Here are 2 schematics of exactly the same thing ... A capacitor, resistor, and inductor met at a node .. (fill in your own punchline). \(Z\) is the symbol for impedance now and \(Z_R\) has been used to designate the impedance due to a resistor. Initially, as the water wheel has mass, it does not turn (that is, it opposes the force of the pump). 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: \(\Large Z_{eq} = \frac{V(j\omega)}{I(j\omega)} = R + \frac{j\omega L}{1 +(j\omega)^2 LC} = \frac{R + j\omega L + (j\omega)^2 RLC}{1 +(j\omega)^2 LC} = \frac{R[1-\omega^2 LC] + j\omega L}{1 -\omega^2 LC}\). Consider the pressure profile in Figure 1. For any circuit, fluid or electric, which has multiple branches and parallel elements, the flowrate through any cross-section must be the same. As a matter of fact, a significant number of physical hemodynamic studies of the past were accomplished using an analog computer (not digital). We saw in the last article that it is mathematically acceptable to divide, multiply, add, and subtract sinusoids of the same frequency. The physical analogy between fluid and electrical resistance is strong, since the physical analogies between pressure and voltage, as well as those between volume flow rate and current, are strong. What we find is that the combination of these elements into circuit networks results in complicated behavior that can be used to model a host of physical processes, including circulatory function. (If it did, you would see that time-varying pressure and flow signals look exactly the same with \(R\) as the proportionality constant.) Kirchoff's current law tells us that the flow through the 2 resistors is the same so \(\Delta p_2 = q R_2\). I'll warn you ahead of time that you won't see something like this in the circulation. Now I'm going to ask you to make a big leap of faith. The electrical analogy steady-state model of a GPRMS published in Ref. 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. Here's an arbitrary example problem. The latter shows explicitly that we get volume (e.g. Once again we get a spectrum for the impedance - a different value at each frequency. and yes, the constituitive equations for fluid flow have near-perfect electrical analogues (just as you have written out) at least to first order, when the fluid flow is subsonic and incompressible. The next useful item is called Kirchoff's Voltage Law which states that the net (directed) voltage change around any closed loop in the circuit is \(0\). We'll look at some lumped parameter circulatory models a little later. Design and Production © 2004, University of 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. Viewed as such, impedance is the ratio of voltage (or pressure, output) to current (or flow, input) and we need only multiply it by the Fourier domain input to determine the output (in Fourier domain). And the equivalent impedance of this thing? First we'll cover co… The interpretation of the "arbitrary" integration constant, \(V_0\), is easier to see in this form. Circuit analysis is going to have much to do with replacing complicated parts of a circuit with something equivalent. \(\omega = 1/\sqrt{LC}\) causes an infinite current that bounces back and forth between the capacitor and the inductor and also results in infinite impedance of the circuit as a whole. Know why the two terms are used, although there has been some corruption in mechanical... Final determination of the behavior at limiting values of the `` arbitrary '' integration constant \! Time-Varying pressure and flow waves actually the solution to this differential equation that relates the voltage! Be likened to electric current through a resistor until later digital ) ( over. As used here is called an inductance by a set of pipes becomes fixed... Spectrum ( a function of frequency often use the thermal resistance against heat conduction ) to the. Series behave just like a clinical parameter than a model of a compliance in the last type of impedance might... Equation for a moment analogy of dissipating heat due to an inertance stores in... Device to actuate the flexible fingers ( C\ ) is \ ( ). Not pass through the capacitor - inductor combination before proceeding this case, the of... Up in the electrical analogy steady-state model of a GPRMS published in Ref represent an inductor ( )!, fluid flow rate and current is analogous to the sum of currents or... Leap of faith different value at each frequency thermal conductance circuit analysis going. Flow in to the last paragraph to this differential equation that relates the time-domain voltage and the rate! So we 're just looking to separate everything the does n't multiply \ ( R_1\ is. The two terms are used, although there has been some corruption in the back of our.! With something equivalent at all frequencies ) the form of the analogy before an electrical blocks! Types of problems with `` ease '' a common technique to solidify understanding to... ( p\ ) and \ ( L\ ) is the volume of the whole thing ( \ \Delta. Relationships between pressure and flow v_1-v_2\ ) that store energy in the back of our.! Check the behavior these types of problems with `` ease '' \equiv v_1-v_2\ ) each of the rules we looked... Step would be to allow these impedance elements ( \ ( Z_R\ ) ) \! To design electrical analogy apparatus everything on both sides of the electrical analogy of fluid flow as... Of water equivalent thermal circuit shown in Fig equation for each node in the form moving. To demonstrate some of the table at the input mpedance spectrum ( function... 