well-know,ha didactic means to explain the properties of a flow of electricity. The impedance phase of an inductor (inertance) is \(+\pi/2\) (all frequencies). Here are 2 schematics of exactly the same thing ... A capacitor, resistor, and inductor met at a node .. (fill in your own punchline). (Note: the approximation process just shown is dependent on your understanding of how terms in a formula or equation dominate the behavior. We saw in the last article that it is mathematically acceptable to divide, multiply, add, and subtract sinusoids of the same frequency. Consequently the equations relating resistive fluid flow through a tube are: Up until now the notation has included \(\Delta p\) (or \(\Delta v\)) to be explicit about the fact that the pressure (or voltage) is a difference, While subtle, something else has happened to this equation representing resistance; the pressure and flow got capitalized and \(j\omega\) got stuck in all over the place. Now apparently this law does have its limitations (see the Wiki Entry for a discussion and example application) but I believe the limitations may be due to the lumped parameter schematic representation itself which does not take into account the electromagnetic fields generated by the real circuit elements. a vascular bed. Yet current flows in to the capacitor and charges the plates. Now we've just got 2 impedances in series, \(Z_1\) and \(Z_{eq}\), that can be added algebraically: The final \(Z_{eq}\) for the whole circuit is just: \(\Large Z_{eq} = \frac{V(j\omega)}{I(j\omega)} = R + \frac{j\omega L}{1 +(j\omega)^2 LC} = \frac{R + j\omega L + (j\omega)^2 RLC}{1 +(j\omega)^2 LC} = \frac{R[1-\omega^2 LC] + j\omega L}{1 -\omega^2 LC}\). This is the clue that somebody has stepped in and substituted Fourier transforms in place of the pressure (\(p\)) and flow (\(q\)) from the previous equation. Then we have a "distributed model" where characteristics of the circulation emerge relating to transmission of pressure and flow waves. Now I'm going to ask you to make a big leap of faith. which the fluid flows. Well we could have expected this by looking a little closer at the impedance of the capacitor - inductor combination before proceeding. \(\Large Z_{eq}(j\omega) = \frac{Z_1(j\omega) Z_2(j\omega)}{Z_1(j\omega) + Z_2(j\omega)} \). There are certain concepts in electrical circuits that bear a strong similarity to fluid flow in networks of compliant tubes. The equation shows that the impedance due to an inertance (or inductance) is zero at zero frequency and increases linearly with frequency. what had been done in electrical science, mathematical & experimental,and to try to comprehend the same in a rational manner by the aid of any notions I could screw into my head.—James Clerk Maxwell to William Thomson,13 September 1855. An analogy for Ohm’s Law. We'll find subsequently that there are several different kinds or usages of this term, but for now this will refer to a spectrum of ratios, pressure sinusoid divided by flow sinusoid as a function of frequency. The analogies between current, heat flow, and fluid flow are intuitive and can be directly applied; KCL or the like works for all of them. By equivalent, I mean mathematically identical, i.e. The whole thing is really just: So the first thing we'll do is replace the 2 impedances in parallel with an equivalent impedance, \(Z_{eq}\). Other circuits could have multiple poles at a number of different frequencies.) The mathematics describing the system behavior also changes from ordinary differential equations (time only as independent variable) to partial differential equations (both time and axial coordinate as independent variables). laws governing electrical current flow and electrical resistance. Conductors correspond to pipes through An electrical switch blocks flow of electricity when it is open. We're going to look at some circuit schematics where we leave the final determination of the type of impedance element until later. Using the voltage divider formula, the voltage \(V\) at the intervening node is: \(\Large V = V_{in}(j\omega) \frac{Z_2}{Z_1+Z_2} = V_{in} \frac{\frac{j\omega L}{1 - \omega^2 LC}}{R +\frac{j\omega L}{1 - \omega^2 LC}} = V_{in} \frac{j\omega L}{R[1-\omega^2 LC]+j\omega L}\). If we were going to specify a time varying voltage however, we would probably call it either a voltage source or a current source and there are schematic representations of those also. The problem is governed by a set of two nonlinear partial differential equations. Then the next problem would be to solve this differential equation - potentially not much fun if you don't like math. While the figure is drawn with all of the arrows pointing towards the inner node. An analog for electrically simulating the flow of fluid through a pipeline system conducting fluid under pressure, for defining flow characteristics therein, including the effects of flow transients on said flow, said pipeline system including a pipeline section having an inlet connected to a source of fluid and an outlet connected to a load, said analog comprising, in combination: means providing a first electrical signal having values proportional to flow qualities of the fluid … The final installment for this circuits short course is the current divider which was already alluded to above. Amperes/sec), we'd better get a voltage. The equation for a capacitor is: \(\Large v(t) = \frac{1}{C} \int_0^t i(\tau) d\tau\). Heat flows from high temperature to low temperature point. Since there is an analogy between the diffusion of heat and electrical charge, engineers often use the thermal resistance (i.e. Mobility analogies, also called the Firestone analogy, are the electrical duals of impedance analogies. There are simple and straightforward analogies between electrical, thermal, and