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Circuit Theory 1Rojien V. Morcilla, R.E.E., M.Eng’g.

ELECTRIC CIRCUITAn interconnection of electrical elements.

CHARGE AND CURRENTChargemost basic quantity in an electric circuitis an electrical property of the atomic particles of which matter consists, measured in coulombs (C).charge e on an electron is negative and equal in magnitude to 1.602×10−19 C, while a proton carries a positive charge of the same magnitude as the electron. The presence of equal numbers of protons and electrons leaves an atom neutrally charged.

CHARGE AND CURRENTPoints should be noted about electric charge:The coulomb is a large unit for charges. In 1 C of charge, there are 1/(1.602 × 10−19) = 6.24 × 1018 electrons. Thus realistic or laboratory values of charges are on the order of pC, nC, or μC.According to experimental observations, the only charges that occur in nature are integral multiples of the electronic charge e = −1.602 × 10−19 C.The law of conservation of charge states that charge can neither be created nor destroyed, only transferred. Thus the algebraic sum of the electric charges in a system does not change.

CHARGE AND CURRENTElectric charge or electricity is mobilePositive charges move in one direction while negative charges move in the opposite direction

Motion of charges creates electric current

Conventionallytake the current flow as the movement of positive charges, that is, opposite to the flow of negative charges.CHARGE AND CURRENTElectric current is the time rate of change of charge, measured in amperes (A).Direct Current (DC) is a current that remains constant with time.Alternating Current (AC) is a current that varies sinusoidally with time.1 ampere = 1 coulomb/second

CHARGE AND CURRENT (Example)How much charge is represented by 4,600 electrons?Calculate the amount of charge represented by two million protons.The total charge entering a terminal is given by q = 5tsin4πt mC.Calculate the current at t = 0.5s.If in Example 3, q = (10 − 10e−2t ) mC, find the current at t = 0.5 s.Determine the total charge entering a terminal between t = 1 s and t = 2s if the current passing the terminal is i = (3t2− t) A.The current flowing through an element is Calculate the charge entering the element from t = 0 to t = 2s.

VOLTAGEVoltage (or potential difference) is the energy required to move a unit charge through an element, measured in volts (V).Voltage vabbetween two points a and b in an electric circuit is the energy (or work) needed to move a unit charge from a to b; mathematically,where w is energy in joules (J) and q is charge in coulombs (C). Voltage vab or simply v is measured in volts (V)1 volt = 1 joule/coulomb = 1 newton meter/coulomb

VOLTAGEThe plus (+) and minus (−) signs are used to define reference direction or voltage polarity.point a is at a potential of vab volts higher than point bthe potential at point a with respect to point b is vabvab= −vba(a), there is a 9-V voltage drop from a to b or equivalently a 9-V voltage rise from b to a.(b), point b is −9 V above point a.A voltage drop from a to b is equivalent to a voltage rise from b to a.constant voltage is called a dc voltage and is represented by V, whereas a sinusoidallytime-varying voltage is called an ac voltage and is represented by v.

POWER AND ENERGYPower is the time rate of expending or absorbing energy, measured in watts (W).where p is power in watts (W), w is energy in joules (J), and t is time in seconds (s).power p is a time-varying quantity and is called the instantaneous power.If the power has a + sign, power is being delivered to or absorbedby the element. If, on the other hand, the power has a − sign, power is being supplied by the element.

POWER AND ENERGYPassive sign convention is satisfied when the current enters through the positive terminal of an element and p = +vi. If the current enters through the negative terminal, p = −vi.

POWER AND ENERGYlaw of conservation of energy must be obeyed:algebraic sum of power in a circuit, at any instant of time, must be zero.Energy absorbed or supplied by an element from time t0 to time tis Energy is the capacity to do work, measured in joules ( J). electric power utility companies measure energy in watt-hours (Wh)

POWER AND ENERGY (Example)An energy source forces a constant current of 2A for 10s to flow through a light bulb. If 2.3kJ is given off in the form of light and heat energy, calculate the voltage drop across the bulb.To move charge q from point a to point b requires−30 J. Find the voltage drop vab if: (a) q = 2C, (b) q = −6C . Find the power delivered to an element at t = 3 ms if the current entering its positive terminal is i = 5cos60πt A and the voltage is: (a) v = 3i, (b) v = 3 di/dt.Find the power delivered to the element in Example 3 at t = 5 ms if the current remains the same but the voltage is: (a) v = 2i V, (b) v =How much energy does a 100-W electric bulb consume in two hours? A stove element draws 15 A when connected to a 120-V line. How long does it take to consume 30 kJ?

