1. 5. Modeling of Electrical Systems
EE3511: Automatic Control Systems
5. Modeling of Electrical Systems
Dr. Ahmed Nassef
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Learning Objective
To derive mathematical models (ODE models or transfer functions) of simple electric circuits involving resistors, capacitors, inductors, current sources and voltage sources.
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Modeling Procedure
Represent the system by a set of idealized elements
Define a set of suitable variables
Write the appropriate element laws
Write the appropriate interaction laws
Combine element and interaction laws to write the model of the system
If the model is nonlinear, select an appropriate equilibrium condition and linearize the system
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Variables The equations are expressed in terms of one or more of the following types of variables
Other variables of interest ( charge, flux, flux linkage )
Variables
Unit
symbol
Electromotive Force (Voltage)
Volt
e, v
Current
Ampere i
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Notation i1 i2
Circuit element
The current entering any circuit element is the same as the current leaving the element and therefore only one of them is shown i1
Circuit element
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Notation i _
Circuit element
The direction of the current is the direction of movement of positive ions entering the element.
Electrons (negative charges) moves opposite to the indicated direction
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Element Laws Resistor
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8. Element Laws Capacitor
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9. Element Laws Capacitor
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Element Laws Inductor
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Element Laws Inductor
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Element Laws Sources 3A 6V − 6V −
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Interaction Laws
Interaction laws describe the way different elements are interconnected
important interaction laws
Kirchhoff’s Current Law (KCL)
Kirchhoff’s Voltage Law (KVL)
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14. Kirchhoff’s Current Law (KCL)
The algebraic sum of all currents at any node is zero at all times.
Current entering the node i1 i2 i3
Current leaving the node
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Applications of KCL
The current through the capacitor is the same as the current through the resistor
The current through the capacitor is the same as the sum of currents through the two resistors
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16. Kirchhoff’s Voltage Law (KVL)
The algebraic sum of voltages across all the elements that make up a loop is zero
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Applications of KVL
The voltage across the capacitor is the same as the voltage across the resistor
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18. Element Laws (t -Domain)
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19. Element Laws (S -Domain) Resistor
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20. Element Laws (S -Domain) Capacitor
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21. Element Laws (S -Domain) Capacitor
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22. Element Laws (S -Domain) Inductor
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23. Element Laws (S -Domain) Inductor
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24. Element Laws (S -Domain) (With zero initial condition)
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25. Obtaining The Model (Method 1) Loop Equation Method
Label loop currents
Express current through all elements in terms of one or more loop currents
Use KVL and element's laws to write the model
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26. Obtaining The Model (Method 2) Node Equation Method
Label all voltage nodes
Express voltage across all elements in terms of node voltages
Use KCL and element's laws to write the model
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Resistive Circuits
You can simplify the resistive circuit by
replacing resistors in parallel by a single resistor
replacing resistors in series by a single resistor
Remember that a resistive circuit does not have any dynamics and it can be modeled by a static model.
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Resistor in series
Equivalent resistance of resistors in series is the sum of resistances
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Resistors in parallel + e − + e −
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Current Divider
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Voltage Divider + e −
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32. Transfer Function (TF)
The transfer function of a linear, time-invariant, differential equation system is defined as the ratio of the Laplace transform of the output (response function) to the Laplace transform of the input (driving function) under the assumption that all initial conditions are zero.
Transfer Function
G(s)
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33. Example 1 Find eo in terms of ei
Two resistors in parallel
Replace
them by
one resistor
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34. Example 1 Find eo in terms of ei
Two resistors in series
Replace
them by
one resistor
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35. Example 1 Find eo in terms of ei
Two resistors in parallel
Replace
them by
one resistor
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36. Example 1 Find eo in terms of ei
Two resistors in series
Replace
them by
one resistor
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37. Example 1 Find eo in terms of ei
Two resistors in parallel
Replace
them by
one resistor
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38. Example 1 Find eo in terms of ei
This can be seen as voltage divider
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39. Example 1 Find eo in terms of ei
This can be seen as voltage divider
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40. Example 1 Find eo in terms of ei
This can be seen as voltage divider
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Example 2
Derive an input-output model
Input ei (t) output e0 (t)
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Example 2 i
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Example 2 i
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44. Example 2 (alternative method)
Convert the circuit to its Laplace form
and use KVL and KCL as the resistive circuits i
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45. Example 3: Find TF model of the circuit
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46. Example 3: Transfer the circuit to Laplace and use KVL
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47. Example 4 (Homework) Derive a state-variable model
Find a model of the circuit
Input ii (t) output e0 (t)
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48. Steady State Response to Constant input
Steady state response to constant input is the response after very long time.
The steady state behavior is governed by a static model obtained from resistive circuit by
Replacing capacitors by open circuits
Replacing inductors by short circuits
Solve to get the steady state response.
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49. Steady State Response to Constant input
Method 1:
After deriving the I/O model, replace derivatives by zero and solve the algebraic model to get steady state response.
Method 2:
Replace Capacitors by open circuits and replace inductors by short circuits and obtain the model from the resistive circuit. Solve to get the steady state response.
Both give the same answer
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Example 5 + 24 − + 24 −
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