Bond graph

A bond graph is a graphical representation of a physical dynamic system. It allows the conversion of the system into a state-space representation. It is similar to a block diagram or signal-flow graph, with the major difference that the arcs in bond graphs represent bi-directional exchange of physical energy, while those in block diagrams and signal-flow graphs represent uni-directional flow of information. Bond graphs are multi-energy domain (e.g. mechanical, electrical, hydraulic, etc.) and domain neutral. This means a bond graph can incorporate multiple domains seamlessly.

The bond graph is composed of the "bonds" which link together "single-port", "double-port" and "multi-port" elements (see below for details). Each bond represents the instantaneous flow of energy ( $dE / dt$ ) or power. The flow in each bond is denoted by a pair of variables called power variables, akin to conjugate variables, whose product is the instantaneous power of the bond. The power variables are broken into two parts: flow and effort. For example, for the bond of an electrical system, the flow is the current, while the effort is the voltage. By multiplying current and voltage in this example you can get the instantaneous power of the bond.

A bond has two other features described briefly here, and discussed in more detail below. One is the "half-arrow" sign convention. This defines the assumed direction of positive energy flow. As with electrical circuit diagrams and free-body diagrams, the choice of positive direction is arbitrary, with the caveat that the analyst must be consistent throughout with the chosen definition. The other feature is the "causality". This is a vertical bar placed on only one end of the bond. It is not arbitrary. As described below, there are rules for assigning the proper causality to a given port, and rules for the precedence among ports. Causality explains the mathematical relationship between effort and flow. The positions of the causalities show which of the power variables are dependent and which are independent.

If the dynamics of the physical system to be modeled operate on widely varying time scales, fast continuous-time behaviors can be modeled as instantaneous phenomena by using a hybrid bond graph. Bond graphs were invented by Henry Paynter.^[1]

Systems for bond graph

Many systems can be expressed in terms used in bond graph. These terms are expressed in the table below.

Conventions for the table below:

$P$ is the active power;
${\hat {X}}$ is a matrix object;
${\vec {x}}$ is a vector object;
$x^{\dagger }$ is the Hermitian conjugate of $x$ ; it is the complex conjugate of the transpose of $x$ . If $x$ is a scalar, then the Hermitian conjugate is the same as the complex conjugate;
$D_{t}^{n}$ is the Euler notation for differentiation, where: $D_{t}^{n}f(t)={\begin{cases}\displaystyle \int _{-\infty }^{t}f(s)\,ds,&n=-1\\[2pt]f(t),&n=0\\[2pt]{\dfrac {\partial ^{n}f(t)}{\partial t^{n}}},&n>0\end{cases}}$
${\begin{cases}\langle x\rangle ^{\alpha }:=|x|^{\alpha }\operatorname {sgn}(x)\\\langle {a}\rangle =k\langle b\rangle ^{\beta }\implies \langle b\rangle =\left({\frac {1}{k}}\langle a\rangle \right)^{1/\beta }\end{cases}}$
Vergent-factor: $\phi _{L}={\begin{cases}{\textrm {Prismatic}}:\ {\dfrac {\textrm {length}}{{\textrm {cross-sectional}}\ {\textrm {area}}}}\\{\textrm {Cylinder}}:\ {\dfrac {\ln \left({\frac {\mathrm {radius_{out}} }{\mathrm {radius_{in}} }}\right)}{2\pi \cdot {\textrm {length}}}}\\{\textrm {Sphere}}:\ {\dfrac {1}{4\pi \left(\mathrm {radius_{in}} \parallel \mathrm {-radius_{out}} \right)}}\end{cases}}$

