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EENG223: CIRCUIT THEORY I DC Circuits: Basic Laws Dr. Hasan Demirel EENG223: CIRCUIT THEORY I Resistance and Ohms Law All materials resist the flow of current.. Resistance R of an element denotes its ability to resist the flow of electric current, which is measured in ohms (). A cylindrical material of length l and cross-sectional area A has the following resistance: AlR R= resistance of the element in ohms () = resistivity of the material in ohm-meters (-m) l = lenghth of the cylindrical material in meters (m) A=Cross sectional area of the material in meters2 (m2) EENG223: CIRCUIT THEORY I Resistance and Ohms Law Resistance: Basic Concepts and Assumptions: Conductors (e.g. Wires) have very low resistance (50 M) that can be usually ignored ( omitted from circuit for analysis). Resistors have a medium range of resistance and must be accounted for the circuit analysis. Conceptually, a light bulb is similar to the resistor. Properties of the bulb control how much current flows and how much power is dissipated (absorbed then emitted as light and heat). As with the circuit elements, we need to know the current through and voltage across the device are ralated. The relationship between the current and voltage can be linear or nonlinear. EENG223: CIRCUIT THEORY I Resistance and Ohms Law Ohms Law states that the voltage v accross a resistor is directly proportional to the current i flowing through the resistor. Ri = voltage in volts (V), i = current in (A), R = resistance in (). Sign is determined by passive sign convention (PSC). Materials with linear relationships between the current and the voltage satisfy the Ohms Law. Resistor symbol Ohmlaw applies Ohms Law does Not apply Resistance R is equal to the slope m in (a). EENG223: CIRCUIT THEORY I Resistance and Ohms Law Resistors and Passive Sign Convention (PSC) Note that the the relationship between current and voltage are sign sensitive. PSC is satisfied if the current enters the positive terminal of an element: f PSC is satisfied : =iR f PSC is not satisfied : =iR PSC satisfied PSC not satisfied PSC not satisfied PSC satisfied ( =iR) ( =iR) ( =iR) ( =iR) EENG223: CIRCUIT THEORY I Equations derived from Ohms Law (1 = 1 V/A) Resistors cannot produce power , so the power absorbed by a resistors will always be positive. 1 = 1 V/A iR iR Ohms Law: RRiip22 Recall: Ri EENG223: CIRCUIT THEORY I Short Circuit and Open Circuit Short Circuit as Zero resistance: An element (or wire) with R =0 is called a short circuit. Short circuit is just drawn as a wire (line). EENG223: CIRCUIT THEORY I Short Circuit and Open Circuit Short Circuit as Voltage Source (0V): An ideal voltage sorce with Vs=0 V is equivalent to a short circuit. Since =iR and R =0, =0 regardless of i. You could draw a source with Vs=0 V, but it is not done in practice. You cannot connect a voltage source to a short circuit. f connected, usually wire wins and the voltage source melts (smoke comes out) if not protected. EENG223: CIRCUIT THEORY I Short Circuit and Open Circuit An element (or wire) with R= is called an open circuit. Such an element is just omitted. EENG223: CIRCUIT THEORY I Short Circuit and Open Circuit Open Circuit as Current Source (0 A): An ideal current source with I=0 A is equivalent to a open circuit. Since =iR and if I=0 A, then R = . You could draw a source with I=0 A, but it is not done in practice. You cannot connect a current source to an open circuit. f connected, usually you blow the current source (smoke comes out) if not protected. The insulator (air) wins. Else, sparks fly. EENG223: CIRCUIT THEORY I Conductance: inverse of resistance Conductance is the ability of an element to conduct electric current . Conductance is the inverse of resistance. iRG 1 Units: siemens (S) or mho ( ) Ri Gi RRiip22 GiGip22 EENG223: CIRCUIT THEORY I Circuit Building Blocks: Nodes, Branches and Loops: In circuit analysis, we need a common language and framework for describing circuits. In this course, circuits are modelled to be the same as networks. Networks are composed of nodes, braches and loops. EENG223: CIRCUIT THEORY I Circuit Building Blocks: Nodes, Branches and Loops: A branch represents a single element such as voltage source, resistor or current source. How many branches? Branch: a single two-terminal circuit element. Wire segments are not counted as branches. Examples: voltage source/current source/resistors. There are 5 branches EENG223: CIRCUIT THEORY I Circuit Building Blocks: Nodes, Branches and Loops: A node is a point of connection between two or more branches. How many nodes? Node: a connection point between two or more branches. May include a portion of circuit (more than a single point). Essential Node: the point of connection between three or more brances. There are 3 nodes (a, b and c) 2 essential nodes (b and c) EENG223: CIRCUIT THEORY I Circuit Building Blocks: Nodes, Branches and Loops: A loop is a closed path in a circuit. How loops? Loop: a closed path in a circuit. Independent Loop: A loop is independent if it contains at least one branch which is not a part of any other independent loop. There are 6 loops. There are 3 independent loops. EENG223: CIRCUIT THEORY I Kirchhoffs Laws: Overview Ohms Law is not sufficient to analyze circuits alone. Kirchhoffs