Basic Electrical Circuits Analysis ECE 221

Basic Electrical Circuits Analysis ECE 221 PhD. Khodr Saaifan http://trsys.faculty.jacobs-university.de k.saaifan@jacobs-university.de 1

2 Reference: Electric Circuits, 8th Edition James W. Nilsson, and Susan A. Riedel Lecturer: Khodr Saaifan (k.saaifan@jacobs-university.de) Research 1, Room 79 Office Phone 0421-200-3107 Office: M,W 1:30-3:30 pm Class Meets: Tue (11:15 am-12:30 pm) and Th (9:45-11:00 am) Grading: Two In-class Exams Laboratory/Practicum Final Exam Homework 50% 20% 20% 10%

Outline 1. Introduction 2. Basic Components and Electrical Circuits 1.Units and Scales 2.Circuit Variables 3.Voltage and Current Sources 4.Ohm's Law 3.Voltage and Current laws 1.Node, Branches, and loops 2.Kirchhoff's Current Law 3.Kirchhoff's Voltage Law 4.The Single-Loop Circuit 5.The Single-Node-Pair Circuit 6.Series and Parallel Connected Sources 7.Resistors Series and Parallel 8.Voltage and Current Division

4.Basic Nodal and Mesh Analysis 1.Units and Scales 2.The Supernode 3.Mesh Analysis 4.The Supermesh 5.Handy Circuit Analysis Techniques 1.Linearity and Superposition 2.Source Transformations 3.Thévenin and Norton Equivalent Circuits 4.Maximum Power Transfer 5.Delta-Wye Conversion 6.A Summary of Various Techniques

5 1. Introduction Electrical engineering is the field of engineering concerned with systems that produce, transmit, and measure electric signals Electrical circuits and systems are networks of electrical components used to supply, transmit and use electric power Circuit analysis is the process of studying and analyzing the various electrical quantities, such as currents, voltages, or powers, associated with each circuit's component Basic electrical circuits analysis covers the following topics: Linear circuit analysis Transient analysis Phasor domain circuit analysis

6 2. Basic Components and Electrical Circuits 2.1 Units and Scales The International System of Units (SI) defines 6 principal units from which the units of all other physical quantities can be derived Table 2.1 SI base units Basic Quantity Length Mass Time Electric current Thermodynamic temperature Luminous intensity Unit meter kilogram second ampere kelvin candela Symbol m kg s A K cd The SI unit of work or energy is the joule (J), which equals to a kg m2 s-2 in SI base units The SI unit of power is equivalent to one joule per second

7 The SI unit uses prefixes based on the power of 10 to relate larger and smaller units to the basic unit Table 2.2 SI prefixes Basic Quantity 1012 109 106 103 10-2 10-3 10-6 10-9 10-12 Name Symbol tera giga mega kilo centi milli micro nano pico T G M k c m n p

2.2 Circuit Variables 2.2.1 Charge Electric charge is an electrical property of the atomic particles of which matter consists, measured in coulombs (C) The charge of an electron is -1.602 X 10-19 C The coulomb is a large unit for charges such that in 1 C of charge, there are 1/(1.602 X 10-19)=6.24 X 1018 electrons The realistic or laboratory values of charges are on the order pc, nc, uc 2.2.2 Current Electric current is a flow of electric charge measured in ampere (A) note that 1 ampere (A) is equal to 1 coulombs per second (C/s) 8

9 The total charge transferred between time t0 and t can be expressed as There are several different types of current Direct current (dc) Sinusoidal current (ac) Exponential current Damped sinusoidal currents Representation of current in circuit analysis

Practice In the wire of shown figure, electrons are moving left to right to create a current of 1 ma. Determine and Ans: the current is in the opposite direction to flow of electrons 2.2.3 Voltage Voltage or potential difference measured in volts (V) is the energy required to move a unit of charge through an element note that 1 volt (V) is equal to 1 joule per coulombs (J/C) 10

Representation of voltage in circuit analysis The plus (+) and minus (-) signs at the points a and b are used to define a reference direction (the voltage polarity) Similar to the electric current, a constant voltage is called a dc voltage, whereas a sinusoidal voltage (time-varying) is called an ac voltage For practical purposes, the power and energy are important measures in circuit analysis 11

12 2.2.4 Power Measured in watts (W) to indicate the average absorbing energy by a circuit element The sign of power + sign: the power is absorbed by the element (resistor) - sign: the power is supplied by the element (?) Since the energy can neither be created or dissipated (only transferred), the algebraic sum of powers in a circuit, at any instant of time, must be zero

13 Determine p1 Ans: 2.2.5 Energy The energy absorbed or supplied by an element from time 0 to t is Electricity bills:: The electric power utility companies measure energy in kilowatt-hours (kwh), where 1 kwh = 3600 kj