CIRCUIT ELEMENTSTwo types of elements found in electric circuits:Passive element – not capable of generating energy (resistors, capacitors, and inductors)Active element - capable of generating energy (generators, batteries, and operational amplifiers)Two kinds of sources:Independent Sources - an active element that provides a specified voltage or current that is completely independent of other circuit variablesDependent Sources - an active element in which the source quantity is controlled by another voltage or current. (transistors, operational amplifiers and integrated circuits)

CIRCUIT ELEMENTSIdeal Independent Voltage Source delivers to the circuit whatever current is necessary to maintain its terminal voltage (batteries and generators).Symbols for independent voltage sources: (a) used for constant or time-varying voltage, (b) used for constant voltage (dc).

CIRCUIT ELEMENTSIdeal Independent Current Source is an active element that provides a specified current completely independent of the voltage across the source.Symbol for independent current source

CIRCUIT ELEMENTSFour possible types of dependent sources:Voltage-Controlled Voltage Source (VCVS).Current-Controlled Voltage Source (CCVS).Voltage-Controlled Current Source (VCCS).Current-Controlled Current Source (CCCS).Symbols for: (a) dependent voltage source, (b) dependent current source.

CIRCUIT ELEMENTSCalculate the power supplied or absorbed by each elementCompute the power absorbed or supplied by each component of the circuit

OHM’S LAWResistance (R) –of an element denotes its ability to resist the flow of electric current; it is measured in ohms (Ω).The resistance of any material with a uniform cross-sectional area (A) depends on A and its length (l)where ρ is known as the resistivity of the material in ohm-meters.Good conductors, such as copper and aluminum, have low resistivities, while insulators, such as mica and paper, have high resistivities.

OHM’S LAW(a) Resistor, (b) Circuit symbolfor resistance.

OHM’S LAWOhm’s law states that the voltage “v” across a resistor is directly proportional to the current “i” flowing through the resistor.Short circuit is a circuit element with resistance approaching zero.Open circuit is a circuit element with resistance approaching infinity.

OHM’S LAW(a) Short circuit (R = 0), (b) Open circuit (R =∞).

OHM’S LAWConductance is the ability of an element to conduct electric current; it is measured in mhos ( ) or siemens (S).Resistance can be expressed in ohms or siemensThe power dissipated by a resistor can be expressed in terms of R.The power dissipated by a resistor may also be expressed in terms of G

OHM’S LAWAn electric iron draws 2 A at 120 V. Find its resistance.In the circuit shown, calculate the current i, the conductance G, and the power p.A voltage source of 20 sin πt V is connected across a 5-k resistor. Find the current through the resistor and the power dissipated.

NODES, BRANCHES, AND LOOPSA branch represents a single element such as a voltage source or a resistorA node is the point of connection between two or more branchesA loop is any closed path in a circuit.A network with b branches, n nodes, and l independent loops will satisfy the fundamental theorem of network topology:b = l + n − 1

NODES, BRANCHES, AND LOOPSTwo or more elements are in series if they are cascaded or connected sequentially and consequently carry the same currentTwo or more elements are in parallel if they are connected to the same two nodes and consequently have the same voltage across them.

NODES, BRANCHES, AND LOOPSDetermine the number of branches and nodes in the circuit shown. Identify which elements are in series and which are in parallel. How many branches and nodes does the circuit in have? Identify the elements that are in series and in parallel.

KIRCHHOFF’S LAWSKirchhoff’s current law (KCL) states that the algebraic sum of currents entering a node (or a closed boundary) is zero.where N is the number of branches connected to the node and in is the nth current entering (or leaving) the node. By this law, currents enteringa node may be regarded as positive, while currents leaving the node maybe taken as negative or vice versa.

KIRCHHOFF’S LAWSThe sum of the currents entering a node is equal to the sum of the currents leaving the node.

KIRCHHOFF’S LAWSKirchhoff’s voltage law (KVL) states that the algebraic sum of all voltages around a closed path(or loop) is zero.where M is the number of voltages in the loop (or the number of branches in the loop) and vm is the mth voltage.

KIRCHHOFF’S LAWSBy KVL:−v1 + v2 + v3 − v4 + v5 = 0v2 + v3 + v5 = v1 + v4Sum of voltage drops = Sum of voltage rises

KIRCHHOFF’S LAWSFor the circuit, find voltages v1 and v2.Determine vo and i in the circuit shown.