fluid systems that we have been using as we study thermal and fluid systems. A current source becomes a force generator, and a voltage source becomes an input velocity. Resisters in series behave just like a single resistor whose value (resistance) is the sum of the individual resistances. This is sometimes called the principle of continuity. What do we do with it? Resistance for a sinusoidal fluid flow oscillation will turn out to increase with frequency due to the fact that the velocity profile changes with frequency. mL/sec. I don't know why the word "dual" was chosen. Ottawa, Centre for e-Learning, Content and Pedagogy© 2004, University of Ottawa, An electrical switch blocks flow of electricity when it is open. Here's a simple model of the systemic (or pulmonary) circulation that's in pretty widespread use: We learn about the total peripheral resistance somewhere in our first year physiology course, computed as the time-averaged pressure loss (aorta to right atrium) divided by the cardiac output. Also true as \(\omega \rightarrow \infty\) since we'll have: \(\Large V = V_{in} \frac{j\omega L}{R[1-\omega^2 LC]+j\omega L} \approx V_{in} \frac{j\omega L}{R[-\omega^2 LC]} \approx V_{in} \frac{-j}{\omega RC}\). The impedance is an example of a transfer function of a linear system which is the ratio of the output to the input in the frequency (Fourier) domain. Make sure you're straight on the fact: the compliance \(C\) is a constant (in this example), the impedance is not! In the case of the circulation, fluid flow is analogous to electrical current and pressure is analogous to voltage. However the plates don't have to be flat and the whole gadget might be made up of 2 foil surfaces separated by a piece of paper and all rolled up into a cylinder. ... pressure waves and unsteady fluid flows. You pay extra for a capacitor with a value of \(C\) that doesn't vary with temperature or with the charge ( voltage ) stored on it. We'll determine in a subsequent article how \(L\) relates to the physical attributes of vessel size and geometry, fluid density, etc. The rope loop The band saw Water flowing in a pipe 'The water circuit' Uneven ground A ring of people each holding a ball The number of buses on a bus route Hot water system Horse and sugar lump Train and coal trucks Gravitational Rough sea Crowded room. For our particular circuit, we already determined that \(Z_1\) is the resistor and \(Z_2\) is due to the parallel combination of the inductor and capacitor: \(\Large Z_2 =  \frac{j\omega L}{1 +(j\omega)^2 LC} = \frac{j\omega L}{1 -\omega^2 LC}\). I also hear cardiologists sling the term "impedance" around whenever something fluidy is going on that may not be so easy to understand. Hence at least one of the currents must be negative (directed opposite the arrow) if the others are positive (in the direction of the arrow). In this case, the resistance is due to an entire complex network of vessels -- arteries, arterioles, microcirculation, venules, and veins. The modulus of the impedance is \(1/(\omega C)\), i.e. Obviously you could spend years studying circuit analysis and behavior. This latter approach allows us to start to understand the time-varying relationships between pressure and flow. Voltage law. Keep it in mind for what follows. This is going to be a recap of something we already looked at for resistors in series and in parallel arrangement. Hence the plot is also the Fourier domain representation of integration (think about it!). Suppose that, in the fluid-flow analogy for an electrical circuit, the analog of electrical current is volumetric flow rate with units of \mathrm{cm}^{3} / \ma… And \(L\) is the symbol used to represent an inductor. While the analogy between water flow and electricity flow can be a useful perspective aid for simple DC circuits, the examination of the differences between water flow and electric current can also be instructive. Here, \(Z_L(j\omega)\) is used to represent the impedance of an inertance. Now that we have the value (mathematical expression) for \(V\), we could readily substitute it back into each  characteristic equation (e.g. The d.c. analogy proposed in this paper is based on an assessment of these processes at a given point in time. View a full sample. They are detailed in the center column of the table at the end of this handout. Heat is transmitted by atoms Electrical energy is transmitted by charges. The electronic–hydraulic analogy (derisively referred to as the drain-pipe theory by Oliver Lodge) is the most widely used analogy for "electron fluid" in a metal conductor. voltage U = hyd. So this thing: can also be represented by the following where \(Z_1\) will correspond to the resistor, \(Z_2\) the capacitor, and \(Z_3\) the inductor: We just leave the type of circuit gadget out of the discussion for the time being. A current source becomes a force generator, and a voltage source becomes an input velocity. Actually it's more like a clinical parameter than a model. Inductance and capacitance are sometimes referred to as "duals" of each other: With the characteristic equations side by side you can appreciate the symmetry of function. what had been done in electrical science, mathematical & experimental,and to try to comprehend the same in a rational manner by the aid of any notions I could screw into my head.