	Generalized flow	Generalized displacement	Generalized effort	Generalized momentum	Generalized power (in watts for power systems)	Generalized energy (in joules for power systems)
Name	${\vec {f}}(t)$	${\vec {q}}(t)$	${\vec {e}}(t)$	${\vec {p}}(t)$	$P={\vec {f}}(t)^{\dagger }{\vec {e}}(t)$	$E={\vec {q}}(t)^{\dagger }{\vec {e}}(t)$
Description	Time derivative of displacement	A quality related to static behaviour.	The energy per unit of displacement	Time integral of effort	Transformation of energy from one to another form	Conserved quantity in closed systems
Elements
Name	Hyperance $H$ , hyperrigitance $P=H^{-1}$	Compliance $C$ , rigitance $K=C^{-1}$	Resistance $R$	Inertance $I$ (or $L$ )	Abrahance $A$	Magnance $M$
Properties	Power dissipative element	Charge storage element (State variable: displacement) (Costate variable: effort)	Power dissipative element	Momentum storage element (State variable: momentum) (Costate variable: flow)	Power dissipative element	Power dissipative element
Quantitative behaviour	For 1-dimension systems (linear): $P=H\cdot \left(D_{t}^{0}q(t)\right)^{2}$ For 1-dimension systems: $e(t)=P\gamma \left[D_{t}^{-1}q(t)\right]$ Impedance: $Z(s)={\frac {1}{s^{2}H}}={\frac {1}{s^{2}}}P$	Potential energy for N-dimension systems: ${\begin{array}{lcl}V&=&{\frac {1}{2}}{\vec {q}}(t)^{\dagger }{\vec {e}}(t)\\{\vec {q}}(t)&=&{\hat {C}}{\vec {e}}(t)\end{array}}$ Potential energy: $V=\int _{0}^{q}e(q)\,dq$ Potential coenergy: ${\overline {V}}=\int _{0}^{e}q(e)de$ For 1-dimension systems: $f(t)=C\cdot {\frac {de}{dt}}+e{\frac {dC}{dt}}$ Impedance: $Z(s)={\frac {1}{sC}}={\frac {1}{s}}k$	For 1-dimension systems (linear): $P=R\cdot \left(D_{t}q(t)\right)^{2}$ Power for 1-dimension non-linear resistances ( $e(f)$ is the effort developed by the element): $P=e(f)f$ Rayleigh power: ${\mathfrak {R}}={\frac {1}{2}}R\cdot f(t)^{2}$ Rayleigh oower for non-linear resistances: ${\mathfrak {R}}=\int _{0}^{f}e(f)\,df$ Rayleigh effort: $e_{\mathfrak {R}}={\frac {d{\mathfrak {R}}}{df}}=e(f)$ For N-dimension systems: ${\begin{array}{lcr}P&=&{\vec {f}}(t)^{\dagger }{\vec {e}}(t)\\{\vec {e}}(t)&=&{\hat {R}}{\vec {f}}(t)\end{array}}$ For 1-dimension systems: $e(t)=R\cdot \gamma \left[D_{t}^{1}q(t)\right]$ Impedance: $Z(s)=R$	Kinetic energy for N-dimension systems: ${\begin{array}{lcl}T={\frac {1}{2}}{\vec {\rho }}(t)^{\dagger }{\vec {f}}(t)\\{\vec {\rho }}(t)={\hat {L}}{\vec {f}}(t)\end{array}}$ Kinetic energy: $T=\int _{0}^{\rho }f(\rho )\,d\rho$ Kinetic coenergy: ${\overline {T}}=\int _{0}^{f}\rho (f)\,df$ For 1-dimension systems: $e(t)=L\cdot {\frac {df}{dt}}+f\cdot {\frac {dL}{dt}}$ Impedance: $Z(s)=sL$	For 1-dimension systems (linear): $P=A\cdot \left(D_{t}^{2}q(t)\right)^{2}$ For 1-dimension systems: $e(t)=A\cdot \gamma \left[D_{t}^{3}q(t)\right]$ Impedance $Z(s)=s^{2}A$	For 1-dimension systems (linear) $P=A\cdot \left(D_{t}^{3}q(t)\right)^{2}$ For 1-dimension systems $e(t)=M\cdot \gamma \left[D_{t}^{5}q(t)\right]$ Impedance $Z(s)=s^{4}M$
Generalized behaviour	Energy from active effort sources: $W=\int _{0}^{q}{e_{\text{source}}}\,dq$ Lagrangian: ${\mathfrak {L}}=T-(V-W)$ Hamiltonian: ${\mathfrak {H}}=T+(V-W)$ Hamiltonian effort: $e_{\mathfrak {H}}={\frac {d{\mathfrak {H}}}{dq}}$ Lagrangian effort: $e_{\mathfrak {L}}={\frac {d{\mathfrak {L}}}{dq}}$ Passive effort: $e_{\mathfrak {LH}}={\frac {d}{dt}}{\frac {d{\mathfrak {L}}}{df}}$ Power equation: ${\frac {d{\mathfrak {L}}}{dt}}+{\frac {d{\mathfrak {H}}}{dt}}=0$ Effort equation: $e_{\mathfrak {L}}+e_{\mathfrak {H}}=0$ Lagrangian equation: $e_{\mathfrak {R}}+e_{\mathfrak {LH}}=e_{\mathfrak {L}}$ Hamiltonian equation: $e_{\mathfrak {R}}+e_{\mathfrak {LH}}=-e_{\mathfrak {H}}$ If ${\overline {W}}$ is the coenergy, $W$ is the energy, $S$ is the state variable and ${\overline {S}}$ is the costate variable, ${\begin{aligned}{\overline {W}}+W=S\cdot {\overline {S}}\\{\overline {W}}=\int _{0}^{\overline {S}}S({\overline {S}})\,d{\overline {S}}\\W=\int _{0}^{S}{\overline {S}}(S)dS\end{aligned}}$ For linear elements: ${\overline {W}}=W={\frac {1}{2}}S\cdot {\overline {S}}$