14 2.3 Voltage and Current Sources There are two types of circuit elements: Active elements (supplying energies), e.g., electric generator, batteries Passive elements (absorbing energy), e.g., resistors, capacitors, and inductors The passive elements can be classified according to the relationship of the current through it to the voltage Resistor, Capacitor, Inductor, Voltage and current sources: Voltage sources provides the circuit with a specified voltage Current source provides the circuit with a specified current

15 Independent voltage source The source is characterized by a terminal voltage which is completely independent of the current through it dc voltage source ac voltage source Independent current source The current through the element is completely independent of the voltage across it dc current source ac current source

16 Dependent sources The value of dependent sources depends on a voltage or currents of some other elements There are 4 different types of dependent sources current-controlled current source voltage-controlled current source Find vl Ans: voltage-controlled voltage source current-controlled voltage source

17 Dependent sources The value of dependent sources depends on a voltage or currents of some other elements There are 4 different types of dependent sources current-controlled current source voltage-controlled current source voltage-controlled voltage source current-controlled voltage source Find the power absorbed by each element in the circuit Ans:

18 2.4 Ohm's Law Ohm's law states the voltage across conducting materials is directly proportional to the current flowing through the material, or where R is the resistance slope=r (V/A) The unit of the resistance is Ohm (Ω) The Resistor has a linear relation between the applied voltage and the current The current goes from a higher potential to a lower potential The power absorbed by the resistor can be expressed as The resistor is a passive element that cannot deliver or store energy Find i and R, if v=-10 V and R is absorbing 0.1 W Ans:

19 The resistance of any cylindrical object is given as l Material with resistivity r A For a linear resistor, the ratio of the current to the voltage is called the conductance The SI unit of the electrical conductance G is siemens (S) Homework Assignments P2.11, P2.12, P2.15, P2.17, P2.20, P2.22, P2.23, P2.26, P2.31, P2.32, P2.33, and P2.35

20 3. Voltage and Current laws 3.1 Node, Branches, and loops A branch represents a single element such as a voltage source or a resistor A node is the point of the connection between two or more elements (branches) It is usually indicated by a dot in a circuit If a connecting wire (short circuit) connects two nodes, the two nodes constitute a single nodes A loop is any closed path in a circuit A closed path is formed by starting at a node, passing through a set of nodes and returning to the start node without passing through any node more than once branch loop

3.2 Kirchhoff's Current Law Kirchhoff's Current Law (KCL) is based on the law of conservation of charge The algebraic sum of the currents entering any node is zero An alternative form of KCL is the current entering any node = the current leaving that node KCL can be applied to any closed boundary (closed path) 21

22 3.3 Kirchhoff's Voltage Law Kirchhoff's voltage Law (KVL) is based on the law of conservation of energy The algebraic sum of the voltages around any closed path is zero KVL When voltage sources are connected in series, KVL can be applied to obtain the total voltage a a = b b

Determine vx in the circuit Ans: 23

Determine vx in the circuit Ans: 24

3.4 The Single-Loop Circuit Single-loop circuits Elements are connected in series All elements carry the same current We shall determine The current through each element The voltage across each element The power absorbed by each element 25

26 3.4 The Single-Loop Circuit Single-loop circuits Elements are connected in series All elements carry the same current We shall determine The current through each element The voltage across each element The power absorbed by each element We apply the following steps 1) Assign a reference direction for the unknown current 2) Assign voltage references to the elements 3) Apply KVL to the closed loop path 4) Use Ohm's law where needed to get an equation in i 5) Solve for i KVL

27 Find i and p for all elements in the circuit Ans: KVL 1) Assign a reference direction for the unknown current 2) Assign voltage references to the elements (note that va=-v2) 3) Apply KVL to the closed loop path 4) Use Ohms law where needed to get an equation in i 5) Solve for i

28 Find i and p for all elements in the circuit Ans: KVL Computing the power absorbed by each element The total power absorbed by all elements

3.5 The Single-Node-Pair Circuit Single-node-pair circuits Elements are connected in parallel All elements have a common voltage We shall determine The current through each element The voltage across each element The power absorbed by each element 29

3.5 The Single-Node-Pair Circuit Single-node-pair circuits Elements are connected in parallel All elements have a common voltage We shall determine The current through each element The voltage across each element The power absorbed by each element We apply the following steps 1) Define the voltage v and arbitrary select its polarity 2) Use passive sign convention to determine the currents directions 3) Apply KCL at the node 4) Use Ohm's law where needed to get an equation in v 5) Solve for v 30

Find v and p supplied by the independent source Ans: 1) Assign an arbitrary sign for the unknown voltage 2) passive sign convention to find the currents directions (note that ix=-i2) 3) Apply KCL to the nodes 4) Use Ohm's law where needed to get an equation in v 5) Solve for v 31

HW: Find i1, i2, i3, and i4 32

3.6 Series and Parallel Connected Sources Series-connected voltage sources can be replaced by a single source Parallel current sources can be replaced by a single source 33

