Unit2.0 Circuit Element Theory Dr. Anurag Srivastava Associate Professor ABVIIITM, Gwalior Circuit Theory Overview Of Circuit Theory; Lumped Circuit Elements; Topology Of Circuits; Resistors; KCL and KVL; Resistors in Series and Parallel; Energy Storage Elements; FirstOrder Circuits 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 firstorder circuits. Anurag Srivastava
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. 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. A distributed circuit is one where the terminal voltages and currents are functions of position as well as time. Transmission lines are distributed circuit elements. 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. Anurag Srivastava 2
Overview of Circuit Theory Current is moving positive electrical charge. Measured in Amperes (A) = 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). 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): J of energy is needed to move C of charge through a V potential difference. 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 upper case symbols V and I are usually used to denote voltages and currents that are DC or AC steadystate voltages and currents. Anurag Srivastava 3
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. 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: ( ) ( ) ( ) VI p t = v t i t or P = Sign convention: positive power means that energy is being absorbed; negative power means that power is being supplied. Overview of Circuit Theory Rest of circuit 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. Circuit element under consideration Anurag Srivastava 4
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. 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. Independent and Dependent Sources An independent source (voltage or current) may be DC (constant) or timevarying; 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. Anurag Srivastava 5
Independent Sources v s i s Voltage Source Current Source Dependent Sources v=f(v x ) v=f(i x ) Voltage Controlled Voltage Source (VCVS) Current Controlled Voltage Source (CCVS) Dependent Sources I=f(V x ) I=f(I x ) Voltage Controlled Current Source (VCCS) Current Controlled Current Source (CCCS) Anurag Srivastava 6
Passive Lumped Circuit Elements Resistors R Capacitors C Inductors L 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. 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. Anurag Srivastava 7
Topology of Circuits 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. 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. 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. Anurag Srivastava 8
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. 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. Topology of Circuits N N2 N3 N4 M M2 N0 In this example there are 5 nodes and 2 meshes. In addition to the meshes, there is one additional loop (following the outer perimeter of the circuit). Anurag Srivastava 9
Resistors A resistor is a circuit element that dissipates electrical energy (usually as heat). Realworld 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. Resistors The Rest of the Circuit i(t) R v(t) Ri v = Resistance is measured in Ohms (Ω) 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 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. Anurag Srivastava 0
KCL and KVL KCL 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. i (t) i 5 (t) i 2 (t) i 4 (t) i 3 (t) n j= i j = 0 KCL and KVL KVL The sum of voltages around any loop in a circuit is zero. n j= v j = 0 v (t) v 2 (t) v 3 (t) In KVL: KCL and 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) Anurag Srivastava
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. Resistors in Series Two elements are in series if the current that flows through one must also flow through the other. R R 2 Series Resistors in Series Consider two resistors in series with a voltage v(t) across them: i(t) R v(t) R 2 v (t) v 2 (t) Voltage division: v v = v( t) 2 = v( t) R R R 2 R2 R R 2 Anurag Srivastava 2
Resistors in Series If we wish to replace the two series resistors with a single equivalent resistor whose voltagecurrent relationship is the same, the equivalent resistor has a value given by R eq = R R 2 Resistors in Series For N resistors in series, the equivalent resistor has a value given by R R 2 R eq R 3 R = R R R L eq 2 3 R N 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. Anurag Srivastava 3
Resistors in Parallel Consider two resistors in parallel with a voltage v(t) across them: i(t) v(t) i (t) i 2 (t) R R 2 Current division: i i = i( t) 2 = i( t) R2 R R R R R 2 2 Resistors in Parallel If we wish to replace the two parallel resistors with a single equivalent resistor whose voltagecurrent relationship is the same, the equivalent resistor has a value given by R RR 2 = eq R R 2 Resistors in Parallel For N resistors in parallel, the equivalent resistor has a value given by R R 2 R 3 R eq R eq = R R 2 L R R 3 N Anurag Srivastava 4