Longitudinal mechanical power system^[2]
Passive elements
Flow-related variables	$D_{t}^{6}x$ : pounce / pop $\mathrm {[m/s^{6}]}$	$D_{t}^{5}x$ : flounce / crackle $\mathrm {[m/s^{5}]}$	$D_{t}^{4}x$ : jounce / snap $\mathrm {[m/s^{4}]}$	$D_{t}^{3}x$ : jerk $\mathrm {[m/s^{3}]}$
	$D_{t}^{2}x$ : acceleration $(x_{2t})\ \mathrm {[m/s^{2}]}$	$D_{t}^{1}x$ : velocity $(x_{t})\ \mathrm {[m/s]}$ (flow)	$D_{t}^{0}x$ : displacement $(x)\ \mathrm {[m]}$ (displacement)	$D_{t}^{-1}x$ : absement $\mathrm {[m\cdot s]}$
	$D_{t}^{-2}x$ : absity $\mathrm {[m\cdot s^{2}]}$	$D_{t}^{-3}x$ : abseleration $\mathrm {[m\cdot s^{3}]}$	$D_{t}^{-4}x$ : abserk $\mathrm {[m\cdot s^{4}]}$
Effort-related variables	$D_{t}^{1}F$ : yank $\mathrm {[N/s]}$	$D_{t}^{0}F$ : force $(P_{x}^{t})\ \mathrm {[N]}$ (effort)	$D_{t}^{-1}F$ : linear momentum $(P_{x}^{2t})\ \mathrm {[kg\cdot m/s]}$ (momentum)
Compliance (C)	Resistance (R)	Inertance (I)	Abrahance (A)	Magnance (M)
Spring $\langle P_{x}^{t}\rangle =k\langle x\rangle$ where $k$ is the spring stiffness	Damper $\langle P_{x}^{t}\rangle =b\langle x_{t}\rangle$ where $b$ is the damper parameter	Mass $\langle P_{x}^{t}\rangle =m\langle x_{2t}\rangle$ where $m$ is the mass	Abraham–Lorentz force $\langle P_{x}^{t}\rangle ={\frac {\mu _{0}q^{2}}{6\pi c}}\langle x_{3t}\rangle$ where $\mu _{0}$ is the permeability $q$ is the electric charge $c$ is the speed of light	Magnetic radiation reaction force $\langle P_{q}^{t}\rangle ={\frac {\mu _{0}q^{2}R}{24\pi c^{3}}}\langle x_{5t}\rangle$ where $\mu _{0}$ is the permeability $q$ is the electric charge $c$ is the speed of light $R$ is the radius of the magnetic moment
Cantilever $\langle P_{x}^{t}\rangle ={\frac {3EI}{L^{3}}}\langle x\rangle$ $L$ : cantilever length $E$ : Young's modulus $I$ : second moment of area	Cyclotron radiation resistance $\langle P_{x}^{t}\rangle ={\frac {\sigma _{t}B^{2}}{c\mu _{0}}}\langle x_{t}\rangle$ where $\sigma _{t}$ : Thompson cross-section $c$ : speed of light $\mu _{0}$ : permeability $B$ : magnetic field density
Prismatic floater in a wide waterbody $\langle P_{x}^{t}\rangle =\rho _{L}gA\langle x\rangle$ where $\rho _{L}$ : liquid density $A$ : area $g$ : gravitational acceleration	Viscous friction $\langle P_{x}^{t}\rangle =b\langle x_{t}\rangle$ where $b$ is the viscous friction parameter
Elastic rod $\langle P_{x}^{t}\rangle ={\frac {EA}{L}}\langle x\rangle$ where $E$ : Young's modulus $A$ : area $L$ : rod length	Inverse of kinetic mobility $\langle P_{x}^{t}\rangle ={\frac {1}{\mu }}\langle x_{t}\rangle$ where $\mu$ is the kinetic mobility
Newton's law of gravitation $\langle P_{x}^{t}\rangle =GMm\langle x\rangle ^{-2}$ where $G$ : gravitational constant $M$ : mass of body 1 $m$ : mass of body 2	Geometry interacting with air (e.g. drag) $\langle P_{x}^{t}\rangle ={\frac {1}{2}}c\rho A\langle x_{t}\rangle ^{2}$ where $A$ : contact area $c$ : aerodynamic geometry coefficient $\rho$ : fluid density
Coulomb's law $\langle P_{x}^{t}\rangle ={\frac {Qq}{4\pi \varepsilon _{0}}}\langle x\rangle ^{-2}$ where $\varepsilon _{0}$ : permittivity $Q$ : charge of body 1 $q$ : charge of body 2	Absquare^{[clarification needed]} damper $\langle P_{x}^{t}\rangle =B\langle x_{t}\rangle ^{2}$ where $B$ is the Absquare damper parameter
Casimir force $\langle P_{x}^{t}\rangle ={\frac {A\hbar c\pi ^{2}}{240}}\langle x\rangle ^{-4}$ where $A$ : area of plate $\hbar$ : Planck's reduced constant $c$ : speed of light	Dry friction $\langle P_{x}^{t}\rangle =\mu F_{n}\langle x_{t}\rangle ^{0}$ where $\mu$ : kinetic friction coefficient $F_{n}$ : normal force
Biot–Savart law $\langle P_{x}^{t}\rangle ={\frac {\mu _{0}I_{1}I_{2}l}{2\pi }}\langle x\rangle ^{-1}$ where $\mu _{0}$ : permeability $I_{1}$ : current in wire 1 $I_{2}$ : current in wire 2 $l$ : length of wires
Piston pressing fluid inside an adiabatic chamber $P_{x}^{t}=AP_{0}\left(1+{\frac {x}{x_{0}}}\right)^{-\gamma }$ where $A$ : area of piston $P_{0}$ : initial pressure inside $x_{0}$ : initial position of piston $\gamma$ : poisson coefficient

