ARRDEKTA INSTITUTE OF TECHNOLOGY GUIDED BY. GUIDED BY. Prof.Y.B.Vaghela. Prof.Y.B.Vaghela. Asst.prof in electrical Asst.prof in electrical Department Department PREPARED BY. PREPARED BY. GandhiChandani ( ) GandhiChandani ( ) Joshi Ishani Joshi Ishani ( ) ( ) Patel Devangi ( ) Patel Devangi ( ) Rathwa Vaishali ( ) Rathwa Vaishali ( )

Overview of Circuit Theory Electrical circuit elements are idealized models of physical devices that are defined by relationships between their terminal voltages and currents. Circuit elements can have two or more terminals. Electrical circuit elements are idealized models of physical devices that are defined by relationships between their terminal voltages and currents. Circuit elements can have two or more terminals. An electrical circuit is a connection of circuit elements into one or more closed loops. An electrical circuit is a connection of circuit elements into one or more closed loops.

Overview of Circuit Theory Basic quantities are voltage, current, and power. Basic quantities are voltage, current, and power. The sign convention is important in computing power supplied by or absorbed by a circuit element. The sign convention is important in computing power supplied by or absorbed by a circuit element. Circuit elements can be active or passive; active elements are sources. Circuit elements can be active or passive; active elements are sources.

Overview of Circuit Theory Current is moving positive electrical charge. Current is moving positive electrical charge. Measured in Amperes (A) = 1 Coulomb/s Measured in Amperes (A) = 1 Coulomb/s Current is represented by I or i. Current is represented by I or i. In general, current can be an arbitrary function of time. In general, current can be an arbitrary function of time. Constant current is called direct current (DC). Constant current is called direct current (DC). Current that can be represented as a sinusoidal function of time (or in some contexts a sum of sinusoids) is called alternating current (AC). Current that can be represented as a sinusoidal function of time (or in some contexts a sum of sinusoids) is called alternating current (AC).

Overview of Circuit Theory Voltage is electromotive force provided by a source or a potential difference between two points in a circuit. Voltage is electromotive force provided by a source or a potential difference between two points in a circuit. Measured in Volts (V): 1 J of energy is needed to move 1 C of charge through a 1 V potential difference. Measured in Volts (V): 1 J of energy is needed to move 1 C of charge through a 1 V potential difference. Voltage is represented by V or v. Voltage is represented by V or v.

Overview of Circuit Theory The lower case symbols v and i are usually used to denote voltages and currents that are functions of time. The lower case symbols v and i are usually used to denote voltages and currents that are functions of time. The upper case symbols V and I are usually used to denote voltages and currents that are DC or AC steady-state voltages and currents. The upper case symbols V and I are usually used to denote voltages and currents that are DC or AC steady-state voltages and currents.

Overview of Circuit Theory Current has an assumed direction of flow; currents in the direction of assumed current flow have positive values; currents in the opposite direction have negative values. Current has an assumed direction of flow; currents in the direction of assumed current flow have positive values; currents in the opposite direction have negative values. Voltage has an assumed polarity; volt drops in with the assumed polarity have positive values; volt drops of the opposite polarity have negative values. Voltage has an assumed polarity; volt drops in with the assumed polarity have positive values; volt drops of the opposite polarity have negative values. In circuit analysis the assumed polarity of voltages are often defined by the direction of assumed current flow. In circuit analysis the assumed polarity of voltages are often defined by the direction of assumed current flow.

Overview of Circuit Theory Power is the rate at which energy is being absorbed or supplied. Power is the rate at which energy is being absorbed or supplied. Power is computed as the product of voltage and current: Power is computed as the product of voltage and current: Sign convention: positive power means that energy is being absorbed ; negative power means that power is being supplied. Sign convention: positive power means that energy is being absorbed ; negative power means that power is being supplied.

Overview of Circuit Theory + - v(t) i(t) If p(t) > 0, then the circuit element is absorbing power from the rest of the circuit. If p(t) < 0, then the circuit element is supplying power to the rest of the circuit. Rest of circuit Circuit element under consideration

Overview of Circuit Theory If power is positive into a circuit element, it means that the circuit element is absorbing power. If power is positive into a circuit element, it means that the circuit element is absorbing power. If power is negative into a circuit element, it means that the circuit element is supplying power. Only active elements (sources) can supply power to the rest of a circuit. If power is negative into a circuit element, it means that the circuit element is supplying power. Only active elements (sources) can supply power to the rest of a circuit.

Active and Passive Elements Active elements can generate energy. Active elements can generate energy. Examples of active elements are independent and dependent sources. Examples of active elements are independent and dependent sources. Passive elements cannot generate energy. Passive elements cannot generate energy. Examples of passive elements are resistors, capacitors, and inductors. Examples of passive elements are resistors, capacitors, and inductors. In a particular circuit, there can be active elements that absorb power – for example, a battery being charged. In a particular circuit, there can be active elements that absorb power – for example, a battery being charged.

