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EEE 498/598 Overview of Electrical Engineering Lecture 12: Overview Of Circuit Theory; Lumped Circuit Elements; Topology Of Circuits; Resistors; KCL and KVL; Resistors in Series and Parallel; Energy Storage Elements; First-Order Circuits 1 Lecture 12 Objectives To commence our study of circuit theory. To develop an understanding of the concepts of Lumped circuit elements; topology of circuits; resistors; KCL and KVL; resistors in series and parallel; energy storage elements; and first-order circuits. Lecture 12 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. An electrical circuit is a connection of circuit elements into one or more closed loops. Lecture 12 Overview of Circuit Theory A lumped circuit is one where all the terminal voltages and currents are functions of time only. Lumped circuit elements include resistors, capacitors, inductors, independent and dependent sources. An distributed circuit is one where the terminal voltages and currents are functions of position as well as time. Transmission lines are distributed circuit elements. Lecture 12 Overview of Circuit Theory Basic quantities are voltage, current, and power. 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. Lecture 12 Overview of Circuit Theory Current is moving positive electrical charge. Measured in Amperes (A) = 1 Coulomb/s Current is represented by I or i. In general, current can be an arbitrary function of time. 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). Lecture 12 Overview of Circuit Theory 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. Voltage is represented by V or v. Lecture 12 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 upper case symbols V and I are usually used to denote voltages and currents that are DC or AC steady-state voltages and currents. Lecture 12 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. 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. Lecture 12 Overview of Circuit Theory Power is the rate at which energy is being absorbed or supplied. Power is computed as the product of voltage and current: pt vt it or P VI Sign convention: positive power means that energy is being absorbed; negative power means that power is being supplied. Lecture 12 Overview of Circuit Theory • If p(t) > 0, then the circuit element is absorbing power i(t) from the rest of the circuit. Rest of • If p(t) < 0, then the circuit + circuit element is supplying power v(t) to the rest of the circuit. - Circuit element under consideration Lecture 12 Overview of Circuit Theory 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. Lecture 12 Active and Passive Elements Active elements can generate energy. Examples of active elements are independent and dependent sources. Passive elements cannot generate energy. 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. Lecture 12 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. A dependent source has a value that depends on another voltage or current in the circuit. Lecture 12 Independent Sources vs t is t Voltage Source Current Source Lecture 12 Dependent Sources v=f(vx) v=f(ix) + + - - Voltage Current Controlled Controlled Voltage Source Voltage Source (VCVS) (CCVS) Lecture 12 Dependent Sources I=f(Vx) I=f(Ix) Voltage Current Controlled Controlled Current Source Current Source (VCCS) (CCCS) Lecture 12 Passive Lumped Circuit Elements Resistors R Capacitors C Inductors L Lecture 12 Topology of Circuits 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 conductors are perfect. Lecture 12 Topology of Circuits 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. We can rearrange the schematic and have the same circuit as long as the connections between elements remain the same. Lecture 12 Topology of Circuits Example: Schematic of a circuit: “Ground”: a reference point where the voltage (or potential) is assumed to be zero. Lecture 12 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. This circuit element can be removed without affecting functionality. This circuit behaves identically to the previous one. Lecture 12 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. 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. Lecture 12 Topology of Circuits 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. A mesh is a loop that has no other loops inside of it. Lecture 12 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. The same is true if we know every mesh current. Lecture 12 Topology of Circuits N1 N2 N3 N4 • In this example there are 5 nodes and 2 meshes. • In addition to the M1 M2 meshes, there is one additional loop N0 (following the outer perimeter of the circuit). Lecture 12 Resistors 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 Parasitic resistances: many resistors on circuit diagrams model unwanted resistances in transistors, motors, etc. Lecture 12 Resistors i(t) vt Ri t The + Rest of R v(t) the Circuit - Resistance