Angular mechanical power system
Passive elements
Flow-related variables	$D_{t}^{3}\theta$ : angular jerk $\mathrm {[rad/s^{3}]}$	$D_{t}^{2}\theta$ : angular acceleration $(\theta _{2t})\ \mathrm {[rad/s^{2}]}$
Flow-related variables	$D_{t}^{1}\theta$ : angular velocity $(\theta _{t})\ \mathrm {[rad/s]}$ (flow)	$D_{t}^{0}\theta$ : angular displacement $(\theta )\ \mathrm {[rad]}$ (displacement)
Effort-related variables	$D_{t}^{1}\tau$ : rotatum $\mathrm {[W/rad]}$	$D_{t}^{0}\tau$ : rorque $(P_{\theta }^{t})\ \mathrm {[J/rad]}$ (effort)
Effort-related variables	$D_{t}^{-1}\tau$ : angular momentum $(P_{\theta }^{2t})\ \mathrm {[J\cdot s/rad]}$ (momentum)
Compliance (C)	Resistance (R)	Inertance (I)
Inverse of the angular spring constant $\langle P_{\theta }^{t}\rangle =k_{r}\langle \theta \rangle$ where $k_{r}$ is the angular spring constant	Angular damping $\langle P_{\theta }^{t}\rangle =R\langle \theta _{t}\rangle$ where $R$ is the damping constant	Mass moment of inertia Type $\langle P_{\theta }^{t}\rangle =I\langle \theta _{2t}\rangle$ where $I$ is the mass moment of inertia
Rod torsion $\langle P_{\theta }^{t}\rangle ={\frac {GJ}{L}}\langle \theta \rangle$ where $G$ : shear modulus $J$ : polar moment of area $L$ : rod length	Governor (e.g. used in music boxes) $\langle P_{\theta }^{t}\rangle =R\langle \theta _{t}\rangle ^{2}$ where $R$ is the governor constant
Bending moment (cantilever) $\langle P_{\theta }^{t}\rangle ={\frac {EI}{L}}\langle \theta \rangle$ where $E$ : Young modulus $I$ : second moment of area $L$ : beam length
Parallel force field $\langle P_{\theta }^{t}\rangle =\langle P_{x}^{t}\rangle \sin \left(\alpha -\theta \right)$ where $\alpha$ : angle (counter-clockwise positive) between the polar axis and the force field $\theta$ : current angle of the object (e.g. pendulum)

