Day: September 24, 2014

It is already time for remembrance day propaganda

September 24, 2014 Incoherent ramblings No comments , , , ,

 

The propaganda for celebration of war, destruction, killing, and rape is already starting.  I was greeted today on facebook by the serene and smiling image of a world war II vet, along with his seemingly innocent request for “likes”

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This is probably supposed to make people feel patriotism, the worship of an artificial boundary on a map and its associated label system.  Patriotism and national pride worship are designed to ensure that no rational thought opposes the psychopathic figureheads that run our modern “democracies”.

When we see vet worship propaganda like this, you have to think about the interests that encouraged the war in the first place.  These are power brokers like the Carnegies that requested that Wilson not end the war (WWI) too soon (and who had concluded in their minutes before that there was no better institution for profits than the business of war).  These are the armaments and munitions  manufacturers.  These are the companies that built the tanks and the planes that bombed uncounted civilians in Germany and the UK.  These are the makers of weapons that use white phosphorus and depleted uranium, and the makers of cluster bombs that shred children into hamburger meat before and after they are deployed.

I would describe us as being in world war III now.  This is the series of undeclared and covert wars that the USA and NATO have waged since world war II.  This is a war that happens with our tacit approval.  This approval that is assumed because there is no widespread disapproval.  Unfortunately, so much of this modern and current war happens without our even knowing about it, that nobody thinks to object.  This war is disconnected from our immediate attention, and most of the time we don’t even realize that we are funding it.  As always, I don’t want my tax dollars, money stolen from me at the point of a gun, to be used to fund the slaughter of people in Yugoslavia or Libya or <enter your favorite target in some little known corner of the world>, but I can’t control this.  It happens again and again, and is truly disgusting.

I am happy to see that this poor fellow survived.  However, if he had fought a real war, it would have been the war against the propaganda and brainwashing that involved him in the combat that got most of his compatriots killed, and against the same combat that killed uncountable numbers of civilians and “enemies”, and against the combat that encouraged these “enemies” to do the same.

I will not be liking this picture.

ECE1254H Modeling of Multiphysics Systems. Lecture 3: Nodal Analysis. Taught by Prof. Piero Triverio

September 24, 2014 ece1254 No comments , , ,

[Click here for a PDF of this post with nicer formatting]

Disclaimer

Peeter’s lecture notes from class. These may be incoherent and rough.

Nodal Analysis

Avoiding branch currents can reduce the scope of the computational problem. Consider the same circuit fig. 1, this time introducing only node voltages as unknowns

fig. 1.  Resistive circuit with current sources

fig. 1. Resistive circuit with current sources

Unknowns: node voltages: \(V_1, V_2, \cdots V_4\)

Equations are KCL at each node except \(0\).

  1. \(
    \frac{V_1 – 0}{R_A} +
    \frac{V_1 – V_2}{R_B} + i_{S,A} = 0
    \)

  2. \(
    \frac{V_2 – 0}{R_E} +
    \frac{V_2 – V_1}{R_B} + i_{S,B} + i_{S,C} = 0
    \)

  3. \(
    \frac{V_3 – V_4}{R_C} – i_{S,C} = 0
    \)

  4. \(
    \frac{V_4 – 0}{R_D}
    +\frac{V_4 – V_3}{R_C}
    – i_{S,A} – i_{S,B} = 0
    \)

In matrix form this is

\begin{equation}\label{eqn:multiphysicsL3:20}
\begin{bmatrix}
\inv{R_A} + \inv{R_B} & – \inv{R_B} & 0 & 0 \\
-\inv{R_B} & \inv{R_B} + \inv{R_E} & 0 & 0 \\
0 & 0 & \inv{R_C} & -\inv{R_C} \\
0 & 0 & -\inv{R_C} & \inv{R_C} + \inv{R_D}
\end{bmatrix}
\begin{bmatrix}
V_1 \\
V_2 \\
V_3 \\
V_4 \\
\end{bmatrix}
=
\begin{bmatrix}
-i_{S,A} \\
-i_{S,B} – i_{S,C} \\
i_{S,C} \\
i_{S,A} + i_{S,B}
\end{bmatrix}
\end{equation}

Introducing the nodal matrix

\begin{equation}\label{eqn:multiphysicsL3:40}
G \overline{{V}}_N = \overline{{I}}_S
\end{equation}

We identify the {stamp} for a resister of value \(R\) between nodes \(n_1\) and \(n_2\)

stamp matrix

stamp matrix

where we have a set of rows and columns for each of the node voltages \(n_1, n_2\).

Note that some care is required to use this nodal analysis method since we required the invertible relationship \(i = V/R\). We also cannot handle short circuits \(V = 0\), or voltage sources \(V = 5\) (say). We will also have trouble with differential terms like inductors.

Recap of node branch equations

We had

  • KCL: \( A \cdot \overline{{I}}_B = \overline{{I}}_S\)
  • Constitutive: \( \overline{{I}}_B = \alpha A^\T \overline{{V}}_N \),
  • Nodal equations: \( A \alpha A^\T \overline{{V}}_N = \overline{{I}}_S \)

where \(\overline{{I}}_B\) was the branch currents, \(A\) was the incidence matrix, and \(\alpha = \begin{bmatrix}\inv{R_1} & & \\ & \inv{R_2} & \\ & & \ddots \end{bmatrix} \).

The stamp can be observed in the multiplication of the contribution for a single resistor, where we see that the incidence matrix has the form \( G = A \alpha A^\T \)

stamp factor

stamp factor

Theoretical facts

Noting that \(\lr{ A B }^\T = B^\T A^\T \), it is clear that the nodal matrix \(G = A \alpha A^\T \) is symmetric

\begin{equation}\label{eqn:multiphysicsL3:80}
G^\T
=
\lr{ A \alpha A^\T }^\T
=
\lr{ A^\T }^\T \alpha^\T A^\T
=
A \alpha A^\T
= G
\end{equation}

Modified nodal analysis (MNA)

This is the method that we find in software such as spice.

To illustrate the method, consider the same circuit, augmented with an additional voltage sources as in fig. 4.

fig. 4.  Resistive circuit with current and voltage sources

fig. 4. Resistive circuit with current and voltage sources

We know wish to have the following unknowns

  • node voltages (\(N-1\)): \( V_1, V_2, \cdots V_5 \)
  • branch currents for selected components (\(K\)): \( i_{S,C}, i_{S,D} \)

We will have two less unknowns for this system than with standard nodal analysis. Our equations are

  1. \(
    -\frac{V_5-V_1}{R_A}
    +\frac{V_1-V_2}{R_B}
    + i_{S,A} = 0
    \)

  2. \(
    \frac{V_2-V_5}{R_E}
    +\frac{V_2-V_1}{R_B}
    + i_{S,B}
    + i_{S,C}
    = 0
    \)

  3. \(
    -i_{S,C} +
    \frac{V_3-V_4}{R_C} = 0
    \)

  4. \(
    \frac{V_4-0}{R_D}
    +\frac{V_4-V_3}{R_C}
    – i_{S,A}
    – i_{S,B}
    = 0
    \)

  5. \(
    \frac{V_5-V_2}{R_E}
    +\frac{V_5-V_1}{R_A}
    + i_{S,D} = 0
    \)

Put into giant matrix form, this is

giant matrix

giant matrix

Call the extension to the nodal matrix \(G\), the {voltage incidence matrix} \(A_V\).