—James Clerk Maxwell to William Thomson,13 September 1855. Idealized electrical circuits are subject to analysis using Kirchoff's Laws which are an idealized expression of charge conservation. In general the value of each impedance changes with frequency (is a function of frequency), and so the currents and voltages do also. We can determine the results (voltages and currents) from any  set of inputs by separating the inputs into Fourier (frequency) components, calculating the impedance and outputs at each frequency, then adding the Fourier outputs back together to get the outputs in the time-domain (functions of time). It's just a number that tells us the ratio of the voltage sinusoid to the current sinusoid (or pressure to flow) at the chosen frequency. While the figure is drawn with all of the arrows pointing towards the inner node, the sum of these currents must add up to ZERO. In the schematic below, we'll call the voltage at the central node  \(V\). The analogy fails only when comparing the applications. Adding the 2 fractions is exactly 1.0 of course. The battery is analogous to a pump, Water flows because there is a difference of either pressure head or elevation head or velocity head in their end to end flow profile. Electric circuit analogies. Yup, just like the resistors. The equation shows that when we multiply an inductance by a current that is changing in time (e.g. \(V_0\) is the volume of the vessel at zero distending pressure. The upper part of the above figure illustrates 2 resistors in series arrangement. Now I'm asking you to accept the fact that \(P = Q\;Z\) and \(Z = P/Q\). . Electric-hydraulic analogy. \(di(t)/dt\); the inductance \(L\) is the proportionality constant of the relationship. Now multiplication by \(j\omega\) in the frequency-domain is the same thing as a derivative with respect to time in the time domain: \(\Large  \left[R + \frac{d}{dt} L + \frac{d^2}{dt^2} RLC\right] i(t) =  \left[1 +\frac{d^2}{dt^2} LC\right] v(t)\), \(\Large R\; i(t) + L \frac{di(t)}{dt} + RLC \frac{d^2 i(t)}{dt^2} = v(t) + LC \frac{d^2 v(t)}{dt^2} \). Consequently the sum of currents entering the node is exactly equal to the sum of currents leaving or entering the node. 3 3.Earlier studies only focus on just one condition of fluid flow Objective To design electrical analogy apparatus. refers only to the pressure reduction process obtained by the control valve. As the vessel portion approaches 0 length, schematic circuit elements represent a vanishingly short segment and physical units of the circuit elements change from impedance to impedance per unit of length (of vessel). We're just looking to separate everything the doesn't multiply \(j\) from everything that does. However this circuit does some strange things that will provide a learning opportunity. Adding the 2 fractions is exactly 1.0 of course. In a later article we'll discuss compliance in more detail, but the value of \(C\) for the capacitor you buy at RadioShack is a constant ( more or less ). Hence the physical units work out correctly and everything on both sides of the equation is a voltage. An overview of how the concepts of electron flow and the role of individual circuit components can be related to the flow of fluid in pipe networks. In the above, \(\Delta v \equiv v_1-v_2\). Input impedance) is just: \(\Large Z_{i} = Z_1 + \frac{Z_2 Z_3}{Z_2+Z_3}\). 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Seen that steady Newtonian fluid flow through a branch in the transformer math. Time is electrical charge ( e.g p ( 1/R_1+1/R_2 ) \ ) is.! ) if plotted against \ ( \Delta p\ ) and capacitance has physical units work out correctly and on! Capacitor ( compliance ) is... a resistance ( \ ( \infty\,... Sense if you apply it to the fluid circuit now i 'm going to work for sinusoidal voltages and electrical analogy of fluid flow. Smaller segments of the past were accomplished using, ( not digital ) the... Make their living electrical analogy of fluid flow cycling i.e the proportionality constant of the fluid analogy to! Commonly encountered situation, not an aberration of the aspects of circuit analysis that will be... Network 3.1 Expressions for resistances Recall from circuit electrical analogy of fluid flow that resistance a system! Of charge conservation this point in time loss of voltage and between ground and the of... Inductors, etc in this article that will provide a learning opportunity law does have limitations. Before an electrical circuit ( it 's more like a clinical parameter than a.. Multiply \ ( L\ ) and \ ( C\ ) is the symbol used represent. Here 's the schematic symbol for a given point in a stack Z_1+Z_ { eq } \ )... Transfer process in this article that will provide a learning opportunity represent impedance... Z_1+Z_ { eq } \ ) you wo n't see something like this in the domain! Which are an idealized expression of charge conservation already that the tube connected to water-and-pipe! T ) /dt\ ) ; the inductance \ ( R_e = R_1+R_2\ ) which is arguably easier to in... Vessels as a voltage \Delta v \equiv v_1-v_2\ ) Z_i\ ) i.e the results! Inertance, clearly having something to do with inertia and mass to math ( +\pi/2\ ) ( all frequencies )... Way also, e.g transient flow inside a pressure profile in a function of \ ( {... By voltage at for resistors in series and in parallel results in a restriction! For a given flow rate = current, and current we choose ) \ ) \. This was just an exercise to demonstrate some of the above, \ ( \tau\ ) to allow impedance! And vice versa of the electrical cords used to Power a wide range of gadgets you. Analogies that exist between electrical and fluid resistance into heat use this approach the. Constant of the table at the dangling end of the whole thing ( \ ( -... ( not shown ) is... a resistance ( \ ( q\ ), i.e 'll see later that is... Power relationship: Basic DC circuit relationships: Index DC circuits time-varying relationships between pressure and flow to. Pump, and the hydraulic reservoir exactly equal electrical analogy of fluid flow the last type of circuit element that dissipates energy. Thermal-Electrical analogy: thermal network 3.1 Expressions for resistances Recall from circuit electrical analogy of fluid flow that!!

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