Laws help Ohms law to form the foundation for circuit analysis: The defining equations for circuit elements (Ohms Law). Kirchhoffs Current Law (KCL). Kirchhoffs Voltage Law (KVL). The defining equations (from Ohms law) tell us how the voltage and current within a circuit element are related. Kirchhoffs laws tell us how the voltages and currents in different branches are related. EENG223: CIRCUIT THEORY I Kirchhoffs Current Law Kirchhoffs current law (KCL) states that the algebraic sum of currents entering a node (or a closed boundary) is zero. The sum of currents entering a node is equal to the sum of the currents leaving the node. KCL also applies to a closed boundary: EENG223: CIRCUIT THEORY I Kirchhoffs Current Law Applying KCL to node a: 312321 0IIIIIIIITT321 IIIIT Equivalent circuit can be generated as follows: EENG223: CIRCUIT THEORY I Kirchhoffs Current Law: for Closed Boundaries 42314321 0IIIIIIII KCL applies to a closed boundaries: Closed boundary EENG223: CIRCUIT THEORY I Kirchhoffs Current Law: Example Apply KCL to the each essential node in the circuit. Essential node 1: Essential node 2: Essential node 3: i2 i1 i3 i4 231321 0 iiiiii mA50mA5 4343 iiiimA50mA5 142142 iiiiiiEENG223: CIRCUIT THEORY I Ideal Current Sources: Series Ideal currtent sources cannot be connected in series. Recall: ideal current sources guarantee the current flowing through source is at specified value. Recall: the current entering a circuit element must be equal to the current leaving the circuit element: Iin = Iout. Ideal current sources do not exist. Tecnically allowed if : I1 = I2 , but it is a bad idea. EENG223: CIRCUIT THEORY I Kirchhoffs Voltage Law: KVL Kirchhoffs voltage law (KVL) states that the algebraic sum of voltages around a closed path (or loop) is zero. sum of voltage drops = sum of voltage rises or EENG223: CIRCUIT THEORY I Kirchhoffs Voltage Law: KVL Apply KVL to each loop in the following circuit: Loop 1: Loop 2: Loop 3: Loop 4: Loop 5: Loop 6: EENG223: CIRCUIT THEORY I Kirchhoffs Voltage Law (KVL) Example: Calculate V2, V6 and VI. V50301010V30)106).(105(V10)102).(105(100103363326262IIIVVVVVVVVVEENG223: CIRCUIT THEORY I Applying Basic Laws: Example Calculate vo and i. V48A81620641241260642412000viiiiiivivi Apply KVL around the loop: EENG223: CIRCUIT THEORY I Applying Basic Laws: Example Calculate I2, I3, I7, V3, and VI . EENG223: CIRCUIT THEORY I Series Resistors and Voltage Division The equivalent resistance of any number of resistors connected in series is the sum of individual resistance. Consider the following circuit: Applying KVL 432143214321432111111)(GGGGGRRRRRRIRRRRIIRIRIRIRVeqeqeqsssssss (Resistors in series add) EENG223: CIRCUIT THEORY I Series Resistors and Voltage Division (Applying KVL) voltage accross each resistor, is proportional to its resistance. Larger the resistance, larger the voltage drop on that resistor: voltage division principle (voltage divider circuit) EENG223: CIRCUIT THEORY I Parallel Resistors and Current Division The equivalent resistance of N resistors number of resistors can be calculated by: Consider the following circuit: 43214321432143211111111111)1111(RRRRRRRRRRRVRRRRVIIIIIeqeqeqsssEENG223: CIRCUIT THEORY I Parallel Resistors and Current Division 4321432111111GGGGGRRRRReqeq Resistors in parallel have more complicated relationship It is easier to express in conductance EENG223: CIRCUIT THEORY I Parallel Resistors and Current Division (Applying KCL at node a) Given the total current i entering to node a the current is shared by the resistors by inverse proportion to their resistance: Current division principle (current divider circuit) EENG223: CIRCUIT THEORY I Resistor Networks and Equivalent Resistance Example: Calculate the Req for the following circuit. EENG223: CIRCUIT THEORY I Resistor Networks and Equivalent Resistance Example: Calculate the Req for the following circuit. EENG223: CIRCUIT THEORY I Resistor Networks and Equivalent Resistance Example: Calculate the Rab for the following circuit. EENG223: CIRCUIT THEORY I Resistor Networks and Equivalent Resistance Example: Calculate the Geq for the following circuit. EENG223: CIRCUIT THEORY I Resistor Networks: Wye-Delta Transformations There are cases where the resistors are neither in parallel nor in series. More tools are needed. EENG223: CIRCUIT THEORY I Resistor Networks: Wye-Delta Transformations Three-terminal equivalent networks can be used to simplify the analysis of the circuits. There are two types of three-terminal networks: Wye (Y) Networks Delta() Networks Two equivalent forms of (Y) Network Two equivalent forms of () Network EENG223: CIRCUIT THEORY I Resistor Networks: Wye-Delta Transformations Every Wye (Y) network is functionally equivalent to a Delta () network (vice versa). EENG223: CIRCUIT THEORY I Resistor Networks: Wye-Delta Transformations Example: Convert the following D network to a Y network. EENG223: CIRCUIT THEORY I Resistor Networks: Wye-Delta Transformations Example: Convert the following Ynetwork to a D network. EENG223: CIRCUIT THEORY I Resistor Networks: Wye-Delta Transformations Example: Calculate Req and Power delivered by the source. EENG223: CIRCUIT THEORY I Resistor Networks: Wye-Delta Transformations Example: Calculate Rab and and use it to calculate i.

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