SaM - MEMORIZE (Lumped Parameter Systems)

List of things to memorize:

• General Lumped Parameter System
• Thermal Domain
• Mechanical Domain
• Magnetic Domain
• Acustic Domain

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General Lumped Parameter System

	General lumped parameter system	Electrical System
Effort Quantity	$Y(s)$	Voltage: $V(s)$
Flow Quantity	$X(s)$	Current: $I(s)$
Generated Impedance	$Z(s)=\frac{Y(s)}{X(s)}$	Resistance: $R$

System	Effort Quantity	Flow Quantity	Impedances
Thermal Domain	Temperature Difference:$\Delta T(s)$	Heat Flux:$\dot{Q}(s)$	Thermal Resisitance due to Convection:$R_{\text{TH}}=\frac{1}{hA}$Thermal Resisitance due to Condution:$R_{\text{TH}}=\frac{t}{kA}$Thermal Capacitance:$C_{\text{TH}}=c\cdot V$
Mechanical Domain	Force:$F=ma$	Speed:$\dot{x}$	A Spring is modelled as a Capacitance:$C=\frac{1}{k}$ A Damper is modelled as a Resistance:$R=\lambda$A mass is modelled as an Inductance:$L=m$
Magnetic Domain	Magneto Motive Force:$\text{MMF}=N\cdot I$	Flux of the Magnetic Field:$\varphi (B)=\mu HS$	Reluctances are modelled as a resistance:$\mathcal{R}=\frac{l}{\mu S}$Permanent Magnets are modelled as a Voltage genrator and a relactance, so:$\begin{array}{l}{\mathcal{R}}_M=\frac{l_M}{{\mu }_M{S}_M}\\ \text{MMF}={\mathcal{R}}_M\cdot \varphi (B_r)\\ \varphi (B_r)=S_M\cdot B_r={\mu }_0\cdot S_M\cdot M_r\end{array}$
Acustic Domain	How fast the wave changes form:$\dot{u}$	Stress (for Solids):$\overline{\overline{T}}$Preassure (for Liquids):$p$	Acustic Impedance:$\begin{array}{lc}\text{fluids:}& Z=\frac{p}{{\dot{u}}_1}\\ \text{solids:}& Z=\frac{T_{11}}{{\dot{u}}_1}\end{array}$Or:$Z=\rho \cdot v_l$

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Thermal Domain

• Heat Convection:$\begin{array}{l}{R}_{TH}=\frac{1}{h\cdot A}\\ [h]=\frac{\text{W}}{\text{°K}}\end{array}$
• Heat conduction:$\begin{array}{l}{R}_{TH}=\frac{t}{k\cdot A}\\ \left[k\right]=\frac{\text{W}}{\text{°K}\cdot \text{m}}\end{array}$
• Heat Stored in a Body:$\begin{array}{l}{C}_{TH}=c\cdot V\end{array}$
• Examples for thermal lumped parameter systems:
  • Target + Ambient:
    
  • Sensor + Target:
    
  • Sensor + Target + Ambient:
    
• Terminology:
  • $R$ : thermal resistance.
  • $h$ : convective heat transfer coefficient.
  • $k$ : thermal conductivity.
  • $C$ : thermal capacitance.
  • $c$ : specific heat.
  • $V$ : volume.
  • $T$ : temperature.
  • $\dot{Q}$ : heat flux.

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Mechanical Domain

• Generated Impendaces:
  
• Example:
  
• Terminology:
  • $k$ : elastic constant.
  • $\lambda$ : damping coefficient.
  • $m$ : mass.
  • $F$ : force $(F=ma)$.
  • $\ddot{x}=a$ : acceleration.
  • $\dot{x}$ : velocity.
  • $x$ : displacement.

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Magnetic Domain

• Magnetic Circuit with a Permanent Magnet (PM) (no Coil):
  
• Model with a Coil (no Permanent Magnet):
  
  • Formulas:$\begin{array}{l}{\mathcal{R}}_g=\frac{l_g}{{\mu }_0{S}_g}\\ {\mathcal{R}}_c=\frac{l}{{\mu }_C{S}_C}\\ \varphi (B)=B\cdot S,B=\mu H\\ \varphi (B)={\mu }_0{H}_g{S}_g={\mu }_C{H}_C{S}_C\end{array}$
• Magnetic Circuit with Coil and Permanent Magnet (PM):
  
• Permament Magnet Model:
  
  • Formulas:$\begin{array}{l}{\mathcal{R}}_M=\frac{l_M}{{\mu }_M{S}_M}\\ \text{MMF}={\mathcal{R}}_M\cdot \varphi (B_r)\\ \varphi (B_r)=S_M\cdot B_r={\mu }_0\cdot S_M\cdot M_r\end{array}$
• Complete Model:
  
  • Formula:$\begin{array}{l}v=-N\frac{d\varphi (B)}{dt}=\\ =-\frac{N^2}{{\mathcal{R}}_{TOT}}\frac{dI}{dt}+\left(\frac{N^2}{{\mathcal{R}}_{TOT}^2}I+\frac{N}{{\mathcal{R}}_{TOT}^2}{\text{MMF}}_{eq}\right)\frac{d{\mathcal{R}}_{TOT}}{dt}\end{array}$
• Terminology:
  • ${\mu }_r$ : realtive magnetic permeability.
  • ${\mu }_0$ : vacum magnetic permeability $({\mu }_0=4\pi \times 10^{-7}\frac{\text{T}\cdot \text{m}}{\text{A}})$.
  • $l$ : lenght.
  • $S$ : surface.
  • $\mathcal{R}$ : reluctance.
  • $\text{MMF}$ : magneto motive force.
  • $\varphi (B)$ : flux of the magnetic field.
  • $N$ : number of turns of the coil.
  • $I$ : current flowing in the coil.
  • $B$ : magnetic field density.
  • $H$ : magnetic field intensity $(B=\mu H)$.
  • $M$ : magnetization $\left(M=H\cdot (1+{\mu }_r),B={\mu }_0(H+M)\right)$.
  • $v$ : elettro motive force $(v=-V),V$ : voltage.

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Acustic Domain

• Generated Impedance:$\begin{array}{lc}\text{fluids:}& Z=\frac{p}{{\dot{u}}_1}\\ \text{solids:}& Z=\frac{T_{11}}{{\dot{u}}_1}\\ \text{general:}& Z=\rho \cdot v_l\end{array}$
• Measure Unit:$\left[Z\right]=\text{Rayl}=\frac{\text{kg}}{{\text{m}}^2\text{s}}$
• $v_l=\sqrt{\frac{c_{11}}{\rho }}$
• Relation between ${\dot{u}}_1$ and $v_l$:${\dot{u}}_1=-\frac{T_{11}}{\rho }\frac{1}{v_l}$
• Terminology:
  • $Z$ : acustic impedance.
  • $p$ : preassure.
  • $T_{11}$ : stress.
  • $\dot{u}$ : how fast the wave changes form.
  • $v_l$ : longitudinal propagation velocity of the wave.
  • $\rho$ : resisitivity.
  • $c_{11}$ : elastic constant, specifically this is from the collapsed stiffness tensor.