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. 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. Energy Storage Elements Electrical engineers (and their software tools) usually do not solve the differential equations directly. Instead, they use: LaPlace transforms AC steadystate analysis These techniques covert the solution of differential equations into algebraic problems. Anurag Srivastava 5
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. Capacitors Inventor: Edwald George Von Kleirt 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. q t = C v t ( ) ( ) The proportionality constant C is called capacitance. Capacitance is measured in Farads (F). Capacitors The rest of the circuit i(t) v(t) t i v = ( x) dx C v t) = v( t ( 0 dv( t) i = C dt ) C t t0 i( x) dx Anurag Srivastava 6
Capacitance the measure of the ability of a capacitor to store charge Voltage Equation for a Capacitor v=q/c C = Capacitance in Farads (F) q = Charge on one plate in Coulombs (C) v = Voltage across the capacitor in Volts (V) Example: What is the charge on a 200µF capacitor with 00 Volts across its terminals? Calculus of Capacitors q = Cv i = dq dt i ( t ) = C dv( t dt ) Anurag Srivastava 7
Calculus: Derivatives (slopes) d dt ( t n ) = nt n d dt ( e at ) = ae at d (sin( ω t )) = ω cos( ωt ) dt d (cos( ωt )) = ω sin( ωt ) dt Capacitors The voltage across a capacitor cannot change instantaneously. The energy stored in the capacitors is given by w ( ) Cv 2 C t = 2 Anurag Srivastava 8
Capacitance Equation ε o C A r ε = d ε o = permittivity of air, 8.85 x 0 2 F/m ε r = relative permittivity of the dielectric (Table 3.) A = plate area in square meters (m 2 ) d = distance between plates in meters (m) Anurag Srivastava 9
Capacitors Capacitors Types of Capacitors Ceramic capacitors have a ceramic dielectric. Film and paper capacitors are named for their dielectrics. Aluminum, tantalum and niobium electrolytic capacitors are named after the material used as the anode and the construction of the cathode (electrolyte) Polymer capacitors are aluminum,tantalum or niobium electrolytic capacitors with conductive polymer as electrolyte Supercapacitor is the family name for: Doublelayer capacitors were named for the physical phenomenon of the Helmholtz doublelayer Pseudocapacitors were named for their ability to store electric energy electrochemically with reversible faradaic chargetransfer Hybrid capacitors combine doublelayer and pseudocapacitors to increase power density Silver mica, glass, silicon, airgap and vacuum capacitors are named for their dielectric. Anurag Srivastava 20
The most common dielectrics are: Ceramics Plastic films Oxide layer on metal (Aluminum, Tantalum, Niobium) Natural materials like mica, glass, paper, air, vacuum All of them store their electrical charge statically within an electric field between two (parallel) electrodes. Beneath this conventional capacitors a family of electrochemical capacitors called Supercapacitors was developed. Supercapacitors don't have a conventional dielectric. They store their electrical charge statically in Helmholtz doublelayers and faradaically at the surface of electrodes with static Doublelayer capacitance in a doublelayer capacitor and with pseudocapacitance (faradaic charge transfer) in a Pseudocapacitor or with both storage principles together in hybrid capacitors. Anurag Srivastava 2
Capacitors in Series Draw three capacitors in series with a battery. What is the same for every capacitor in a series? Answer: Charge This leads to the Total Capacitance Equation: C eq = C C 2 C 3 Anurag Srivastava 22
Capacitors in Parallel Draw three capacitors in parallel with a battery. What is the same for every capacitor in parallel? Answer: Voltage This leads to the Total Capacitance Equation: C = C C C eq 2 3 Anurag Srivastava 23
Example Problems What is the total capacitance for a µf, a 2 µf and a 5µF capacitor is series?...in parallel? 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). Inductance Inductor a circuit component that has two terminals connected to a coil of wire Inductors are also called: Solenoids Coils Electromagnets 3D Drawing Circuit Symbols Anurag Srivastava 24
Air Core Inductor Types of Inductor Radio Frequency Inductor Ferromagnetic core inductor Laminated core inductor Ferritecore inductor Toroidal core inductor Choke Variable inductor Anurag Srivastava 25