Electric power system
Elements
Flow-related variables	$D_{t}^{2}q$ : electric inertia $\mathrm {[A/s]}$	$D_{t}^{1}q$ : electric current $(q_{t})\ \mathrm {[A]}$ (flow)	$D_{t}^{0}q$ : electric charge $(q)\ \mathrm {[C]}$ (displacement)
Effort-related variables	$D_{t}^{1}V$ : distension $\mathrm {[V/s]}$	$D_{t}^{0}V$ : voltage $(P_{q}^{t})\ \mathrm {[V]}$ (effort)	$D_{t}^{-1}V$ : flux linkage $(P_{q}^{2t})\ \mathrm {[V\cdot s]} {\text{ or }}\mathrm {[Wb\cdot turn]}$ (momentum)
Hyperance (H)	Compliance (C)	Resistance (R)	Inertance (I)	Abrahance (A)
Frequency-dependent negative resistor (FDNR) $\langle P_{q}^{t}\rangle ={\frac {1}{H}}\langle q^{t}\rangle$	Linear capacitor $\langle P_{q}^{t}\rangle ={\frac {1}{\varepsilon }}\phi _{L}\langle q\rangle$ where $\varepsilon$ : Permittivity $\phi _{L}$ : vergent factor	Linear resistor $\langle P_{q}^{t}\rangle =\rho \phi _{L}(1+\alpha (T-T_{0}))\langle q_{t}\rangle$ where $\rho$ : resistivity $\phi _{L}$ : vergent factor $\alpha$ : thermal coefficient $T$ : current temperature $T_{0}$ : reference temperature	Linear inductor (solenoid) $\langle P_{q}^{t}\rangle ={\frac {\mu _{0}N^{2}A}{L}}\langle q_{2t}\rangle$ where $\mu _{0}$ : permeability $N$ : number of turns $A$ : area $L$ : length	Frequency-dependent negative conductance (FDNC) Type $\gamma ^{1}\iff V=RD_{t}^{2}i\quad [H\cdot s]$
		Diode $P_{q}^{t}=nV_{T}\ln \left(1+{\frac {q_{t}}{I_{s}}}\right)$ where $n$ : ideality factor $V_{T}$ : thermal voltage $I_{s}$ : leakage current	Toroid $\langle P_{q}^{t}\rangle ={\frac {\mu N^{2}A}{2\pi r}}\langle q_{2t}\rangle$ where $\mu$ : permeability $N$ : number of turns $A$ : cross-sectional area $r$ : toroid radius to centerline
Intra-gyrator			Inter-gyrator
Compliant gyrator	Resistive gyrator	Inertant gyrator	Inter-gyrator
	Hall effect device ${\begin{aligned}e_{x}=R_{H}{\frac {B_{z}}{t_{z}}}i_{y}\\e_{y}=R_{H}{\frac {B_{z}}{t_{z}}}i_{x}\end{aligned}}$ where $R_{H}$ : Hall coefficient $B_{z}$ : vertical magnetic induction field $t_{z}$ : vertical thickness	Induction motor ${\begin{aligned}e_{r}^{1}=M\cos(\theta )D_{t}f_{s}^{1}+f_{s}^{1}D_{t}\left[M\cos(\theta )\right]\\e_{s}^{1}=M\cos(\theta )D_{t}f_{r}^{1}+f_{r}^{1}D_{t}\left[M\cos(\theta )\right]\end{aligned}}$ where $\theta$ : electrical angle $e_{r}^{1}$ : voltage at rotor bar $e_{s}^{1}$ : voltage at stator bar $f_{s}^{1}$ : flow through stator bar $f_{r}^{1}$ : flow through rotor bar	DC motor ${\begin{aligned}\tau _{e}=k_{a}\phi _{a}(i_{e})i_{a}\\\omega _{e}={\frac {1}{k_{a}\phi _{a}(i_{e})}}e_{a}\end{aligned}}$ where $\tau _{e}$ : electromagnetic torque $\omega _{e}$ : axis angular velocity $i_{e}$ : field current $i_{a}$ : armature current	Faraday gyrator ${\begin{aligned}F&=Bli\\V&={\frac {1}{Bl}}v\end{aligned}}$ where $F$ : force $v$ : rod velocity $V$ : rod voltage $B$ : magnetic induction field $l$ : rod length
			Faraday disk ${\begin{aligned}V&={\frac {1}{2}}Br^{2}\omega \\\tau &={\frac {1}{2}}Br^{2}i\end{aligned}}$ where $B$ : magnetic induction field $r$ : disk radius
Intra-transformer			Inter-transformer
Electrical transformer (only for AC signals) ${\begin{aligned}V_{2}&={\frac {N_{2}}{N_{1}}}V_{1}\\f_{2}&={\frac {N_{1}}{N_{2}}}f_{1}\end{aligned}}$