is measured in Ohms (W) 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 Lecture 12 KCL and KVL 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. KVL is the basis of mesh analysis – in which the unknowns are the currents flowing in each of the meshes of the circuit. Lecture 12 KCL and KVL KCL i1(t) i5(t) The sum of all currents i2(t) i4(t) entering a node is zero, or i3(t) The sum of currents n entering node is equal to sum of currents i (t ) 0 j 1 j leaving node. Lecture 12 KCL and KVL KVL The sum of voltages n around any loop in a v j 1 j (t ) 0 circuit is zero. - + v2(t) - + v1(t) v3(t) + - Lecture 12 KCL and KVL In KVL: A voltage encountered + to - is positive. A voltage encountered - to + is negative. Arrows are sometimes used to represent voltage differences; they point from low to high voltage. + v(t) ≡ v(t) - Lecture 12 Resistors in Series 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. Lecture 12 Resistors in Series Two elements are in series if the current that flows through one must also flow through the other. Series R1 R2 Lecture 12 Resistors in Series Consider two resistors in series with a voltage v(t) across them: Voltage division: i(t) R1 + + v1 (t ) v(t ) R1 v1(t) R1 R2 - v(t) R2 + v2 (t ) v(t ) R2 v2(t) R1 R2 - - Lecture 12 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 Req R1 R2 Lecture 12 Resistors in Series For N resistors in series, the equivalent resistor has a value given by R1 R2 Req R3 Req R1 R2 R3 RN Lecture 12 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. Lecture 12 Resistors in Parallel Consider two resistors in parallel with a voltage v(t) across them: Current division: i(t) R2 + i1(t) i2(t) i1 (t ) i (t ) R1 R2 v(t) R1 R2 R1 i2 (t ) i (t ) R1 R2 - Lecture 12 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 R1 R2 Req R1 R2 Lecture 12 Resistors in Parallel For N resistors in parallel, the equivalent resistor has a value given by R1 R2 R3 Req 1 Req 1 1 1 1 R1 R2 R3 RN Lecture 12 Energy Storage Elements Capacitors store energy in an electric field. Inductors store energy in a magnetic field. Capacitors and inductors are passive elements: Can store energy supplied by circuit Can return stored energy to circuit Cannot supply more energy to circuit than is stored. Lecture 12 Energy Storage Elements 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. Lecture 12 Energy Storage Elements Electrical engineers (and their software tools) usually do not solve the differential equations directly. Instead, they use: LaPlace transforms AC steady-state analysis These techniques covert the solution of differential equations into algebraic problems. Lecture 12 Energy Storage Elements Energy storage elements model electrical loads: Capacitors model computers and other electronics (power supplies). Inductors model motors. Capacitors and inductors are used to build filters and amplifiers with desired frequency responses. Capacitors are used in A/D converters to hold a sampled signal until it can be converted into bits. Lecture 12 Capacitors Capacitance occurs when two conductors are separated by a dielectric (insulator). Charge on the two conductors creates an electric field that stores energy. The voltage difference between the two conductors is proportional to the charge. qt C vt The proportionality constant C is called capacitance. Capacitance is measured in Farads (F). Lecture 12 Capacitors The + dv(t ) rest i(t) of v(t) i (t ) C the dt - circuit t 1 v(t ) i ( x)dx C t 1 v(t ) v(t 0 ) i ( x)dx C t0 Lecture 12 Capacitors The voltage across a capacitor cannot change instantaneously. The energy stored in the capacitors is given by 1 2 wC (t ) Cv (t ) 2 Lecture 12 Inductors 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 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. Inductance is measured in Henrys (H). Lecture 12 Inductors The + di(t ) rest i(t) v(t ) L of L v(t) dt the - circuit t 1 i (t ) v( x)dx L t 1 i (t ) i (t 0 ) v( x)dx L t0 Lecture 12 Inductors The current through an inductor cannot change instantaneously. The energy stored in the inductor is given by 1 2 wL (t ) Li (t ) 2 Lecture 12 Analysis of Circuits Containing Energy Storage Elements Need to determine: The order of the circuit. Forced (particular) and natural (complementary/homogeneous) responses. Transient and steady state responses. 1st order circuits - the time constant. 