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ECEN 215, Principles of Electrical Engineering Dr. Mina Rahimian Lecture#

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Overview of Electrical Engineering Electrical Engineers design systems that have two main objectives: 1. To gather, store, process and present information. 2. To distribute, store and convert energy between various forms. Major Classifications: a) Communication systems generate, transmit and distribute information. Examples: telephone systems, radar systems, sattelite systems,.. b) Computer systems process and store information in digital form. Examples: A typical modern automobile contains tens of special purpose computers. Many electronic devices, household appliances, chemical processes and railroad switching yards routinely controlled through computers. c) Signal processing systems transform signals and the information contained in them into a the useful form. Examples: Image processing, voice recognition, fault detection,... d) Power systems generate, transmit and distribute electric power. Electric power is generate in large quantity in power plants by nuclear, hydroelectric and thermal (coal-, oil-, or gas fired) generators. Power is transmitted and distributed across the country by transmission lines and power grid. e) Control systems gather information with sensors and use electrical energy to control a physical process. Examples: Temperature control in heating/cooling system, temperature, pressure and flow rate control in oil refinery, autopilot systems in airplanes,...

All these branches have something in common which is Electric Circuit Theory. Fundamental Concepts Electrical phenomena are defined based on the concept of electric charge. Charactrerictics of electric charge: 1. Charge is bipolar. 2. Charge exists in descrete quantities, which are the integral multiples of electronic charge. ( 1.6 x10 ^ -19 colulombs) 3. Electrical effects are attributed to both separation of charge and charge in motion.

Voltage is the energy per unit charge created by separation of positive and negative charges.

Current is the rate of charge flow.

Ex. The current flowing through an element is

Calculate the charge entering the element from t=0 to t = 2 s.

Ex. The total charge entering a terminal is q=10 -10e^-2t (mc). Find the current at t= 0. 5 s.

Ideal circuit element: 1. It has only two terminals . 2. It is described mathematically in terms of its voltage and current. 3. It cannot be subdivided into other circuit elements.

Passive sign convention ( or, Passive Reference Configuration): when the current enters through the positive terminal of an element, passive sign convention is satisfied. In this case, use a positive sign in any expression that relates the voltage to the current. Otherwise, use a negative sign.

Power and Energy

If P > 0, the circuit element absorbs power and if p < 0, the circuit element delivers power. In any circuit sum of the element powers must equal zero.

Ex. Assuming a 20 V drop from terminal b to a, and a current of 4 A enters terminal b. Find the value of v, i and p for different polarity references.

Ex. The numerical values of voltages and currents are given in the table. Find the power of each element.

ECEN 215, Principles of Electrical Engineering Dr. Mina Rahimian Lecture#

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Ex. The power associated with element "a" and "b" are given. Find the power of element "c".

Five Basic Circuit Elements: - voltage source - current source - resistor - inductor - capacitor

Ideal Sources There are two types of ideal sources: dependent sources and independent sources. An independent source provides a specified voltage or current independent of the other circuit elements connected to it. independent voltage source: maintains the specified voltage regardless of the current passing through it.

Independent current source: maintains the specified current regardless of the voltage across it.

The relationship between the voltage and current of an independent source is determined by the other elements connected to it.

Dependent Sources The value of a dependent source is controlled by another voltage or current in the circuit. There are 4 types of dependent sources:

a) voltge controlled voltage source b) current controlled voltage source c) voltage controlled current source d) current controlled current source

Ex. Which interconnection is valid?

Ex. Compute the power associated with each element.

Resistor Resistance is the capacity of material to resist the flow of electric current. It is denoted by R and measured in ohms.

Resistor is a circuit element used to model the current resistivity of the resistor.

Basic Laws: Ohm's law, Kirchhoff's current law (KCL), Kirchhoff's voltage law (KVL). Ohm's Law: The ratio of voltage to current of a resistor is constant and equals the resistance of the resistor.

Conductance is the reciprocal of resistance, denoted by G and measured in Siemens (S).

Power for resistors

Ex. Calculate the unknown voltage or current and determine the power dissipated in each resistor.

