11.1. Network analysis of linear circuits#

Dynamical equations of a linear circuit can be written as a general linear state-space model

\[\begin{split}\begin{cases} \mathbf{M} \dot{\mathbf{x}} = \mathbf{A} \mathbf{x} + \mathbf{B} \mathbf{u} \\ \mathbf{y} = \mathbf{C} \mathbf{x} + \mathbf{D} \mathbf{u} \\ \end{cases}\end{split}\]

The mathematical problem is a system of DAE (dynamical-algebraic equations), as it includes:

  • constitutive equations of the linear components

  • Kirchhoff laws for current at nodes and voltage in loops

Thus matrix \(\mathbf{M}\) is likely to be singular, here vector \(\mathbf{x}\) contains both dynamical (like voltage across a capacitor or current through an inductor) and algebraic grid variables, current and voltages whose time derivative doesn’t appear explicitly in the system of DAE.

Different representations. Possible choices of the unknowns:

  1. current through any side, voltage at any node

  2. loop currents, voltage drops across any side.

  3. … any other (linear) combination on the physical quantities

11.1.1. Thevenin equivalent#

One-port. Thevenin’s theorem states that any linear circuit can be reduced to a single voltage source and a single impedance in series.

11.1.1.1. One-port circuit#

As the goal of Thevenin’s theorem is to find the constitutive equation of the network as \(v(i)\), the network is connected to an external current generator that prescribes \(i\) and the voltage \(v\) at the port is evaluated.

The input of the extended network is

\[\mathbf{u} = ( \mathbf{u}_{gen}, i ) \ ,\]

while the output is, or at least contains, the voltage \(v\)

\[\mathbf{y} = \mathbf{C} \mathbf{x} + \mathbf{D} \mathbf{u} \ .\]

The linear system can be written in Laplace domain as

\[\begin{split}\begin{cases} s \mathbf{M} \mathbf{x} - \mathbf{M} \mathbf{x}_0 = \mathbf{A} \mathbf{x} + \mathbf{B} \mathbf{u} \\ \mathbf{y} = \mathbf{C} \mathbf{x} + \mathbf{D} \mathbf{u} \\ \end{cases}\end{split}\]

The state and the output are the sum of the free response to non-zero initial conditions and forced response,

\[\begin{split}\begin{cases} \mathbf{x} = (s \mathbf{M} - \mathbf{A})^{-1} \mathbf{M} \mathbf{x}_0 + (s \mathbf{M} - \mathbf{A})^{-1} \mathbf{B} \mathbf{u} \\ \mathbf{y} = \mathbf{C} (s \mathbf{M} - \mathbf{A})^{-1} \mathbf{M} \mathbf{x}_0 + \left[ \mathbf{C} (s \mathbf{M} - \mathbf{A})^{-1} \mathbf{B} + \mathbf{D} \right] \mathbf{u} \\ \end{cases}\end{split}\]

Forced response can be further manipulated exploiting PSCE, evaluating the effect of one input at a time, setting all the other inputs equal to zero.

  • the effect of setting the input of the external current generator, \(i = 0\), is equivalent to evaluate the system with an open circuit at the port

  • the effect of setting equal to zero a tension generator, \(e = 0\), is equivalent to a short-circuit on the same side

  • the effect of setting equal to zero a current generator, \(a = 0\), is equivalent to an open circuit on the same side

If the system is asymptotically stable, the free response is approximately zero when the transient dynamics is over, and the output equals the forced output. Introducing the transfer function

\[\mathbf{G}(s) = [ \ \mathbf{G}_{gen}(s) \quad \mathbf{G}_i(s) \ ] \ ,\]

the input-output relation reads

\[\begin{split}\begin{aligned} v = \mathbf{G}(s) \mathbf{u} & = \mathbf{G}_{gen}(s) \mathbf{u}_{gen} + G_i(s) \, i = \\ & = v_{Th}(s) - Z_{Th}(s) i(s) \ , \end{aligned}\end{split}\]

having recast it as Thevenin’s theorem defining the voltage \(v_{Th}\) and the impedance \(Z_{Th}\) of the equivalent circuit,

\[\begin{split}\begin{aligned} v_{Th} & := \mathbf{G}_{gen}(s) \mathbf{u}_{gen}(s) \\ Z_{Th}(s) & := - G_i(s) \end{aligned}\end{split}\]

11.1.1.2. Many-port circuit#

\[\mathbf{v} = \mathbf{G}_{gen}(s) \mathbf{u}_{gen} + \mathbf{G}_i(s) \mathbf{i} = \mathbf{v}_{Th} - \mathbf{Z}_{Th} \mathbf{i} \ .\]

11.1.2. Norton equivalent#