2nd order circuits - the natural frequency and the damping ratio. Lecture 12 Analysis of Circuits Containing Energy Storage Elements The number and configuration of the energy storage elements determines the order of the circuit. n # of energy storage elements Every voltage and current is the solution to a differential equation. In a circuit of order n, these differential equations are linear constant coefficient and have order n. Lecture 12 Analysis of Circuits Containing Energy Storage Elements Any voltage or current in an nth order circuit is the solution to a differential equation of the form d n v(t ) d n1v(t ) n an1 n 1 ... a0v(t ) f (t ) dt dt as well as initial conditions derived from the capacitor voltages and inductor currents at t = 0-. Lecture 12 Analysis of Circuits Containing Energy Storage Elements The solution to any differential equation consists of two parts: v(t) = vp(t) + vc(t) Particular (forced) solution is vp(t) Response particular to the source Complementary/homogeneous (natural) solution is vc(t) Response common to all sources Lecture 12 Analysis of Circuits Containing Energy Storage Elements The particular solution vp(t) is typically a weighted sum of f(t) and its first n derivatives. If f(t) is constant, then vp(t) is constant. If f(t) is sinusoidal, then vp(t) is sinusoidal. Lecture 12 Analysis of Circuits Containing Energy Storage Elements The complementary solution is the solution to n n 1 d v(t ) d v(t ) n an 1 n 1 ... a0v(t ) 0 dt dt The complementary solution has the form n vc (t ) K i e sit i 1 Lecture 12 Analysis of Circuits Containing Energy Storage Elements s1through sn are the roots of the characteristic equation n 1 s an1s n ... a1s a0 0 Lecture 12 Analysis of Circuits Containing Energy Storage Elements If si is a real root, it corresponds to a decaying exponential term Ki e s t , si 0 i If si is a complex root, there is another complex root that is its complex conjugate, and together they correspond to an exponentially decaying sinusoidal term e it Ai cos d t Bi sin d t Lecture 12 Analysis of Circuits Containing Energy Storage Elements The steady state (SS) response of a circuit is the waveform after a long time has passed. DC SS if response approaches a constant. AC SS if response approaches a sinusoid. The transient response is the circuit response minus the steady state response. Lecture 12 Analysis of Circuits Containing Energy Storage Elements Transients usually are associated with the complementary solution. The actual form of transients usually depends on initial capacitor voltages and inductor currents. Steady state responses usually are associated with the particular solution. Lecture 12 First-Order Circuits Any circuit with a single energy storage element, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of 1st order. Any voltage or current in such a circuit is the solution to a 1st order differential equation. Lecture 12 First-Order Circuits Examples of 1st order circuits: Computer RAM A dynamic RAM stores ones as charge on a capacitor. The charge leaks out through transistors modeled by large resistances. The charge must be periodically refreshed. Lecture 12 First-Order Circuits Examples of 1st order circuits (Cont’d): The RC low-pass filter for an envelope detector in a superheterodyne AM receiver. Sample-and-hold circuit: The capacitor is charged to the voltage of a waveform to be sampled. The capacitor holds this voltage until an A/D converter can convert it to bits. The windings in an electric motor or generator can be modeled as an RL 1st order circuit. Lecture 12 First-Order Circuits vR(t) - + R + + vS(t) C vC(t) - - 1st Order Circuit: One capacitor and one resistor The source and resistor may be equivalent to a circuit with many resistors and sources. Lecture 12 First-Order Circuits vR(t) + - R + + vS(t) C vC(t) - - i(t) Let’s derive the (1st order) differential equation for the mesh current i(t). Lecture 12 First-Order Circuits KVL around the loop: vS t vR t vC t We have vR t Ri t t vC t vC 0 ix dx 1 C0 Lecture 12 First-Order Circuits The KVL equation becomes: t Ri t vC 0 ix dx vS t 1 C0 Differentiating both sides w.r.t. t, we have dit 1 dvS t R it dt C dt or dit 1 1 dvS t it dt RC R dt Lecture 12 First-Order Circuits iL(t) iR(t) + iS(t) R L v(t) - 1st Order Circuit: One inductor and one resistor The source and resistor may be equivalent to a circuit with many resistors and sources. Lecture 12 First-Order Circuits iL(t) iR(t) + iS(t) R L v(t) - Let’s derive the (1st order) differential equation for the node voltage v(t). Lecture 12 First-Order Circuits KCL at the top node: iS t iR t iL t We have vt iR t R t iL t iL 0 v x dx 1 L0 Lecture 12 First-Order Circuits The KVL equation becomes: vt t iL 0 vx dx iS t 1 R L0 Differentiating both sides w.r.t. t, we have 1 dvt 1 diS t vt R dt L dt or dvt R diS t vt R dt L dt Lecture 12 First-Order Circuits For all 1st order circuits, the diff. eq. can be written as dvt 1 vt f t dt The complementary solution is given by vC t Ke t where K is evaluated from the initial conditions. Lecture 12 First-Order Circuits The time constant of the complementary response is . For an RC circuit, = RC For an RL circuit, = L/R is the amount of time necessary for an exponential to decay to 36.7% of its initial value. Lecture 12 First-Order Circuits The particular solution vp(t) is usually a weighted sum of f(t) and its first derivative. If f(t) is constant, then vp(t) is constant. If f(t) is sinusoidal, then vp(t) is sinusoidal. Lecture 12