L15

10/09/26 Thu 8:03am

Lumped and Distributed parameters

While modelling systems always use lumped parameters
distributed parameters are data at each point of the pipe

Lumped parameters assume that physical properties like resistance, capacitance, and inductance are concentrated at specific points, ignoring wave propagation and time delays.  This lumped parameter model is suitable for low-frequency circuits where component dimensions are much smaller than the signal wavelength, allowing the system to be described by ordinary differential equations (ODEs).

Distributed parameters assume these properties are spread continuously throughout a structure, such as in transmission lines or flexible structures.  This distributed parameter system accounts for spatial and temporal variations, requiring partial differential equations (PDEs) or delay differential equations to model.  It is essential for high-frequency circuits or physical systems where dimensions are comparable to the signal wavelength.

Modeling of Flow Systems

$$Flow=\frac{\text{Level difference}}{\text{Resistance}}⇝q=\frac{\Delta h}{R}$$ $$\text{What comes in}-\text{What goes out}=\text{What accumulates}$$

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$$q_{12}=\frac{h_1-h_2}{R_1}$$ $$q_{out}=\frac{h_2}{R_2}$$

Dynamic equation for tank 1

$$q_{in}-q_{12}=A_1\frac{\delta h_1}{\delta t}\to 1$$

Dynamic equation for tank 2

$$q_{12}-q_{out}=A_2\frac{\delta h_2}{\delta t}\to 2$$

on substituting the knows relations

$$A_1\frac{\delta h_1}{\delta t}=q_{in}-\frac{h_1-h_2}{R_1}$$ $$A_2\frac{\delta h_2}{\delta t}=\frac{h_1-h_2}{R_1}-\frac{h_2}{R_2}$$

Dividing those with their respective Areas

$$\dot{h_1}=\frac{q_{in}}{A_1}-\frac{h_1-h_2}{A_1{R}_1}$$ $$\dot{h_2}=\frac{h_1-h_2}{A_2{R}_1}-\frac{h_2}{A_2{R}_2}$$

now the state space equation

$$\dot{x}=Ax+Bu$$ $$\left[\begin{array}{c}\dot{h_1}\\ \dot{h_2}\end{array}\right]=\left[\begin{array}{cc}\left(-\frac{1}{A_1{R}_1}\right)& \frac{1}{A_1{R}_1}\\ \frac{1}{A_2{R}_2}& -\left(\frac{1}{A_2{R}_1}+\frac{1}{A_2{R}_2}\right)\end{array}\right]\left[\begin{array}{c}{h}_1\\ h_2\end{array}\right]+\left[\begin{array}{c}\frac{1}{A_1}\\ 0\end{array}\right]q_{in}$$ $$y=Cx+Du$$ $$q_{out}=\left[\begin{array}{cc}0& \frac{1}{R_2}\end{array}\right]\left[\begin{array}{c}{h}_1\\ h_2\end{array}\right]+\left[\begin{array}{c}0\end{array}\right]q_{in}$$

At steady state the system is in equilibrium

$$\therefore q_{in}=q_{12}=q_{out}=q$$ $$q_{12}=\frac{h_1-h_2}{R_1}$$ $$q_{out}=\frac{h_2}{R_2}$$ $$h_2=q\times R_2$$ $$h_1=q\times (R_1+R_2)$$

Causal Model

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