1 Introduction to Model Order Reduction Thanks to Jacob White, Kin Sou, Deepak Ramaswamy, Michal Rewienski, and Karen Veroy I.2.a – Assembling Models from MNA Modified Nodal Analysis Luca Daniel

2 Power Distribution for a VLSI Circuit Cache ALU Decoder v Power Supply Main power wires Select topology and metal widths & lengths so that a) Voltage across every function block > 3 volts b) Minimize the area used for the metal wires

3 Heat Conducting Bar Demonstration Example Output of Interest Lamp Input of Interest Select the shape (e.g. thickness) so that a) The temperature does not get too high b) Minimize the metal used.

4 Load Bearing Space Frame Attachment to the ground Joint Beam Vehicle Cargo Droop Select topology and Strut widths and lengths so that a) Droop is small enough b) Minimize the metal used.

5 Assembling Systems from MNA Formulating Equations –Circuit Example –Heat Conducting Bar Example –Struts and Joints Example Modified Nodal Analysis Stamping Procedure –Nodal Analysis (NA) –Modified Nodal Analysis (MNA) From MNA to State Space Models –e.g. circuits –e.g. struts and joints

6 Given the topology and metal widths & lengths determine a)the voltage across the ALU, Cache and Decoder b)the temperature distribution in the engine block c)the droop of the space frame under load. First Step - Analysis Tools Droop Cache ALU Decode r v Lamp

7 Cache ALU Decoder v Modeling VLSI circuit Power Distribution Power supply provide current at a certain voltage. Functional blocks draw current. The wire resistance generates losses.

8 Modeling the Circuit Supply becomes A Voltage Source + + Power supply Physical Symbol + Voltage current Current element Constitutive Equation

9 Modeling the Circuit Functional blocks become Current Sources + - ALU Physical Symbol Circuit Element Constitutive Equation

10 Modeling the Circuit Metal lines become Resistors + - Physical Symbol Circuit model Constitutive Equation (Ohm’s Law) Material Property Design Parameters

11 Modeling VLSI Power Distribution Cache ALU Decoder +-+- ICIC IDID I ALU Power Supply voltage source Functional Blocks current sources Wires become resistors Result is a schematic Putting it all together

12 Formulating Equations from Schematics Circuit Example Step 1: Identifying Unknowns Assign each node a voltage, with one node as

13 Formulating Equations from Schematics Circuit Example Assign each element a current Step 1: Identifying Unknowns

14 Formulating Equations from Schematics Circuit Example Sum of currents = 0 (Kirchoff’s current law) Step 2: Conservation Laws

15 Formulating Equations from Schematics Circuit Example Use Constitutive Equations to relate branch currents to node voltages Step 3: Constitutive Equations

16 Assembling Systems from MNA Formulating Equations –Circuit Example –Heat Conducting Bar Example –Struts and Joints Example Modified Nodal Analysis Stamping Procedure –Nodal Analysis (NA) –Modified Nodal Analysis (MNA) From MNA to State Space Models –e.g. circuits –e.g. struts and joints

17 Heat Conducting Bar Demonstration Example Output of Interest Lamp Input of Interest

18 Conservation Laws and Constitutive Equations Heat Flow 1-D Example Unit Length Rod Near End Temperature Far End Temperature Question: What is the temperature distribution along the bar x T Incoming Heat

19 Conservation Laws and Constitutive Equations Heat Flow Discrete Representation 1) Cut the bar into short sections 2) Assign each cut a temperature

20 Conservation Laws and Constitutive Equations Heat Flow Constitutive Relation Heat Flow through one section

21 Conservation Laws and Constitutive Equations Heat Flow Conservation Law Heat in from left Heat out from right Incoming heat per unit length Net Heat Flow into Control Volume = 0 ~ ~ “control volume”

22 Conservation Laws and Constitutive Equations Heat Flow Circuit Analogy Temperature analogous to Voltage Heat Flow analogous to Current ~

23 Assembling Systems from MNA Formulating Equations –Circuit Example –Heat Conducting Bar Example –Struts and Joints Example Modified Nodal Analysis Stamping Procedure –Nodal Analysis (NA) –Modified Nodal Analysis (MNA) From MNA to State Space Models –e.g. circuits –e.g. struts and joints

24 Oscillations in a Space Frame Application Problems What is the oscillation amplitude?

Ground Bolts Struts Load Example Simplified for Illustration Application Problems Simplified Structure Oscillations in a Space Frame

Application Problems Modeling with Struts, Joints and Point Masses Oscillations in a Space Frame Point Mass Strut Replace cargo with point mass. Constructing the Model Replace Metal Beams with Struts. 1:20

27 Strut Example To Demonstrate Sign convention Two Struts Aligned with the X axis Conservation Law

28 Strut Example To Demonstrate Sign convention Two Struts Aligned with the X axis Constitutive Equations

29 Strut Example To Demonstrate Sign convention Two Struts Aligned with the X axis Reduced (Nodal) Equations

30 Strut Example To Demonstrate Sign convention Two Struts Aligned with the X axis Solution of Nodal Equations

31 Strut Example To Demonstrate Sign convention Two Struts Aligned with the X axis Notice the signs of the forces

32 Formulating Equations from Schematics Struts Example C D A B Assign each joint an X,Y position, with one joint as zero. Y X hinged Step 1: Identifying Unknowns