Air Core Inductor The term air core coil describes an inductor that does not use a magnetic core made of a ferromagnetic material. The term refers to coils wound on plastic, ceramic, or other nonmagnetic forms, as well as those that have only air inside the windings. Air core coils have lower inductance than ferromagnetic core coils, but are often used at high frequencies because they are free from energy losses called core losses that occur in ferromagnetic cores, which increase with frequency. Radio frequency inductor At high frequencies, particularly radio frequencies (RF), inductors have higher resistance and other losses. In addition to causing power loss, in resonant circuits this can reduce the Q factor of the circuit, broadening the bandwidth. In RF inductors, which are mostly air core types, specialized construction techniques are used to minimize these losses. The losses are due to various surface and other effects: Skin effect, Proximity effect Dielectric losses, Parasitic capacitance Skin effect: The resistance of a wire to high frequency current is higher than its resistance to direct current because of skin effect. Radio frequency alternating current does not penetrate far into the body of a conductor but travels along its surface. Therefore, in a solid wire, most of the cross sectional area of the wire is not used to conduct the current, which is in a narrow annulus on the surface. This effect increases the resistance of the wire in the coil, which may already have a relatively high resistance due to its length and small diameter. Anurag Srivastava 26
Proximity effect: Another similar effect that also increases the resistance of the wire at high frequencies is proximity effect, which occurs in parallel wires that lie close to each other. The individual magnetic field of adjacent turns induces eddy currents in the wire of the coil, which causes the current in the conductor to be concentrated in a thin strip on the side near the adjacent wire. Like skin effect, this reduces the effective crosssectional area of the wire conducting current, increasing its resistance. Dielectric losses: The high frequency electric field near the conductors in a tank coil can cause the motion of polar molecules in nearby insulating materials, dissipating energy as heat. So coils used for tuned circuits are often not wound on coil forms but are suspended in air, supported by narrow plastic or ceramic strips. Parasitic capacitance: The capacitance between individual wire turns of the coil, called parasitic capacitance, does not cause energy losses but can change the behavior of the coil. Each turn of the coil is at a slightly different potential, so the electric field between neighboring turns stores charge on the wire, so the coil acts as if it has a capacitor in parallel with it. At a high enough frequency this capacitance can resonate with the inductance of the coil forming a tuned circuit, causing the coil to become selfresonant. Anurag Srivastava 27
Q factor An ideal inductor would have no resistance or energy losses. However, real inductors have winding resistance from the metal wire forming the coils. Since the winding resistance appears as a resistance in series with the inductor, it is often called the series resistance. The inductor's series resistance converts electric current through the coils into heat, thus causing a loss of inductive quality. Q factor The quality factor (or Q) of an inductor is the ratio (Q=ωL/R) of its inductive reactance to its resistance at a given frequency, and is a measure of its efficiency. The higher the Q factor of the inductor, the closer it approaches the behavior of an ideal, lossless, inductor. High Q inductors are used with capacitors to make resonant circuits in radio transmitters and receivers. The higher the Q is, the narrower the bandwidth of the resonant circuit. Notice that Q increases linearly with frequency if L and R are constant. Although they are constant at low frequencies, the parameters vary with frequency. For example, skin effect, proximity effect, and core losses increase R with frequency; winding capacitance and variations in permeability with frequency affect L. Anurag Srivastava 28
Ferromagneticcore or ironcore inductors Ferromagneticcore or ironcore inductors use a magnetic core made of ferromagnetic or ferrimagnetic material such as iron or ferrite to increase the inductance. A magnetic core can increase the inductance of a coil by a factor of several thousand, by increasing the magnetic field due to its higher magnetic permeability. However the magnetic properties of the core material cause several side effects which alter the behavior of the inductor and require special construction: Laminated core inductor: Lowfrequency inductors are often made with laminated cores to prevent eddy currents, using construction similar to transformers. The core is made of stacks of thin steel sheets or laminations oriented parallel to the field, with an insulating coating on the surface. The insulation prevents eddy currents between the sheets, so any remaining currents must be within the cross sectional area of the individual laminations, reducing the area of the loop and thus reducing the energy losses greatly. The laminations are made of lowcoercivity silicon steel, to reduce hysteresis losses. Anurag Srivastava 29
