Intro_Course_2_2

Intro Course 2.2: Nodal Analysis

Nodal Analysis is the mathematical method at the heart of GUNNS.

• Wikipedia has a good article on it here.
• We have a derivation and example of how it’s used to solve an electrical circuit here, see section IV: Elecrical Refresher & GunnShow Course.
  • Note this is an old document that refers to our old network drawing tool GunnShow, which has been replaced with GunnsDraw. Most GunnShow concepts apply to GunnsDraw, just with slightly different looking graphics and procedures. We will cover GunnsDraw later in this course; you should focus on the electrical refresher and nodal analysis parts of this document instead.
• There are varying forms of Nodal Anlysis, such as Modified Nodal Analysis. However, GUNNS uses the most basic form.

To summarize the method:

• Kirchoff’s Current Law states that the net current into a node must be zero. The total current into a node must equal the total current out of the node.
• Physical laws describe various relationships between current and potential for hardware devices. These can be modeled with a concise set of generic “effects” (Conductance, Capacitance, etc). These effects are always written in terms of current.
• Combining multiple effects on the nodes, re-arranging in terms of current, and setting the total current on each node = 0, we get a system of simultaneous equations.
• The system can be arranged in matrix form and voltages solved via linear algebra. This is our system of equations, where [A] is the Admittance Matrix, {p} is the Potential Vector (the voltages), and {w} is the Source Vector:

[A]{p} = {w}

• The network links contribute to [A] and {w} via their effects (Conductance, Capacitance, etc).
• The system is then solved by GUNNS for the new state of {p}, thus propagating the system in the sim.

There are different types of Effects. Some effects are time-dependent, such as Capacitance & Inductance. Others, such as Source & Conductance, are not. Therefore, systems that contain time-dependent effects are themselves time-dependent systems. Non-time-dependent systems solve to the same state regardless of time.

Each type of effect (Conductance, Capacitance, etc.) has a different and unique footprint in the system of equations. The important thing is that they are all additive. So we simply add each effect’s [A] and {w} terms to the total network’s [A] and {w} system, and then solve. The solver, nodes & links do all this adding & solving for you. An illustration of this adding-up of multiple link’s effects here. All you have to do is arrange & configure the links that implement the effects you want to represent the modelled system, and off you go!

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