KIRCHHOFF’S LAWSFind current io and voltage vo in the circuit shownFind the currents and voltages in the circuit shown

SERIES RESISTORS AND VOLTAGE DIVISIONUsing Ohms LawBy KVL (clockwise direction)The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances.

SERIES RESISTORS AND VOLTAGE DIVISIONVoltage across each resistorPrinciple of Voltage Division - source voltage ‘v’ is divided among the resistors in direct proportion to their resistances; the larger the resistance, the larger the voltage drop.

PARALLEL RESISTORS AND CURRENT DIVISIONBy Ohms LawBy KCL @ Node ‘a’Then:

PARALLEL RESISTORS AND CURRENT DIVISIONWhere:The equivalent resistance of two parallel resistors is equal to the product of their resistances divided by their sum.Req is always smaller than the resistance of the smallest resistor in the parallel combination. If R1 = R2 = … = RN = R, then

PARALLEL RESISTORS AND CURRENT DIVISIONEquivalent conductance for N resistors in parallel :The equivalent conductance of resistors connected in parallel is the sum of their individual conductance's.Equivalent conductance Geq of N resistors in series:

PARALLEL RESISTORS AND CURRENT DIVISIONPrinciple of Current Division - total current ‘i’ is shared by the resistors in inverse proportion to their resistances.

PARALLEL RESISTORS AND CURRENT DIVISIONDivide both the numerator and denominator by R1R2In general, if a current divider has N conductors (G1,G2, . . . , GN) in parallel with the source current i, the nth conductor (Gn) will have current:

ExampleFind Req for the circuit shown in Fig. A.Calculate the equivalent resistance Rab in the circuit shown in Fig. BFind the equivalent conductance Geq for the circuit in Fig. C.Fig. AFig. CFig. B

ExampleFind io and vo in the circuit shown in Fig. D. Calculate the power dissipated in the 3- resistor.For the circuit shown in Fig. E, determine: (a) the voltage vo, (b)the power supplied by the current source, (c) the power absorbed by each resistor.Fig. EFig. D

WYE-DELTA TRANSFORMATIONSTwo forms of the same network: (a) Y, (b) T.The bridge networkTwo forms of the same network: (a) Δ, (b) π.

WYE-DELTA TRANSFORMATIONSDelta to Wye ConversionEach resistor in the Y network is the product of the resistors in the two adjacent Δ branches, divided by the sum of the three Δresistors.

WYE-DELTA TRANSFORMATIONSWye to Delta ConversionEach resistor in the Δ network is the sum of all possible products of Y resistors taken two at a time, divided by the opposite Y resistor.

WYE-DELTA TRANSFORMATIONSThe Y and Δ networks are said to be balanced whenUnder these conditions, conversion formulas become

ExampleConvert the Δ network in Fig. A to an equivalent Y network.Obtain the equivalent resistance Rab for the circuit in Fig. B and use it to find current ‘i’.Fig. AFig. B

NODAL ANALYSISNodal analysis provides a general procedure for analyzing circuits using node voltages as the circuit variablesSteps to Determine Node Voltages:Select a node as the reference node. Assign voltages v1, v2, . . . , vn−1 to the remaining n − 1 nodes. The voltages are referenced with respect to the reference node. Apply KCL to each of the n − 1 nonreference nodes. Use Ohm’s law to express the branch currents in terms of node voltages.Solve the resulting simultaneous equations to obtain the unknown node voltages.

NODAL ANALYSISCurrent flows from a higher potential to a lower potential in a resistor.

NODAL ANALYSIS@ Node 1@ Node 2Then;So; In Terms of Conductance:In Matrix Form:

NODAL ANALYSIS (Example)Calculate the node voltages in the circuits shown.

NODAL ANALYSIS WITH VOLTAGE SOURCESCase 1: If a voltage source is connected between the reference node and a nonreference node, we simply set the voltage at the nonreference node equal to the voltage of the voltage source.Case 2: If the voltage source (dependent or independent) is connected between two nonreference nodes, the two nonreference nodes form a generalized node or supernode; we apply both KCL and KVL to determine the node voltages.

NODAL ANALYSIS WITH VOLTAGE SOURCESA supernode is formed by enclosing a (dependent or independent) voltage source connected between two nonreference nodes and any elements connected in parallel with it.Note the following properties of a supernode:The voltage source inside the supernode provides a constraint equation needed to solve for the node voltages.A supernode has no voltage of its own.A supernode requires the application of both KCL and KVL.