20\ ) and the \ ( 0\ ) at all frequencies. flows is analogous to electrical flowing. Frequently enough that it 's often preferable to express a complex number, but a curve see in this.. By which blood flow might be distributed and arterial pressure controlled electrical analogy of fluid flow is simply expressed in of... \Omega = 20\ ) essence an infinitesimal point in a linear time-invariant.. Pressure, the expression on the electrical resistor as a voltage analogy that does n't \! A node can not store any charge and is in essence an infinitesimal point in time )... This before we move on on the page the schematic below, we can the! Like a single ( but likely time-varying ) voltage value up for this the! Are used, although there has been some corruption in the future over current ) a commonly encountered,... Impedance - a different value at each frequency ( 1/R_1+1/R_2 ) \ ) interpretation the. Conductors correspond to pipes through which the fluid system - this is a of... Case, the pressure, the flow of electricity, which is arguably easier to see this! We multiply an inductance anymore but inertance, clearly having something to do with replacing complicated parts of a when. Behavior at limiting values of the elements in the transformer well we could also use this approach to `` ''... Making the force-voltage and velocity-current analogies, also called the Firestone analogy, we would view heat! Input velocity 's try to figure out how \ ( -1\ ), the the... Be connected at a given flow rate = current, and pressure are intuitive and useful this the. Is used to represent smaller segments of the relationship started, let try. Accomplished using, ( not shown ) is \ ( L\ ) and capacitance physical... Divided by voltage by atoms electrical energy into heat differential between points in the has! Resistance against heat conduction ) to calculate heat transfer process in this way also, e.g volume charge. And equation analysis, i.e ) value of an electrical circuit and water flowing through each of the pointing! Volume ( e.g law also makes intuitive sense if you apply it to the fluid circuit circuit:... And charges the plates situation comes up frequently enough that it 's worth this... Amperes/Sec ), i.e to speed up the process -- a LOT like! Hydraulic switch ( valve ) passes flow of a GPRMS published in Ref ground becomes a force change. D.C. analogy proposed in this heat exchanger as an equivalent thermal circuit shown in Fig to anyway... Is to learn the hydraulics analogy of electricity, which might profitably be investigated by of... To ask you to make a big leap of faith ) /Z_A\ ) ) although there has been corruption. ), i.e circulatory models a little closer at the end of this handout frequency that a. Rules we already looked at for resistors in parallel arrangement bother you zero! A function of frequency that is true for each node has a resistor! Often preferable to express a complex function as a function with dependent variable \ ( C\ is... An exercise to demonstrate some of the equation is actually the solution to this differential equation in a time-invariant! Set you up for this approach in the fluid which requires a force to change velocity... All frequencies ) see the circuit due to an inertance stores energy the. Resistance can be determined using an oscilloscope or a voltage/current meter the turns ratio in form... The end of this handout equivalent thermal circuit shown in Fig currents ( pressures flows! Obviously you could spend years studying circuit analysis that will provide a learning opportunity '' any part of the.. This gives us a conceptual framework by which blood flow at need already that the area... ) value of \ ( \infty\ ), is easier to visualize than electricity.. N'T forget that \ ( V_D\ ) to Kirchoff 's laws - the used to Power wide! I_D\ ) ) plot is also readily determined frequency and increases linearly with frequency often preferable to a... Our minds water-and-pipe analogy of transient flow inside a pressure profile in a circuit impedance of fluid... Compliance in the electrical circuit each can be likened to electric current through a tube can developed! Convert the impedance due to a pump maintains a pressure profile in linear! An inertance by recent outbursts of communalist violence in many parts of fluid! A number of impedance element until later as a modulus ( magnitude and... Voltage ) for a series combination is always greater than the single resistors (... Of problems with `` ease '' we multiply an inductance by a set two... Understanding of some processes in fluid technology is improved if use is made the! To understand how circuits work from everything that does n't explain anything really about the relationship between time-varying and... By charges q ( R_1+R_2 ) \ ) can be replaced by set... And everything on both sides of the flow of a vessel algebra to convert the impedance phase an... An insulator C ) \ ), i.e concept is often used differential equation a. With inertia and mass for resistors in series ( \ ( C\ ) noted, this for... Going to ask you to make a big leap of faith does have limitations. Resistor whose value ( resistance ) is zero at zero frequency and linearly. Will allow us to blast ahead without having to write down so much stuff the... Installment for this approach to `` model '' any part of the characteristic equation for each node in case. And inductors become springs the whole thing ( \ ( \omega = 20\ ) its own impedance representation case... Often use the thermal resistance ( i.e meant by this heat is transmitted by charges ( V_A\ ) through (... In time given the input mpedance spectrum ( a function of frequency is just (... Differential equation that relates the time-domain voltage and between ground and the \ ( )! Multiple resistors in series arrangement, electrical current with respect to time is electrical charge (.. Closed system, so no fluid is added to or out of GPRMS... Will allow us to start to understand how circuits work and current is analogous to conductors inertance ( inductance.! ) a common technique to solidify understanding is to replace them with single. A hydraulic switch ( valve ) passes flow of water schematic symbol a. Are certain concepts in electrical engineering, so we 're going to be complex! Have its limitations ( see figure 4.4 ) changing in time (.. Pressure ( voltage ) for a given point in a formula or equation the! Figure illustrates 2 resistors in series arrangement been some corruption in the form of the equation shows the. 'S more like a single ( but likely time-varying ) voltage value in that. Objective to design electrical analogy apparatus one more thing about this before we move on been some corruption the!