Energy Stored on a Inductor Inductors also do not dissipate energy like resistors. They store energy in the form of a magnetic field. w ( t ) = 2 2 Li Anurag Srivastava 30
Inductors in Series What is the same for every inductor in a series? Answer: Current This leads to the Total Inductance Equation: L = L L L eq 2 3 Inductors in Parallel What is the same for every inductor in parallel? Answer: Voltage This leads to the Total Inductance Equation: L eq = L L 2 L 3 Anurag Srivastava 3
Inductors The rest of the circuit i(t) L v(t) di( t) v = L dt t i = v( x) dx L i t) = i( t ( 0 ) L t t0 v( x) dx Anurag Srivastava 32
Inductors The current through an inductor cannot change instantaneously. The energy stored in the inductor is given by w ( ) Li 2 L t = 2 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. st order circuits the time constant. 2nd order circuits the natural frequency and the damping ratio. 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. Anurag Srivastava 33
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 n n d v( t) d v( t) a... a0v( t) f n n = n dt dt as well as initial conditions derived from the capacitor voltages and inductor currents at t = 0. Analysis of Circuits Containing Energy Storage Elements The solution to any differential equation consists of two parts: v(t) = v p (t) v c (t) Particular (forced) solution is v p (t) Response particular to the source Complementary/homogeneous (natural) solution is v c (t) Response common to all sources Analysis of Circuits Containing Energy Storage Elements The particular solution v p (t) is typically a weighted sum of f(t) and its first n derivatives. If f(t) is constant, then v p (t) is constant. If f(t) is sinusoidal, then v p (t) is sinusoidal. Anurag Srivastava 34
Analysis of Circuits Containing Energy Storage Elements The complementary solution is the solution to n d v( t) n dt n d v( t)... a0v( t) = 0 n dt an The complementary solution has the form v = c n i= K e sit i Analysis of Circuits Containing Energy Storage Elements s through s n are the roots of the characteristic equation n s a n n s... as a0 = 0 Analysis of Circuits Containing Energy Storage Elements If s i is a real root, it corresponds to a s t decaying exponential term K e i, s < 0 If s i 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 σ it ( A cos ω t B sin ω t) i d i i d i Anurag Srivastava 35
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. 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. FirstOrder Circuits Any circuit with a single energy storage element, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of st order. Any voltage or current in such a circuit is the solution to a st order differential equation. Anurag Srivastava 36
FirstOrder Circuits Examples of st 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. FirstOrder Circuits Examples of st order circuits (Cont d): The RC lowpass filter for an envelope detector in a superheterodyne AM receiver. Sampleandhold 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 st order circuit. FirstOrder Circuits v R (t) v S (t) R C v C (t) st Order Circuit: One capacitor and one resistor The source and resistor may be equivalent to a circuit with many resistors and sources. Anurag Srivastava 37
FirstOrder Circuits v R (t) v S (t) R i(t) C v C (t) Let s derive the ( st order) differential equation for the mesh current i(t). FirstOrder Circuits KVL around the loop: We have v v v = v v S R R C = Ri = vc ( 0) i( x) C C t 0 dx FirstOrder Circuits The KVL equation becomes: Ri vc ( 0) i( x) dx v C = S 0 Differentiating both sides w.r.t. t, we have or di R dt i C di i dt RC t dvs = dt dvs = R dt Anurag Srivastava 38
FirstOrder Circuits i L (t) i R (t) i S (t) R L v(t) st Order Circuit: One inductor and one resistor The source and resistor may be equivalent to a circuit with many resistors and sources. FirstOrder Circuits i L (t) i R (t) i S (t) R L v(t) Let s derive the ( st order) differential equation for the node voltage v(t). FirstOrder Circuits KCL at the top node: We have i i L i R = i i S R v = R = il( 0) v( x) L L t 0 dx Anurag Srivastava 39
FirstOrder Circuits The KVL equation becomes: v t il ( 0) v( x) dx i R L = S 0 Differentiating both sides w.r.t. t, we have or ( ) dv R dt dv dt v L R v L t dis = dt dis = R dt FirstOrder Circuits For all st order circuits, the diff. eq. can be written as dv v = f dt τ The complementary solution is given by v C τ = Ke t where K is evaluated from the initial conditions. FirstOrder Circuits The time constant of the complementary response is τ. For an RC circuit, τ = RC For an RL circuit, τ = L/R τ is the amount of time necessary for an exponential to decay to 36.7% of its initial value. Anurag Srivastava 40
FirstOrder Circuits The particular solution v p (t) is usually a weighted sum of f(t) and its first derivative. If f(t) is constant, then v p (t) is constant. If f(t) is sinusoidal, then v p (t) is sinusoidal. Anurag Srivastava 4