Hydraulic / pneumatic power system
Elements
Flow-related variables	$D_{t}^{1}V$ : volumetric flow rate $(V_{t})\ \mathrm {[m^{3}/s]}$ (flow)	$D_{t}^{0}V$ : volume $(V)\ \mathrm {[m^{3}]}$ (displacement)
Effort-related variables	$D_{t}^{0}P$ : pressure $(P_{V}^{t})\ \mathrm {[Pa]}$ (effort)	$D_{t}^{-1}P$ : fluid momentum $(P_{V}^{2t})\ \mathrm {[Pa\cdot s]}$ (momentum)
Compliance (C)	Resistance (R)	Inertance (I)
Pipe elasticity $\langle P_{V}^{t}\rangle =\left({\frac {t_{W}E}{2r_{0}V_{0}}}\right)\langle V\rangle$ where $r_{0}$ : nominal pipe radius $V_{0}$ : pipe volume (without stress) $t_{W}$ : wall thickness $E$ : Young's modulus	Darcy sponge $\langle P_{V}^{t}\rangle =\left({\frac {\mu }{k}}\phi _{L}\right)\langle V_{t}\rangle$ where $\mu$ : dynamic viscosity $k$ : permeability $\phi _{L}$ : vergent factor	Fluid inertia in pipes $\langle P_{V}^{t}\rangle =\left(\rho \phi _{L}\right)\langle V_{2t}\rangle$ where $\rho$ : fluid density $\phi _{L}$ : vergent factor
Compressible fluid (approximation) $\langle P_{V}^{t}\rangle =\left({\frac {B}{V_{0}}}\right)\langle V\rangle =\underbrace {\left({\frac {\rho _{0}c^{2}}{V_{0}}}\right)} _{\begin{array}{c}{\text{Acoustic}}\\{\text{Approximation}}\end{array}}\langle V\rangle$ where $V_{0}$ : pipe volume (without stress) $B$ : bulk modulus $\rho _{0}$ : gas density at reference pressure $c$ : speed of sound	Valve $\langle P_{V}^{t}\rangle =\left({\frac {\rho }{2C_{d}^{2}A_{0}^{2}}}\right)\langle V_{t}\rangle ^{2}$ $C_{d}$ : discharge coefficient $\rho$ : fluid density $A_{0}$ : smallest area for fluid passage
Tank with area ${\textstyle A{\frac {1}{[m]^{n-1}}}\left(h+h_{0}\right)^{n-1}}$ : $P_{V}^{t}=\rho gh_{0}\left[\left(1+{\frac {nV}{A{\frac {1}{[m]^{n-1}}}h_{0}^{n}}}\right)^{1/n}-1\right]$ where $h$ : height with respect to the ground $h_{0}$ : inlet height $\rho$ : fluid density $A$ : curve scaling (dimension of area) $[m]^{n-1}$ : correction of dimension $n$ : curve parameter $g$ : acceleration of gravity Case $n=1$ (prismatic case): $P_{V}^{t}={\frac {\rho g}{A}}V$ Case $n=0$ (current diode case): $P_{V}^{t}=\rho gh_{0}\left(e^{\frac {V}{A[m]}}-1\right)$	Poiseuille resistance for cylinders $\langle P_{V}^{t}\rangle =\left({\frac {8\mu L}{\pi R^{4}}}\right)\langle V_{t}\rangle$ where $\mu$ : dynamic viscosity $L$ : pipe length $R$ : pipe radius
Isothermal chamber $\langle P_{V}^{t}\rangle =k\langle V_{t}\rangle ^{-1}$ where $k$ is chamber's constant	Turbulence resistance $\langle P_{V}^{t}\rangle =a_{t}\langle V_{t}\rangle ^{\frac {7}{4}}$ where $a_{t}$ is an empirical parameter
Compressible fluid $P_{V}^{t}=-B\ln V$ where $B$ is the bulk modulus	Nozzle $\langle P_{V}^{t}\rangle ={\frac {\rho }{2\left(A_{\text{out}}^{2}\parallel -A_{\text{in}}^{2}\right)}}\langle V_{t}\rangle ^{2}$ where $\rho$ : fluid density $A_{\text{out}}$ : outlet area $A_{\text{in}}$ : inlet area
Adiabatic bladder $P_{V}^{t}=P_{V,0}^{t}\left(1-{\frac {V}{V_{0}}}\right)^{-\gamma }$ where $P_{V,0}^{t}$ : reference pressure $V_{0}$ : reference volume $\gamma$ : Poisson coefficient	Check valve $P_{V}^{t}=k\ln \left(1+{\frac {V_{t}}{V_{t,0}}}\right)$ where $k$ : empirical constant $V_{t,0}$ : reverse bias absolute flow

