1.2: First-Order ODE Models

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1.2 First-Order ODE Models

Electrical, mechanical, thermal, and fluid systems that contain a single energy storage element are described by first-order ODE models, described in terms of the the output of the energy storage element. This is illustrated in the following examples.

Example 1.1: A series RC network

We consider a series RC network connected across a constant voltage source, \(V_

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\) (Figure 1.1). Kirchhoff’s voltage law (KVL) is used to model the circuit behavior as: \(v_

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+v_

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=V_

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\), where the capital letters are used to represent constant values and small letters represent time-varying quantities. By substituting: \(v_

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=v_{0}\), the ciruit output, and \(v_

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=iR=RC\frac{{\rm d}v_{0} }{{\rm d}t}\), we obtain a first-order ODE model that describes the circuit behavior as:

image1 \(RC\frac

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.\)

Figure 1: An RC circuit.

Example 1.2: A parallel RL network

We consider a parallel RL network connected across a constant current source, \(I_

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\) (Figure 1.2). The circuit is modeled by a first-order ODE, where the variable of interest is the inductor current, \(i_{L}\), and Kirchhoff’s current law (KCL) is applied at either of the nodes, to obtain: \(i_

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+i_

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=I_

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\). By substituting \(i_

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=\frac{v}{R} =\frac{L}{R} \frac{{\rm d}i_{L} }{{\rm d}t}\) we obtain the ODE dscription of the RL circuit as:

image2

\(\frac{L}{R} \frac

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(t)=I_

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Figure 2: An RL circuit.

We note that the constant multiplier appearing with the derivative term in the above RC and RL circuits defines the time constant of the circuit, i.e., the time when the system output in response to a constant input rises to 63.2% of its final value. The time constant is denoted by \(\tau\) and is measured in \(\left[sec\right]\). In particular, \(\tau =RC\) for the RC circuit, and \(\tau =L/R\) for the RL circuit.

Example 1.3: Inertial mass acted upon by a force

The motion of an inertial mass, \(m\), acted by a force, \(f(t)\), in the presence of kinetic friction represented by \(b\) is governed by Newton’s second law of motion. The friction opposing motion is represented as \(bv\), where \(v\) is the velocity variable and \(b\) is the friction constant. The resultant force on the mass element is \(f-bv\). The resulting first-order ODE model for the system is given as: \[m\frac{{\rm d}v(t)}{{\rm d}t} +bv(t)=f(t).\]

Figure 3: Motion of an inertial mass with surface friction.

The time constant for the mechanical model is: \(\tau =\frac{m}{b}\), and desribes the rate at which the velocity builds up in response to a constant force input.

To generalize, let \(u(t)\) denote a generic input, \(y(t)\) denote a generic output, and \(\tau\) denote a time constant; then, a generic first-order ODE model is expressed as: \[\tau \frac{{\rm d}y(t)}{{\rm d}t} +y(t)=u(t).\] Further examples of first-order models include the thermal and fluid systems. Theses systems may be modeled using balance equations applied to a control volume as illustrated below.

Example 1.4: A model for room heating

In order to model the room heating process, we assume that the heat flow into the room is denoted as \(q_{i}\), the thermal capacity of the room is \(C_

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\), the temperature of the room is \(\theta _

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\), the ambient temperature is \(\theta _

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\), and the wall insulation is represented by a thermal resistance, \(R_

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\). Then, from the heat energy balance, we can write: \[C_

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\frac{{\rm d}\theta _{{\rm r}} }{{\rm d}t} +\frac{\theta _{{\rm r}} -\theta _{{\rm a}} }{R_{{\rm w}} } =q_{i} .\] In terms of the temperature differential, \(\Delta \theta =\theta _

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-\theta _

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\), the govering differential equation is: \[R_

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\frac

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q_{i} .\] The temperature is measures in \([{}^\circ C]\), heat flow is measured in \([W]\), thermal capacitance is measured in \(\left[\frac{J}{{}^\circ C}\right]\), and thermal resistance is measured in \(\left[\frac{{}^\circ C}{W}\right]\).

Figure 4: Room heating with heatflow through walls.

We may note the similarity of the room heating model with the general first-order system model, \(\tau \frac{{\rm d}y}{{\rm d}t} +y=u\), where the thermal time constant is given by \(\tau =R_

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\).

Example 1.5: A hydraulic reservoir

We consider a cylindrical reservoir filled with an incompressible fluid with a controlled exit at the bottom (Figure 1.5).

To proceed further, let \(P\) denote the hydraulic pressure, \(A\) denote the area of the reservoir, \(h\) denote the height, \(V\) denote the volume, \(\rho\) denote the mass density, \(R_{l}\) denote the valve resistance to the fluid flow; \(q_{{\rm i}n} ,\; q_{{\rm o}ut}\) denote the volumetric flow rates, and \(g\) denote the gravitational constant. Then, the base pressure in the reservoir is obtained as: \[P=P_{{\rm a}tm} +\rho gh=P_{{\rm a}tm} +\frac{\rho g}{A} V.\] Using reservoir capacitance, defined as: \(C_{h} =\frac{{\rm d}V}{{\rm d}P} =\frac{A}{\rho g}\), the governing equation of the hydraulic flow through the reservoir is given as: \[C_{h} \frac{{\rm d}P}{{\rm d}t} =q_{{\rm i}n} -\frac{P-P_{{\rm a}tm} }{R_{l} }\] In terms of the pressure difference, the equation is written as: \[R_{l} C_{h} \frac{{\rm d}\Delta P}{{\rm d}t} +\Delta P=R_{l} q_{in} .\] The above equation matches the standard first-order system model with \(\tau =R_{l} C_{h}\). Further, using \(\mathrm{\Delta }P=\rho gh\) and \(C_h=\frac{A}{\rho g}\), we can equivalently express the governing equation in terms of the liquid height, \(h\left(t\right)\), in the reservoir as: \[AR_l\frac{dh}{dt}+\rho gh=R_lq_{in}\] In the above, the hydraulic pressure is measures in \(\left[\frac{N}{m^2}\right]\), volumetric flow is measured in \(\left[\frac{m^3}{s}\right]\), hydraulic capacitance is measured in \(\left[\frac{m^5}{N}\right]\), and flow resistance is measured in \(\left[\frac{Ns}{m^5}\right]\).

Figure 5: Fluid reservoir with constricted outflow.