Electric Circuit Models The circuit elements are used to construct the circuit model for practical systems. Example: circuit model of a flashlight

Analogy of fluid flow system to electric circuits: Voltage source ==> pump Conductor ==> pipe Current ==> flow rate Voltage ==> pressure difference Switch ==> valve resistance ==> constriction

ECEN 215, Principles of Electrical Engineering Dr. Mina Rahimian Lecture#

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Definition of some terms Node: The connection point between two or more circuit elements is called a node. Loop: Any closed path in the circuit is called a loop. Series connected elements: When two or more elements are connected sequentially, they have the same current and are said to be in series.

Parallel connected elements: When two or more elements are connected between the same node pair, they have the same voltage and are said to be in parallel.

Ex. Identify the nodes, series elements and parallel elements.

Kirchhoff's Curent Law (KCL) : The algebric sum of all currents entering a node (or a closed boundary) is zero.

Kirchhoff's Voltage Law (KVL): The algebric sum of all voltages around any loop in the circuit is zero.

Ex. Write KCL at each node.

kCL at a closed boundary

Ex. Use the basic laws to find i and check your answer using power balance.

Ex. Find V x, Vy, Vz.

To find the voltage from a to b you can choose any path from a to b and add all voltage in that path.

Ex. Find v0 and i0.

Chapter 2, Resistive Circuits Equivalent circuits: Two circuits are said to be equivalent if they have the same v-i characteristics at the specified pair of terminals.

ECEN 215, Principles of Electrical Engineering Dr. Mina Rahimian Lecture#

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EX. Position sensor based on the voltage divider rule

Ex. Combine the resistors and find the equivalent resistance.

Practical Sources

Ex. A 12V, 10 W light bulb is connected to a 12V battery with an internal resistance of 2 ohms. What is the voltage across the light bulb?

Ex. Design an attenuator circuit to suply a 5V, 1W light bulb from a 9V battery.

Ex: Find the power dissipated in the 6 ohm resistor.

The equivalent circuit of a network is different at different terminals. Ex. Find The equivalent resistance looking into terminals AB, AC, BC, and BD.

Combining Sources We can combine parallel connected current sources and series connected voltge sources.

Ex. Three light bulbs are connected to a 9v battery. find a) total current supplied by the battery. b) the current in each light bulb. c) the resistance of each bulb.

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Techniques of Circuit Analysis Node Voltage Method (Nodal Analysis) Nodal analysis is a systematic method of circuit analysis. In this method node voltages are taken as the circuit variables. A) For a circuit with n nodes and without any voltage sources, take these steps: 1. Select one node as the reference point, assign node voltages for the remaining nodes. (These are the voltages of the nodes with respect to the reference node). 2. Apply KCL at each node. Use Ohm's law to express all currents in terms of the node voltages. 3. Solve the KCL equations to find the node voltages. (#of unknowns= n-1)

Ex. Find the node voltages.

Ex. Find the node voltages.

B) when a voltage source exists in the circuit, there are two possibilities: case1: If the voltage source is connected between a node and the reference node, the voltage of that node is known and equals the voltage of the source.

Ex. Find the power associated with the 20v source using nodal analysis.

case 2: If the voltage source is connected between two non-reference nodes, combine the two nodes to form a super node. Apply KCL at the super node. The difference between the two voltages equals the voltage of the source (constraint equation). A supernode should enclose the voltage source and any other elements in parallel with it.

Ex. Find iφ .

Ex. Find the node voltages. Try different refference nodes.

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Source Transformation Source transformation is a simplification technique. It allows a voltage source in series with a resistor to be replaced by a current source in parallel with the same resistor, or vice versa.

Ex. Use source transformation to find the power in the 6v source.

Ex. Find v x using source transformation.

Thevenin Equivalent Circuit Thevenin equivalent circuit is a voltage source in series with a resistor which replaces a resistive network.

Norton Equivalent Circuit Norton equivalent circuit is a current source in parallel with a resistor which replaces a resistive network.