Gyrator–capacitor power system
Elements
Flow-related variables	$D_{t}^{0}\varphi$ : magnetic flux $(\varphi )\ \mathrm {[Wb]}$ (displacement)
Effort-related variables	$D_{t}^{0}{\mathcal {F}}$ : magnetomotive force $({\mathcal {F}})\ \mathrm {[A\cdot turn]}$ (momentum)
Compliance (C)	Resistance (R)	Inertance (I)
Permeance ( ${\mathcal {P}}$ ) $\langle {\mathcal {F}}\rangle ={\frac {1}{\mu }}\phi _{L}\langle \varphi \rangle \quad \mathrm {[H/turn^{2}]}$ where $\mu$ : permeability $\phi _{L}$ : vergent factor	Magnetic complex impedance ( $Z_{M}$ ) ${\mathcal {F}}=Z_{M}{\varphi _{t}}\quad \mathrm {[{turn}^{2}/\Omega ]}$	Magnetic complex inductance ( $L_{M}$ ) ${\mathcal {F}}=L_{M}{\varphi _{2t}}\quad \mathrm {[{turn}^{2}F]}$

Gravitational power system
Elements
Flow-related variables	$D_{t}^{0}i_{g}$ : gravitational current Loop $\left(i_{g}=2\varepsilon _{g}v_{\text{orbit}}^{3}\right)\ \mathrm {[kg/s]}$ (flow)	$D_{t}^{-1}i_{g}$ : gravitational charge $(M)\ \mathrm {[kg]}$ (displacement)
Effort-related variables	$D_{t}^{0}V_{g}$ : gravitational voltage $\left(V_{g}={\frac {1}{2}}{\frac {v_{\text{orbit}}^{4}}{c^{2}}}\right)\ \mathrm {[m^{2}/s^{2}]}$ (effort)	$D_{t}^{-1}V_{g}$ : gravitational momentum $\left(\phi _{g}={\frac {\pi v_{\text{orbit}}^{3}r}{c^{2}}}\right)\ \mathrm {[m^{2}/s]}$ (momentum)
Compliance (C)	Resistance (R)	Inertance (I)
Gravitational capacitance Type $\gamma ^{1}\iff M_{g}=CV_{g}\quad \mathrm {[kg/m^{2}]}$ $C_{g}={\frac {2c^{2}r^{2}}{GM}}$	Gravitational orbital resistance Type $\gamma ^{1}\iff V_{g}=R_{g}i_{g}\quad \mathrm {[m^{2}/kg\cdot s]}$ $R_{g}={\frac {\mu _{g}}{4}}v_{\text{orbit}}$	Gravitational inductance Type $\gamma ^{1}\iff \phi _{g}=Ii_{g}\quad \mathrm {[m^{2}/kg\cdot s^{2}]}$ $L_{g}={\frac {2\pi ^{2}G}{c^{2}}}r$

Electric field volumetric power density system
Elements
Flow-related variables	$D_{t}^{0}J$ : current density $(J)\ \mathrm {[A/m^{2}]}$ (flow)	$D_{t}^{-1}J$ : electric displacement field $(D)\ \mathrm {[C/m^{2}]}$ (displacement)
Effort-related variables	$D_{t}^{0}E$ : electric field $(E)\ \mathrm {[V/m]}$ (effort)	$D_{t}^{-1}E$ : magnetic potential vector $(A)\ \mathrm {[V\cdot s/m]}$ (momentum)
Compliance (C)	Resistance (R)	Volumetric power density inertance ( $I_{V}$ )
Electrical permittivity Type $\gamma ^{1}\iff D=\epsilon _{0}E\quad \mathrm {[F/m]}$	Electrical resistivity Type $\gamma ^{1}\iff E=\rho J\quad \mathrm {[\Omega m]}$	Magnetic permeability $\mu _{0}\ \mathrm {[H/m]}$

