NODICITY One of the important ways that physical system behavior differs between domains is the way elements may be connected. Electric circuit elements may be connected in series or in parallel — networks of arbitrary structure may be assembled This not true of all domains Electric circuit elements have a special property called nodicity Mod. Sim. Dyn. Sys. Nodicity page 1NODICITY Nodicity means that a network element (or sub-network) has behavior analogous to a “node” in an electric circuit. Consider a “delta” network of resistors Assuming linear inductors, the relation between currents and voltages may be written as ⎡⎢⎢⎣ i1 i2 i3 ⎤⎥⎥⎦ = ⎡⎢⎢⎣ G1+G2 –G2 –G1 –G2 G2+G3 –G3 –G1 –G3 G1+G3 ⎡⎢⎢⎣ ⎤⎥⎥⎦ e1 e2 e3 ⎤⎥⎥⎦ where Gi is conductance ii = Gi ei 1Gi = Ri Note that this conductance matrix is singular — the sum down each column is zero. i.e., the sum of currents flowing into the network is zero i1 + i2 + i3 = 0 Mod. Sim. Dyn. Sys. Nodicity page 2As a result, the three constitutive equations describing the currents may be re-arranged as follows ⎡⎢⎢⎣ i1 i2 ⎤⎥⎥⎦ = ⎡⎢⎢⎣ G1+G2 –G2 –G2 G2+G3 ⎡⎢⎢⎣ ⎤⎥⎥⎦ e1–e3 e2–e3 ⎤⎥⎥⎦ and i3 = –(i1 + i3) POINTS TO NOTE: • any one effort may be used as reference • the reference effort may vary arbitrarily • the constitutive equations depend only on input effort differences Mod. Sim. Dyn. Sys. Nodicity page 3Thus the “delta” network has the properties of a circuit node. Kirchhoff’s current law applies This constraint arises from charge continuity It applies when any or all of the resistors are replaced by inductors or capacitors It applies when any or all of the element constitutive equations are non-linear. In fact Kirchhoff’s current law applies to any sub-network of an electric circuit (i.e., any cut set of the network) That is, networks of electric elements (electric circuits) are nodic. Mod. Sim. Dyn. Sys. Nodicity page 4NON-NODIC NETWORKS Network models in other domains may not be nodic. TWO ASPECTS: 1. choice of reference may not be arbitrary example: in the constitutive equation of a translational mass dF = dp = dt (mv)dtvelocity must be referenced to an inertial (non-accelerating) frame — if not, the constitutive equations must be modified to include coriolis and/or centrifugal accelerations example: in the ideal gas equation PV = mRT • temperature must be referenced to absolute zero • pressure must be referenced to vacuum • volume must be referenced to absolute zero volume • mass must be referenced to absolute zero mass Mod. Sim. Dyn. Sys. Nodicity page 5note that linearized approximations may appear nodic example: linearize the ideal gas equation P∆V + V∆P ≈ ∆mRT + mr∆T a fixed mass of gas at constant temperature may be described as –∆V ≈ ⎛⎜⎜⎝ V P ⎞⎟⎟⎠ ∆P = Cfluid∆P in this case pressure may be referenced to any convenient value, e.g., ambient pressure i.e., gauge pressure may be used But this is only an approximation In contrast, the nodicity of electric circuits is fundamental Mod. Sim. Dyn. Sys. Nodicity page 6THE SECOND ASPECT: 2. constitutive equations may not depend input effort differences example Stefan-Boltzmann Law of radiative heat transfer ˙Q = σ(T14 - T24) ˙Q : heat flow rate σ: radiative heat transfer coefficient. T1 and T2: absolute temperatures. example choked (supersonic) flow through an orifice: ˙N = CdAt ⎝ ⎜ ⎜ ⎛ ⎠ ⎟ ⎟2 ⎞ γ+1 1/(γ–1) 2 γ γ+1 ρuPu ˙N : mass flow rate Cd: discharge coefficient At: area at orifice throat γ: ratio of specific heats ρu: upstream density Pu: upstream pressure (source: Handbook of Hydraulic Resistance, 3rd Edition, I.E. Idelchik, 1994.) Mod. Sim. Dyn. Sys. Nodicity page 7NODICITY a formal definition An element (or subsystem) is nodic if efforts and flows at its ports satisfy two conditions: (1) FLOW CONTINUITY: The (signed) sum of flows into the element is zero. i.e., a generalization of Kirchhoff’s current law applies (2) EFFORT RELATIVITY: The element’s constitutive equations depend only on a difference of efforts. If the same effort is added to all inputs, the output is unchanged. Mod. Sim. Dyn. Sys. Nodicity page 8WHY DOES NODICITY MATTER? The analogy between network elements in different domains is not complete arbitrary connections of non-nodic elements may be impossible (or have no physical meaning) example: a “delta” or “wye” network of electrical capacitors may be assembled if we assume an electrical capacitor is analogous to a gas-filled pressure vessel what physical system corresponds to a “delta” or “wye” network of pressure vessels? example: a “bridged-tee” network of inductors can be assembled if we assume an inductor is analogous to a translational mass what physical system corresponds to a “bridged-tee” network of masses? Mod. Sim. Dyn. Sys. Nodicity page