To find Thevenin equivalent circuit: 1. Find the Thevevin voltage by calculating the open circuit voltage. 2. Find the Thevevin resistance by calculationg the ratio of the open circuit voltage which is the Thevenin voltage found in part 1, to the short circuit current. If the circuit contains no dependent sources, the easiest way is to turn off all independent sources (To turn off an idependent source, replace the independent current source with an open circuit and replace the independent voltage source with a short circuit).Then combine the resistors. The Thevenin resistance equals the equivalent resistance.

Ex. Find the Thevenin equivalent circuit.

Ex. Find the Thevenin equivalent circuit.

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Maximum Power Transfer To maximize the power transferred from a circuit to a load, we can replace the circuit by its Thevenin equivalent circuit and connect it to an adjustable load and find the load value for which the power is maximum.

VTh and RTh are fixed. To find the value of RL which maximizes the power, we need to differentiate P with respect to RL and set it equal to zero.

Ex: Find RL for maximum power transfer. Find the maximum power.

Ex. Find RL for maximum power transfer. Find the maximum power.

Ex. Find RL for maximum power transfer. Find the maximum power.

Principle of Superposition A linear system obeys the principle of super position, which states that whenever a system is excited by more than one independent sources, the total response is the sum of the individual responses. Keep only one independent source at a time and turn off the remaining independent sources. Find the response to that source and repeat this for all independent sources. The total response is the sum of the responses. A linear system is one whose output is linearly proportional to its input.

Ex. Find i1 using superposition.

Ex. Find i, using superposition.

Ex. Find v ab using superposition.

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Chapter 3, Capacitance and Inductance In this chapter, we consider energy storage elements. Similar to resistors these elements are passive; that is, they do not supply power. This chapter describes the relationship between voltage, current, power and energy for inductors and capacitors. In addition, it explains the effect of connecting these components in series and in parallel.

The Capacitor A capacitor is an electrical component that consists of two conductors separated by an insulator. When a voltage is applied between the two conductors, a positive charge is deposited on one conductor and a negative charge on the other. The amount of charge stored in the conductors is proportional to the voltage.

Fluid flow analogy for a capacitor

In terms of fluid flow analogy, a capacitor represent a reservoir with an elastic membrane separating the inlet and outlet. As he fluid flows into the inlet, the membrane is stretched, creating a force (analogous to capacitor voltage) that opposes further flow. The displaced fluid volume is analogous to the charge stored in the capacitor.

voltage in terms of current

Power and Energy

Important properties of capacitors

1. A capacitor behaves as an open circuit when its voltage is constant. 2. The voltage across a capacitor cannot change instantaneously. (Capacitor voltage is continuous).

Ex. At t=0, a 2-mA source is applied to an uncharged 2 micro-farad capacitor. Determine a) the capacitor voltage at t=30 ms. b) the energy stored in the capacitor at t= 30 ms.

Ex: The voltage of a 0.4 microfarad capacitor is:

and the current at t=0 is 90 mA. a) Find the initial energy stored in the capacitor. b) Find A1 and A2.

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Ex. The current pulse is applied to a 25 µfcapacitor with no initial voltage. a) Find the capacitor voltage at t=50 µs. b) How much energy is stored in the capacitor by this pulse.

series and parallel combination of capacitors

Ex. Find the equivalent capacitance.

The Inductor : An inductor is an energy storage element composed of a coil of wire wound around a supporting core. The behavior of inductor is based on magnetic field which is created by charge in motion (current). A time varying current creates a time varying magnetic field which induces voltage in any conductor linked by the magnetic field. The parameter that relates the induced voltage to the current is called inductance, denoted by L and measured in Henrys (H).

current in terms of voltage

Power and Energy

Important properties of inductors

1. An inductor behaves as a short circuit when its current is constant. 2. The current in an inductor cannot change instantaneously. (Inductor current is continuous).

series combination of inductors

parallel combination of inductors

Ex. If v(t) = -1800 e-20t , i1(0)= 4A and i2(0)= -16A, a) Find i(t) b) Find i 1 (t) c) i2(t) d) Find the initial energy stored in the parallel inductors. e) Find energy delivered to the source within 0...