Independent and Dependent Sources An independent source (voltage or current) may be DC (constant) or time- varying; its value does not depend on other voltages or currents in the circuit. An independent source (voltage or current) may be DC (constant) or time- varying; its value does not depend on other voltages or currents in the circuit. A dependent source has a value that depends on another voltage or current in the circuit. A dependent source has a value that depends on another voltage or current in the circuit.

Independent Sources Voltage SourceCurrent Source

Dependent Sources + - v=f(v x ) Voltage Controlled Voltage Source (VCVS) + - v=f(i x ) Current Controlled Voltage Source (CCVS)

Dependent Sources I=f(V x ) Voltage Controlled Current Source (VCCS) I=f(I x ) Current Controlled Current Source (CCCS)

Passive Lumped Circuit Elements Resistors Resistors Capacitors Capacitors Inductors Inductors

Topology of Circuits A lumped circuit is composed of lumped elements (sources, resistors, capacitors, inductors) and conductors (wires). A lumped circuit is composed of lumped elements (sources, resistors, capacitors, inductors) and conductors (wires). All the elements are assumed to be lumped, i.e., the entire circuit is of negligible dimensions. All the elements are assumed to be lumped, i.e., the entire circuit is of negligible dimensions. All conductors are perfect. All conductors are perfect.

Topology of Circuits A schematic diagram is an electrical representation of a circuit. A schematic diagram is an electrical representation of a circuit. The location of a circuit element in a schematic may have no relationship to its physical location. The location of a circuit element in a schematic may have no relationship to its physical location. We can rearrange the schematic and have the same circuit as long as the connections between elements remain the same. We can rearrange the schematic and have the same circuit as long as the connections between elements remain the same.

Topology of Circuits Example: Schematic of a circuit: Example: Schematic of a circuit: “Ground”: a reference point where the voltage (or potential) is assumed to be zero.

Topology of Circuits Only circuit elements that are in closed loops (i.e., where a current path exists) contribute to the functionality of a circuit. Only circuit elements that are in closed loops (i.e., where a current path exists) contribute to the functionality of a circuit. This circuit element can be removed without affecting functionality. This circuit behaves identically to the previous one.

Topology of Circuits A node is an equipotential point in a circuit. It is a topological concept – in other words, even if the circuit elements change values, the node remains an equipotential point. A node is an equipotential point in a circuit. It is a topological concept – in other words, even if the circuit elements change values, the node remains an equipotential point. To find a node, start at a point in the circuit. From this point, everywhere you can travel by moving only along perfect conductors is part of a single node. To find a node, start at a point in the circuit. From this point, everywhere you can travel by moving only along perfect conductors is part of a single node.

Topology of Circuits A loop is any closed path through a circuit in which no node is encountered more than once. A loop is any closed path through a circuit in which no node is encountered more than once. To find a loop, start at a node in the circuit. From this node, travel along a path back to the same node ensuring that you do not encounter any node more than once. To find a loop, start at a node in the circuit. From this node, travel along a path back to the same node ensuring that you do not encounter any node more than once. A mesh is a loop that has no other loops inside of it. A mesh is a loop that has no other loops inside of it.

Topology of Circuits If we know the voltage at every node of a circuit relative to a reference node ( ground ), then we know everything about the circuit – i.e., we can determine any other voltage or current in the circuit. If we know the voltage at every node of a circuit relative to a reference node ( ground ), then we know everything about the circuit – i.e., we can determine any other voltage or current in the circuit. The same is true if we know every mesh current. The same is true if we know every mesh current.

Resistors A resistor is a circuit element that dissipates electrical energy (usually as heat). A resistor is a circuit element that dissipates electrical energy (usually as heat). Real-world devices that are modeled by resistors: incandescent light bulb, heating elements (stoves, heaters, etc.), long wires Real-world devices that are modeled by resistors: incandescent light bulb, heating elements (stoves, heaters, etc.), long wires Parasitic resistances: many resistors on circuit diagrams model unwanted resistances in transistors, motors, etc. Parasitic resistances: many resistors on circuit diagrams model unwanted resistances in transistors, motors, etc.

Resistors Resistance is measured in Ohms (  ) Resistance is measured in Ohms (  ) The relationship between terminal voltage and current is governed by Ohm’s law The relationship between terminal voltage and current is governed by Ohm’s law Ohm’s law tells us that the volt drop in the direction of assumed current flow is Ri Ohm’s law tells us that the volt drop in the direction of assumed current flow is Ri The Rest of the Circuit Rv(t)v(t) i(t)i(t) + -

KCL and KVL Kirchhoff’s Current Law ( KCL ) and Kirchhoff’s Voltage Law ( KVL ) are the fundamental laws of circuit analysis. Kirchhoff’s Current Law ( KCL ) and Kirchhoff’s Voltage Law ( KVL ) are the fundamental laws of circuit analysis. KCL is the basis of nodal analysis – in which the unknowns are the voltages at each of the nodes of the circuit. KCL is the basis of nodal analysis – in which the unknowns are the voltages at each of the nodes of the circuit. KVL is the basis of mesh analysis – in which the unknowns are the currents flowing in each of the meshes of the circuit. KVL is the basis of mesh analysis – in which the unknowns are the currents flowing in each of the meshes of the circuit.