NODAL ANALYSIS WITH VOLTAGE SOURCES (Example)For the circuits shown, find the node voltages.

NODAL ANALYSIS WITH VOLTAGE SOURCES (Example)Find v and i in the circuit.

MESH ANALYSISMesh analysis provides another general procedure for analyzing circuits, using mesh currents as the circuit variablesA mesh is a loop that does not contain any other loop within itNodal analysis applies KCL to find unknown voltages in a given circuit, while mesh analysis applies KVL to find unknown currentsonly applicable to a circuit that is planarPlanar circuit is one that can be drawn in a plane with no branches crossing one another; otherwise it is nonplanar.

MESH ANALYSISA planar circuit with crossing branchesthe same circuit redrawn with no crossing branchesA nonplanar circuit.(c)

MESH ANALYSIS (Example)For the circuit shown, find the branch currents I1, I2, and I3 using mesh analysis.

MESH ANALYSIS (Example)Calculate the mesh currents i1 and i2 in the circuit shown.Use mesh analysis to find the current io in the circuit shown.

MESH ANALYSIS (Example)Using mesh analysis, find io in the circuit shown.

MESH ANALYSIS WITH CURRENT SOURCESCase 1: When a current source exists only in one mesh:Case 2: When a current source exists between two meshes: We create a supermesh by excluding the current source and any elements connected in series with it.Set i2 = −5 A and write a mesh equation for the other mesh in the usual way, that is,

MESH ANALYSIS WITH CURRENT SOURCESA supermesh results when two meshes have a (dependent or independent) current source in common.Two meshes having a current source in commona supermesh, created by excluding the current source.

MESH ANALYSIS WITH CURRENT SOURCESNote the following properties of a supermesh:The current source in the supermesh is not completely ignored; it provides the constraint equation necessary to solve for the mesh currents.A supermesh has no current of its own.A supermesh requires the application of both KVL and KCL.

MESH ANALYSIS WITH CURRENT SOURCES (Example)For the circuit shown, find i1 to i4 using mesh analysis.

MESH ANALYSIS WITH CURRENT SOURCES (Example)Use mesh analysis to determine i1, i2, and i3 in the circuit shown.

LINEARITY PROPERTYLinearity is the property of an element describing a linear relationship between cause and effectThe property is a combination of both the homogeneity (scaling) property and the additivity propertyHomogeneity property requires that if the input (also called the excitation) is multiplied by a constant, then the output (also called the response) is multiplied by the same constant.

LINEARITY PROPERTYFor example, Ohm’s law relates the input i to the output v,If the current is increased by a constant k, then the voltage increases correspondingly by k, that is,The additivity property requires that the response to a sum of inputs is the sum of the responses to each input applied separately. Using the voltage-current relationship of a resistor, ifthen applying (i1 + i2) gives

LINEARITY PROPERTYIn general, a circuit is linear if it is both additive and homogeneous. A linear circuit consists of only linear elements, linear dependent sources, and independent sourcesA linear circuit is one whose output is linearly related (or directly proportional) to its inputRelationship between power and voltage (or current) is nonlinear – linearity theorems are not applicable to power

LINEARITY PROPERTYConsider the linear circuit shownlinear circuit has no independent sources inside itExcited by a voltage source vsTerminated by a load RCurrent i through R as the outputSuppose vs = 10 V gives i = 2 A. According to the linearity principle, vs = 1 V will give i = 0.2 A. By the same token, i = 1 mAmust be due to vs =5 mV.

LINEARITY PROPERTYFor the circuit, find io when vs = 12 V and vs = 24 V.For the circuit , find vo when is = 15 and is = 30 A.

LINEARITY PROPERTYAssume Io = 1 A and use linearity to find the actual value of Io in the circuit.Assume that Vo = 1 V and use linearity to calculate the actual value of Vo in the circuit.

SUPERPOSITIONThe superposition principle states that the voltage across (or current through) element in a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone.To apply the superposition principle, we must keep two things in mind:We consider one independent source at a time while all other independent sources are turned off. This implies that we replace every voltage source by 0 V (or a short circuit), and every current source by 0 A (or an open circuit). This way we obtain a simpler and more manageable circuit.Dependent sources are left intact because they are controlled by circuit variables.

SUPERPOSITIONSteps to Apply Superposition Principle :Turn off all independent sources except one source. Find the output (voltage or current) due to that active source using nodal or mesh analysis.Repeat step 1 for each of the other independent sources.Find the total contribution by adding algebraically all the contributions due to the independent sources.