33 Formulating Equations from Schematics Struts Example C D A B Assign each strut an X and Y force component. Step 1: Identifying Unknowns

34 Formulating Equations from Schematics Struts Example C D A B Force Equilibrium Sum of X-directed forces at a joint = 0 Sum of Y-directed forces at a joint = 0 Step 2: Conservation Laws

Formulating Equations from Schematics Struts Example C D A B Use Constitutive Equations to relate strut forces to joint positions. Step 3: Constitutive Equations

36 Formulating Equations from Schematics Comparing Conservation Laws A B ~

37 Summary of key points Two Types of Unknowns Circuit - Node voltages, element currents Struts - Joint positions, strut forces Bar – Node Temperatures, heat flows Two Types of Equations Conservation/Balance Laws Circuit - Sum of Currents at each node = 0 Struts - Sum of Forces at each joint = 0 Bar - Sum of heat flows into control volume = 0 Constitutive Equation Circuit – current-voltage relationship Struts - force-displacement relationship Bar - temperature drop-heat flow relationship

38 Assembling Systems from MNA Formulating Equations –Heat Conducting Bar Example –Circuit Example –Struts and Joints Example Modified Nodal Analysis Stamping Procedure –Nodal Analysis (NA) –Modified Nodal Analysis (MNA) From MNA to State Space Models –e.g. circuits –e.g. struts and joints

39 0 1) Number the nodes with one node as 0. 2) Write a conservation law at each node. except (0) in terms of the node voltages ! Nodal Formulation Generating Matrices Circuit Example

40 One row per node, one column per node. For each resistor Nodal Formulation Generating Matrices Circuit Example 0

41 Nodal Matrix Generation Algorithm Nodal Formulation Generating Matrices Circuit Example

Sparse Matrices Applications Space Frame X X X X X X X X X X X X X X X X X X X X XX X X XX X Unknowns : Joint positions Equations : forces = 0 X Nodal Matrix X X X X X X X X X X X

43 ( Struts and Joints ) (Resistor Networks) Nodal Formulation Generating Matrices

44 Unknowns : Node Voltages Equations : currents = 0 Sparse Matrices Applications Resistor Grid

45 Nodal Formulation Matrix non-zero locations for 100 x 10 Resistor Grid Sparse Matrices Applications Resistor Grid

46 Nodal Formulation Sparse Matrices Applications Temperature in a cube Temperature known on surface, determine interior temperature Circuit Model

47 Assembling Systems from MNA Formulating Equations –Heat Conducting Bar Example –Circuit Example –Struts and Joints Example Modified Nodal Analysis Stamping Procedure –Nodal Analysis (NA) –Modified Nodal Analysis (MNA) From MNA to State Space Models –e.g. circuits –e.g. struts and joints

48 Nodal Formulation Voltage Source Can form Node-Branch Constitutive Equation with Voltage Sources Problem Element

49 Rigid rod Nodal Formulation Rigid Rod Problem Element constitute equation The constitute equation does not contain forces!

50 Assembling Systems from MNA Formulating Equations –Heat Conducting Bar Example –Circuit Example –Struts and Joints Example Modified Nodal Analysis Stamping Procedure –Nodal Analysis (NA) –Modified Nodal Analysis (MNA) From MNA to State Space Models –e.g. circuits –e.g. struts and joints

51 State-Space Models Linear system of ordinary differential equations (ABCDE form)Linear system of ordinary differential equations (ABCDE form) State Input Output

52 State-Space Model Example: Interconnect Segment Step 1: Identify internal state variablesStep 1: Identify internal state variables –Example : MNA uses node voltages & inductor current

53 State-Space Model Example: Interconnect Segment Step 2: Identify inputs & outputsStep 2: Identify inputs & outputs –Example : For Z-parameter representation, choose port currents inputs and port voltage outputs    

54 State-Space Model Example: Interconnect Segment Step 3: Write state-space & I/O equationsStep 3: Write state-space & I/O equations –Example : KCL + inductor equation    

55 State-Space Model Example: Interconnect Segment Step 4: Identify state variables & matricesStep 4: Identify state variables & matrices

56 State-Space Model: circuits more in general LARGE! KCL/KVL

57 Assembling Systems from MNA Formulating Equations –Heat Conducting Bar Example –Circuit Example –Struts and Joints Example Modified Nodal Analysis Stamping Procedure –Nodal Analysis (NA) –Modified Nodal Analysis (MNA) From MNA to State Space Models –e.g. circuits –e.g. struts and joints

58 Application Problems A 2x2 Example Define v as velocity (du/dt) to yield a 2x2 System Constitutive Equations Conservation Law Struts, Joints and point mass example 1:39

59 Summary MNA formulations Formulating Equations –Heat Conducting Bar Example –Circuit Example –Struts and Joints Example Modified Nodal Analysis Stamping Procedure –Nodal Analysis (NA) –Modified Nodal Analysis (MNA) From MNA to State Space Models –e.g. circuits –e.g. struts and joints