Magnetic field volumetric power density system
Elements
Flow-related variables	$D_{t}^{0}B$ : magnetic flux density $(B)\ \mathrm {[T]} {\text{ or }}\mathrm {[Wb/m^{2}]}$ (displacement)
Effort-related variables	$D_{t}^{0}H$ : magnetic field strength $(H)\ \mathrm {[A\cdot {turn}/m]}$ (effort)
Compliance (C)	Resistance (R)	Volumetric power density inertance ( $I_{V}$ )
Magnetic permeability for magnetic circuits Type $\gamma ^{1}\iff B=\mu H\quad \mathrm {[H/(m\cdot {turn}^{2})]}$

Gravitoelectric volumetric power density system
Elements
Flow-related variables	$D_{t}^{0}J_{g}$ : flux of mass $(J_{g})\ \mathrm {[kg/m^{2}s]}$ (flow)	$D_{t}^{-1}J_{g}$ : accumulated flux of mass $(D_{g})\ \mathrm {[kg/m^{2}]}$ (displacement)
Effort-related variables	$D_{t}^{0}g$ : acceleration of gravity $(g)\ \mathrm {[m/s^{2}]}$ (effort)
Compliance (C)	Resistance (R)	Volumetric power density inertance ( $I_{V}$ )
Gravitational permittivity Type $\gamma ^{1}\iff D_{g}=\epsilon _{g}g\quad \mathrm {[kg\cdot s^{2}/m^{3}]}$ $\varepsilon _{g}={\frac {1}{4\pi G}}$		Gravitational permeability $\mathrm {[m/kg]}$ $I_{V}=\mu _{g}={\frac {4\pi G}{c^{2}}}$

Gravitomagnetic volumetric power density system
Elements
Flow-related variables	$D_{t}^{0}B_{g}$ : gravitomagnetic field $\left(B_{g}=\omega _{\text{orbit}}\left({\frac {v_{\text{orbit}}}{c}}\right)^{2}\right)\ \mathrm {[Hz]}$ (displacement)
Effort-related variables	$D_{t}^{0}H_{g}$ : gravitomagnetic field strength $(H_{g})\ \mathrm {[Pa\cdot s]}$ (effort)
Compliance (C)	Resistance (R)	Volumetric power density inertance ( $I_{V}$ )
Gravitational permeability $-{\frac {4\pi G}{c^{2}}}\ \mathrm {[m/kg]}$

Thermal power-temperature system
Elements
Flow-related variables	$D_{t}^{1}Q$ : heat rate $(\psi _{t})\ \mathrm {[W]}$ (flow)	$D_{t}^{0}Q$ : total heat $(\psi )\ \mathrm {[J]}$ (displacement)
Effort-related variables	$D_{t}^{0}T$ : temperature $(T)\ \mathrm {[K]}$ (Effort)
Compliance (C)	Resistance (R)	Inertance (I)
Isobaric heat $T={\frac {1}{\rho Vc_{p}}}\psi ={\frac {1}{C_{P}}}\psi$ where $\rho$ : object mass density $V$ : volume of object $c_{p}$ : pressure specific heat capacity	Conduction resistance $T={\frac {1}{k}}\phi _{L}\psi _{t}$ where $k$ : thermal conductivity $\phi _{L}$ : vergent factor
Isocoric heat $T={\frac {1}{C_{V}}}\psi$ where $C_{V}$ is the constant-volume heat capacitance	Convection resistance $T={\frac {1}{hA}}\psi _{t}$ where $h$ : convection coefficient $A$ : interface area
Isothermal heat $T={\frac {1}{nR\ln \left({\frac {V_{f}}{V_{i}}}\right)}}\psi$ where $R$ : universal gas constant $n$ : number of moles $V_{f}$ : final volume $V_{i}$ : initial volume	Stefan–Boltzmann law $\langle T\rangle =\left({\frac {1}{e\sigma A}}\right)^{0.25}\langle \psi _{t}\rangle ^{0.25}$ where $e$ : emissivity $\sigma$ : Stefan–Boltzmann constant $A$ : interface area

Continuum mechanics volumetric power density system
Flow-related variables	$D_{t}^{1}\varepsilon$

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