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Midterm Solutions for Continuous-System Modeling | ECE 449, Exams of Electrical and Electronics Engineering

University of Arizona (UA)Electrical and Electronics Engineering

Material Type: Exam; Class: Continuous-System Modeling; Subject: Electrical & Computer Engr; University: University of Arizona; Term: Fall 2002;

Typology: Exams

Pre 2010

Uploaded on 08/31/2009

koofers-user-12e 🇺🇸

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December 2, 2002 Start Presentation

4th Midterm - Solution

•4.1 Switching Bond Graphs

•4.2 Chemical Reactions

•4.3 Logistic Model

•4.4 Lotka-Volterra Model

December 2, 2002 Start Presentation

Switching Bond Graphs I

• Given the circuit:

• The circuit has four diodes that can either be

conducting or blocking.

Discover Exams of Electrical and Electronics Engineering University of Arizona (UA)

Partial preview of the text

Download Midterm Solutions for Continuous-System Modeling | ECE 449 and more Exams Electrical and Electronics Engineering in PDF only on Docsity!

December 2, 2002 (^) Start Presentation

4 th^ Midterm - Solution

• 4.1 Switching Bond Graphs

• 4.2 Chemical Reactions

• 4.3 Logistic Model

• 4.4 Lotka-Volterra Model

December 2, 2002 (^) Start Presentation

Switching Bond Graphs I

• Given the circuit:

• The circuit has four diodes that can either be

conducting or blocking.

December 2, 2002 (^) Start Presentation

Switching Bond Graphs II

• Determine a bond graph describing this circuits.

• There are 16 combinations of conducting/blocking.

Determine, which of those are “legal,” i.e., do not

lead to a causality conflict on the bond graph, and

which of them are “illegal.”

• For each of the “illegal” combinations, explain

what is happening physically in the circuit.

December 2, 2002 (^) Start Presentation

Semi-causal Bond Graph

We create a bond graph with causality strokes, wherever they are fixed, and without causality strokes, where they are free.

December 2, 2002 (^) Start Presentation

D 4 is open.

⇒ D^3 is closed.

December 2, 2002 (^) Start Presentation

D 3 is open.

⇒ D^4 is closed.

December 2, 2002 (^) Start Presentation

D 1 is closed and D 2 is closed.

⇒ D^3 is open^ or D^4 is open.

December 2, 2002 (^) Start Presentation

No further constraints seem to exist. D1 D2 D3 D4 Status open open open open illegal open open open closed illegal open open closed open illegal open open closed closed illegal open closed open open illegal open closed open closed legal open closed closed open legal open closed closed closed legal closed open open open illegal closed open open closed legal closed open closed open legal closed open closed closed legal closed closed open open illegal closed closed open closed legal closed closed closed open legal closed closed closed closed illegal

Illegal combinations : D 1 = D 2 = open D 3 = D 4 = open D 1 = D 2 = D 3 = D 4 = closed

December 2, 2002 (^) Start Presentation

Illegal Combinations

D 1 = D 2 = open

D 3 = D 4 = open

D 1 = D 2 = closed D 3 = D 4 = closed

The source is disconnected.

Two parallel short- circuits. The currents cannot be independently determined.

December 2, 2002 (^) Start Presentation

Chemical Reactions

• On the moon, there are large quantities of

anorthite rock with the chemical formula:

• In a first step of a reduction process, the rock is

allowed to react with aluminum:

• Determine the stoichiometric coefficients of this

reaction.

CaAl 2 Si 2 O (^8)

CaAl 2 Si 2 O 8 + Al → Si + Al 2 O 3 + CaO

December 2, 2002 (^) Start Presentation

x · CaAl 2 Si 2 O 8 + y · Al → z · Si + v · Al 2 O 3 + w · CaO

Ca: x = w Al: 2x + y = 2v Si: 2x = z O: 8x = 3v + w

3 CaAl 2 Si 2 O 8 + 8 Al → 6 Si + 7 Al 2 O 3 + 3 CaO

December 2, 2002 (^) Start Presentation

Logistic Model I

We wish to analyze the continuous-time logistic model:
This is a special case of a Bernoulli equation (it is also a special case of a Riccati equation).

x = a · x + b · x· 2 ; x(t 0 ) = x 0

December 2, 2002 (^) Start Presentation

t^ lim → ∞^ x(t) = x^ ∞^ =^ −^ ab ; a > 0

We can use a symmetry condition.

x(t*) = 0.5 · x ∞ = − 2ba

Using this, we can easily find t*.

December 2, 2002 (^) Start Presentation

Lotka-Volterra Model I

Around Tucson, there are many coyotes who live mostly off cottontail rabbits, which in turn eat grasses and leaves.
We want to assume that: Both the coyotes, x (^) c, and the cottontails, x (^) h, die out naturally if they aren’t fed. The grasses and leaves (x (^) b = biomass) die out also, if they don’t get both sun, xs, and water, xw, but grow linearly if both water and sun are available:

The water level grows with rain fall, x (^) r, and recedes naturally if no rain is falling.

x (^) b ∼ x (^) s · x (^) w

December 2, 2002 (^) Start Presentation

Lotka-Volterra Model II

Sun, x (^) s, and rain fall, x (^) r, are exogenous functions of time. The coyotes (carnivores) feed on cottontail rabbits (herbivores) exclusively, i.e., the model needs to take into account the effect of predation. The rabbits feed on the biomass in a process of grazing. If there are too many coyotes, they starve because of crowding. If there are too many rabbits, they also starve because of crowding.

Postulate a Lotka-Volterra model for this ecosystem.

December 2, 2002 (^) Start Presentation

xs = xs (t) xr = xr (t)

Midterm Solutions for Continuous-System Modeling | ECE 449, Exams of Electrical and Electronics Engineering

Related documents

Partial preview of the text

Download Midterm Solutions for Continuous-System Modeling | ECE 449 and more Exams Electrical and Electronics Engineering in PDF only on Docsity!

4 th^ Midterm - Solution

• 4.1 Switching Bond Graphs

• 4.2 Chemical Reactions

• 4.3 Logistic Model

• 4.4 Lotka-Volterra Model

Switching Bond Graphs I

• Given the circuit:

• The circuit has four diodes that can either be

conducting or blocking.

Switching Bond Graphs II

• Determine a bond graph describing this circuits.

• There are 16 combinations of conducting/blocking.

Determine, which of those are “legal,” i.e., do not

lead to a causality conflict on the bond graph, and

which of them are “illegal.”

• For each of the “illegal” combinations, explain

what is happening physically in the circuit.

Semi-causal Bond Graph

⇒ D^3 is closed.

⇒ D^4 is closed.

⇒ D^3 is open^ or D^4 is open.

Illegal Combinations

Chemical Reactions

• On the moon, there are large quantities of

anorthite rock with the chemical formula:

• In a first step of a reduction process, the rock is

allowed to react with aluminum:

• Determine the stoichiometric coefficients of this

reaction.

Logistic Model I

t^ lim → ∞^ x(t) = x^ ∞^ =^ −^ ab ; a > 0

x(t*) = 0.5 · x ∞ = − 2ba

Lotka-Volterra Model I

Lotka-Volterra Model II

xw = − k 1 xw + k 2 xr

xb = − k3 xb + k 4 xs xw − k5 xb xh

xh = − k6 xh + k 7 xb xh − k8 xh xc − k9 xh^2

xc = − k 10 xc + k 11 xb xc − k12 xc^2