KCL and KVL KCL KCL The sum of all currents entering a node is zero, or The sum of all currents entering a node is zero, or The sum of currents entering node is equal to sum of currents leaving node. The sum of currents entering node is equal to sum of currents leaving node. i 1 (t) i 2 (t)i 4 (t) i 5 (t) i 3 (t)

KCL and KVL KVL KVL The sum of voltages around any loop in a circuit is zero. The sum of voltages around any loop in a circuit is zero. + - v1(t)v1(t) v2(t)v2(t) v3(t)v3(t)

KCL and KVL In KVL: In KVL: A voltage encountered + to - is positive. A voltage encountered + to - is positive. A voltage encountered - to + is negative. A voltage encountered - to + is negative. Arrows are sometimes used to represent voltage differences; they point from low to high voltage. Arrows are sometimes used to represent voltage differences; they point from low to high voltage. + - v(t)v(t) v(t)v(t) ≡

Resistors in Series A single loop circuit is one which has only a single loop. A single loop circuit is one which has only a single loop. The same current flows through each element of the circuit - the elements are in series. The same current flows through each element of the circuit - the elements are in series.

Resistors in Series Two elements are in series if the current that flows through one must also flow through the other. R1R1 R2R2 Series

Resistors in Series If we wish to replace the two series resistors with a single equivalent resistor whose voltage-current relationship is the same, the equivalent resistor has a value given by If we wish to replace the two series resistors with a single equivalent resistor whose voltage-current relationship is the same, the equivalent resistor has a value given by

Resistors in Series For N resistors in series, the equivalent resistor has a value given by For N resistors in series, the equivalent resistor has a value given by R1R1 R3R3 R2R2 R eq

Resistors in Parallel When the terminals of two or more circuit elements are connected to the same two nodes, the circuit elements are said to be in parallel. When the terminals of two or more circuit elements are connected to the same two nodes, the circuit elements are said to be in parallel.

Resistors in Parallel If we wish to replace the two parallel resistors with a single equivalent resistor whose voltage-current relationship is the same, the equivalent resistor has a value given by If we wish to replace the two parallel resistors with a single equivalent resistor whose voltage-current relationship is the same, the equivalent resistor has a value given by

Resistors in Parallel For N resistors in parallel, the equivalent resistor has a value given by For N resistors in parallel, the equivalent resistor has a value given by R eq R3R3 R2R2 R1R1

Energy Storage Elements Capacitors store energy in an electric field. Capacitors store energy in an electric field. Inductors store energy in a magnetic field. Inductors store energy in a magnetic field. Capacitors and inductors are passive elements: Capacitors and inductors are passive elements: Can store energy supplied by circuit Can store energy supplied by circuit Can return stored energy to circuit Can return stored energy to circuit Cannot supply more energy to circuit than is stored. Cannot supply more energy to circuit than is stored.

Energy Storage Elements Voltages and currents in a circuit without energy storage elements are solutions to algebraic equations. Voltages and currents in a circuit without energy storage elements are solutions to algebraic equations. Voltages and currents in a circuit with energy storage elements are solutions to linear, constant coefficient differential equations. Voltages and currents in a circuit with energy storage elements are solutions to linear, constant coefficient differential equations.

Capacitors Capacitance occurs when two conductors are separated by a dielectric (insulator). Capacitance occurs when two conductors are separated by a dielectric (insulator). Charge on the two conductors creates an electric field that stores energy. Charge on the two conductors creates an electric field that stores energy. The voltage difference between the two conductors is proportional to the charge. The voltage difference between the two conductors is proportional to the charge. The proportionality constant C is called capacitance. The proportionality constant C is called capacitance. Capacitance is measured in Farads (F). Capacitance is measured in Farads (F).

Capacitors The voltage across a capacitor cannot change instantaneously. The voltage across a capacitor cannot change instantaneously. The energy stored in the capacitors is given by The energy stored in the capacitors is given by

Inductors Inductance occurs when current flows through a (real) conductor. Inductance occurs when current flows through a (real) conductor. The current flowing through the conductor sets up a magnetic field that is proportional to the current. The current flowing through the conductor sets up a magnetic field that is proportional to the current. The voltage difference across the conductor is proportional to the rate of change of the magnetic flux. The voltage difference across the conductor is proportional to the rate of change of the magnetic flux. The proportionality constant is called the inductance, denoted L. The proportionality constant is called the inductance, denoted L. Inductance is measured in Henrys (H). Inductance is measured in Henrys (H).