3.7 Resistors Series and Parallel Series connection KVL Parallel connection 34

Find the voltage and the power of the independent source 1) Apply KCL at the top node 2) Use Ohm's law for (i1=vx/6) and (vx=3i3) 3) Solve i3 and vx 35

36 3.8 Voltage and Current Division Voltage divider: is a passive linear circuit that produces an output voltage (vout) that is a fraction of its input voltage (vin) Easily solved with KCL, KVL, & equivalent resistances Then, Generally, assume we have The voltage vn can be given as Easy to find the other voltages, too

37

38 Current divider: is a simple linear circuit that produces an output current (iout) that is a fraction of its input current (iin) Easily solved with Since For n=2, we have The circuit divider reduces to

Use resistance combination methods and current division to find i1 and i2 and vx Ans: We note i1 goes to the following equivalent resistor Use current divider, we have 39

We note i2 goes to the following equivalent resistor Use current divider, we have HW: Solve vx 40

We note i2 goes to the following equivalent resistor Use current divider, we have HW: Solve vx Homework Assignments P3.6, P3.7, P3.13, P3.15, P3.16, P3.19, P3.20, P3.21, P3.30, P3.31, P3.35, P3.39, P3.73, P3.75 and P3.82 41

42 4. Basic Nodal and Mesh Analysis This chapter introduces two basic circuit analysis techniques named nodal analysis and mesh analysis 4.1 Nodal Analysis For a simple circuit with two nodes, we often have one unknown voltage between two nodes To solve the unknown, applying KCL at this node gives Adding a node should provide an additional unknown, three-node circuit has 2 unknown N-node circuit has (N-1) voltages with (N-1) equations.

Nodal technique applies the following step 1- Count the number of nodes (N) 2- Designate a reference node 3- Label the nodal voltages (we have N-1 voltages) 43

44 4- Write KCL equations for the non-reference nodes (currents in = currents out) 5- Express any additional unknowns in terms of nodal voltages 6- Organize the equations (1) (2) 7- Solve the system of equations for the nodal voltages

7- Solve the system of equations for the nodal voltages using a Cramer's rule and determinants, we have 45

46 Compute the voltages at each node Ans: Write KCL equations for the three nodes Organize the equations (1) (2) (3)

47 Compute the voltage at each node Ans: Solve the system of equations for the nodal voltages HW: use a Cramer's rule and determinants to solve the system

48 4.2 Nodal Analysis with Supernode A supernode is formed when a voltage source is the only element connected between two essential nodes 1- Define a current through the source and write KCL equations for the two nodes 2- We note that there is no need to determine ivs to solve the circuit (1) 3- Apply KVL between the two nodes (2) Thus, the KCL at the supernode is directly given by

49 Determines the node-to reference voltages. Node 1 to reference is supernode Node 2 Node 3 & node 4 Express vx=v2-v1 and vy=v4-v1 in terms of nodal voltages and organize the equations (1) (2) (3) Solve to get

50 4.3 Mesh Analysis In nodal analysis, circuit variables are node voltages Nodal analysis applies KCL to find unknown voltages In mesh analysis, circuit variables are mesh currents Mesh analysis applies KVL to find unknown currents Both methods result in a system of linear equations Mesh analysis is only applicable to a circuit that is planar Planar vs. Non-planar Circuits Planar circuit: it can be drawn on a plane surface where no branch cross any other branch (element) Non-planar circuit there is no way to redraw it and avoid the branches crossing Planar circuit Non planar circuit

51 Mesh & mesh current A mesh is a property of a planar circuit and it is defined a loop that does not contain any other loops within it The current through a mesh is known as a mesh current mesh mesh

52 4.3 Mesh Analysis 1. Determine if the circuit is a planar circuit. If not, perform nodal analysis instead. 2. Count the number of meshes (M) 3. Label each of the M mesh currents (defining all mesh currents to flow clockwise results in a simpler analysis) 4. Write a KVL equation around each mesh For mesh 1, we have or (1) For mesh 2, we have or The solution is easily obtained (2)

53 Determine the power supplied by the 2 V source. i1 We first define two clockwise mesh currents For mesh 1, we write the following KVL equation The same for mesh 2, we write i2

Rearranging and grouping terms, we have and Solve the both equation yields i1=1.132 A and i2=-0.1053 A The 2 v source supplies (2)(i1-i2)=2.4 W 54

55 4.4 The Supermesh Similar to the supernode in a node voltage analysis A supermesh is formed when a current source is the only element connected between two meshes 1- Define a voltage across the source and write KVL equations for the two meshes and 2- We do not need to evaluate vcs to solve the circuit 3- This leads us to create a supermesh whose interior is that of mesh 1 and mesh 2 4- Finally, the source current is related to the mesh currents,

56 Determine the three mesh currents. i1 i2 i1 i2 i1-i2 i1-i2 i3-i2 i3 i3-i2 i3 i1-i3 The 7 A independent current source forms a supermesh between mesh 1 and mesh 3 Applying KVL over the supermesh gives or KVL for mesh 2 or

Homework Assignments 3 57