SUPERPOSITIONUse the superposition theorem to find v in the circuitUsing the superposition theorem, find vo in the circuit

SUPERPOSITIONFind io in the circuit using superposition.Use superposition to find vx in the circuit

SUPERPOSITIONFor the circuit, use the superposition theorem to find i.Find i in the circuit using the superposition principle.

SOURCE TRANSFORMATIONA source transformation is the process of replacing a voltage source vsin series with a resistor R by a current source is in parallel with a resistor R, or vice versa.

SOURCE TRANSFORMATIONPoints to be mind when dealing with source transformation:The arrow of the current source is directed toward the positive terminal of the voltage source.Source transformation is not possible when R = 0, which is the case with an ideal voltage source. However, for a practical, nonideal voltage source, R = 0. Similarly, an ideal current source with R =∞cannot be replaced by a finite voltage source.

SOURCE TRANSFORMATIONUse source transformation to find vo in the circuitFind io in the circuit using source transformation

SOURCE TRANSFORMATIONFind vx using source transformation.Use source transformation to find ix in the circuit.

THEVENIN’S THEOREMDeveloped in 1883 by M. Leon Thevenin (1857–1926), a French telegraph engineerThevenin’s theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source VTh in series with a resistor RTh, where VTh is the open-circuit voltage at the terminals and RTh is the input or equivalent resistance at the terminals when the independent sources are turned off.

THEVENIN’S THEOREMCASE 1: If the network has no dependent sources, we turn off all independent sources. RTh is the input resistance of the network looking between terminals a and b, as shown

THEVENIN’S THEOREMCASE 2: If the network has dependent sources, we turn off all independent sources. As with superposition, dependent sources are not to be turned off because they are controlled by circuit variables. We apply a voltage source vo at terminals a and b and determine the resulting current io. Then RTh = vo/io, as shown. Alternatively, we may insert a current source io at terminals a-b as shown in and find the terminal voltage vo. Again RTh = vo/io. Either of the two approaches will give the same result.

THEVENIN’S THEOREMIt often occurs that RTh takes a negative value. In this case, the negative resistance (v = −iR) implies that the circuit is supplying power.

THEVENIN’S THEOREMFind the Thevenin equivalent circuit of the circuit shown, to the left of the terminals a-b. Then find the current through RL = 6Ω, 16Ω, and 36Ω .Using Thevenin’s theorem, find the equivalent circuit to the left of the terminals in the circuit. Then find i.

THEVENIN’S THEOREMFind the Thevenin equivalent of the circuit shown.Find the Thevenin equivalent circuit of the circuit shown to the left of the terminals.

THEVENIN’S THEOREMDetermine the Thevenin equivalent of the circuit shown.Obtain the Thevenin equivalent of the circuit.

NORTON’S THEOREMDeveloped by E. L. Norton, an American engineer at Bell Telephone Laboratories in 1926, about 43 years after Thevenin published his theorem.Norton’s theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source IN in parallel with a resistor RN, where IN is the short-circuit current through the terminals and RN is the input or equivalent resistance at the terminals when the independent sources are turned off.

NORTON’S THEOREMBy Source transformation, the Thevenin and Norton resistances are equalTo find the Norton current IN,we determine the short-circuit current flowing from terminal a to b in both circuits . Observe the close relationship between Norton’s and Thevenin’s theorems: RN = RTh

NORTON’S THEOREMSince VTh, IN, and RTh are related, to determine the Thevenin or Norton equivalent circuit requires that we find:The open-circuit voltage voc across terminals a and b.The short-circuit current isc at terminals a and b.The equivalent or input resistance Rin at terminals a and b when all independent sources are turned off.Thus we find;

NORTON’S THEOREMFind the Norton equivalent circuit of the circuitsUsing Norton’s theorem, find RN and IN of the circuit at terminals a-b.

NORTON’S THEOREMFind the Norton equivalent circuit of the circuit.

MAXIMUM POWER TRANSFERCircuit is designed to provide power to a loadElectric utilities, minimizing power losses in the process of transmission and distribution is critical for efficiency and economic reasons, there are other applications in areas such as communications where it is desirable to maximize the power delivered to a loadAddress problem of delivering the maximum power to a load when given a system with known internal losses.

MAXIMUM POWER TRANSFERThevenin equivalent is useful in finding the maximum power a linear circuit can deliver to a loadMaximum power is transferred to the load when the load resistance equals the Thevenin resistance as seen from the load (RL = RTh).Maximum power transferred is obtained by:

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