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& Thev Nort Solving Techniques For any Linear Resistive Network there exists: Thevenin Equiv. Network Norton Equivalent Network VOC - open-circuit voltage appearing across the terminals of the network, and RTH - Thevenin equivalent resistance when all independent sources are deactivated. ISC current through load when replacing load with a short circuit Equations Solve for RTH 1. Deactivate all ind. srcs (leave dep. srcs) 2. Remove the load RL in the circuit and look at resistance seen through nodes A & B *In circuits w. dependent sources, may need to apply a voltage or current source between A & B, then apply the definition RTH = VOC / ISC Finding VTH 1. Remove load RL in original circuit & relate voltage VAB to rest of network. a. Sources remain unchanged. b. Use methods such as superposition, mesh, nodal, V-Division, etc Finding ISC 1. Replace load RL in original circuit with a short circuit. a. Sources remain unchanged. Coefficient Approach V-I Char. Of Thev: VAB = RTHIA + VOC V-I Char. Of Nort: IA = (1/RTH)VAB ISC 1. Obtain an equation in one of the above forms allows us to match the coefficients of the above equations to determine RTH, VOC, IA, or GTH. Measured Data 1. Substitute the given data into V-I Char. Eqs: Thev: VAB = RTHIA + VOC or Nort: IA = (1/RTH)VAB - ISC 2. Put into matrix form 3. Solve for remaining variables. General Solving Procedures 1. Have ways to find VOC, ISC, RTH directly using above techniques 2. After solving for 2 of the 3, below are listed ways to solve for 3rd Thev and Nort equiv w/ Dep. Srcs & w/out Ind. Srcs *Extra condition for constructing Thev and Nort equivalents for active networks all controlling voltages or currents must be within the 2-terminal network whose Thev/Nort equiv are being sought. 1. Since there are no ind. internal srcs, the Thev equiv. consists of a single resistance, R TH (VOC = ISC = 0) 2. Write an equation(s) to relate the terms in the 2-terminal network via previous solving techniques (KCL, KVL, Mesh, Nodal, etc) 3. If one equation, match coeficients with the V-I char. equations of a Thev. Or Nort. Equiv. network. 4. If set of equations, solve using matrix operations. Maximum Power Transfer Theorems, Requirements, Conditions, Assumptions, and Definitions 1. Always put circuit in Thevenin or Norton Equivalent form 2. Fixed RTH Equations Always True Under M.P.T. Inductors Theorems, Requirements, Conditions, Assumptions, and Definitions 1. Inductor device whose voltage is proportional to the time rate of change of its current with a constant of proportionality (L), called Inductance. a. Inductance measures the magnitude of the voltage induced by a change in the current through an inductor Symbol: Units: Henry (H) 2. Continuity Property of Inductors If the voltage VL(t) across an inductor is bounded over the time interval t1 < t0 < t2, then iL(t0-) = iL(t0+), even when VL(t0-) VL(t0+) Physics Explanation / Proof 1. A time varying current flow through a wire creates a time varying magnetic field around the wire a. The magnetic flux density, B, in a coil of wire is strongest inside the coil and directed along the axis of the coil 3. The magnetic field in turn sets up a time varying electric field (i.e. electric potential, or voltage). 4. A changing current causes a change in the storage of energy in the magnetic field surrounding the inductor. 5. The energy transferred to the mag. field requires work, and, hence, power. 6. Power is the product of voltage and current which implies that there is an induced voltage between ends of the wire. *If a second wire is immersed in the changing magnetic field of the first wire, a voltage will be* induced between ends of the second wire. Equations Power & Energy - Inductors 1. Energy is the integral of the instantaneous power over a given time interval 2. When current (iL) waveform is bounded, net energy stored in inductor over interval [t1, t0] depends on IL(t0) & IL(t1). a. If current waveform is periodic for some constant T > 0 Over any time interval of length T Net stored energy in inductor = 0 Finding Instantaneous Energy Stored in Inductor 1. Solve for IL(t) through Inductor for each interval specified and/or for each unique interval 2. Apply Energy Stored in Mag. Field over interval equation resulting in integral values of current 3. Keep solutions in terms of t, specifying any variations for different intervals. Inductor Combinations Series Inductors add in series to form equivalent inductance Parallel Inductors in parallel behave like resistors in parallel. Capacitors Theorems, Requirements, Conditions, Assumptions, and Definitions 1. Capacitor An energy storage device composed of two metal plates separated by a dielectric (an insulating material) whose current is proportional to the time rate of change of its voltage a. Capacitance a measure of the capacitors potential to store energy in an electric field. b. Capacitance measures the devices ability to produce a current from changes in the voltage across it. Symbol: Units: Farads ( F) 2. Continuity Property of Capacitors If the current IC(t) through a capacitor is bounded over the time interval t1 < t0 < t2, then VL(t0-) = VL(t0+), even when IL(t0-) IL(t0+) a. Exception - When two charged capacitors or one charged an one uncharged capacitor are instantaneously connected in parallel b. - Exception When capacitors and some indep. V-Srcs form a loop. 3. Principle of Conservation of Charge q transferred into / out of a junction = 0 a. Direct consequence of KCL which relates Voltages, Capacitances, and Charge Transport b. Applies / useful for switches opening & closing / circuit changes between time intervals Physics Explanation / Proof 1. Placing a voltage across plates of the capacitor will cause positive charge to accumulate on the top plate and an equal amount of negative charge on the bottom plate 2. This charge accumulation generates an electric field between the plates that stores energy Equations Solving for VC(t) Given a Periodic Current Src 1. When working with periodic currents (repeats itself), must work on a segment-bysegment basis. 2. Since is periodic, each segment will be a shifted version of the first segment. 3. Right shifting is achieved by replacing (t) with (t-2). Solving Instantaneous || Connection of 2 Charged or 1 Charged / 1 Uncharged Capacitors 1. KVL take precedence 2. Applying KVL forces an instantaneous equality in the capacitor voltages, which are subject to the principle of conservation of charge. Relationship of Charge & Capacitor Voltage / Current 1. Integral of IC(t) over interval [t1, t0] represents amount of charge passing through the top wire over [t1, t0]. 2. By KCL, if IC(t) flows into the top plate, then -IC(t) must flow into the bottom plate, which causes a charge of q(t) to be deposited on the bottom plate. 3. Pos. & Neg. charges on plates, separated by the diaelectric, produce a voltage drop VC(t) from the top plate to the bottom plate. Using Conservation of Charge with Capacitors 1. Given capacitor voltages at time t(0-), find capacitor voltages after switch has been closed for time t > 0 2. Due to conservation of charge, the charge stored on capacitors at t(0-) = charged stored on capacitors at t > 0 3. For t > 0 KVL requires that VLoop = 0 4. If capacitors hold unequal amounts of charge at time t(0-), some charge must be transferred between capacitors to equalize the voltages 5. q(t(0-)) = q(t(0+)) CVC(t(0-)) = CVC(t(0+)) Physical Interpretation of the Integral Relation 1. VC(t) Capacitor Voltage at time t0 2. - Represents additional charge transferred to the capacitor during the interval [t0, t] 3. - Additional voltage attained by the capacitor during the interval [t0, t] Energy Storage in Capacitors 1. Energy stored or utilized in a capacitor is the integral of the power absorbed by the capacitor. 2. Change in energy stored in capacitor over interval [t0, t1] depends only on the values of the capacitor voltage at times t0 & t1 3. Change in stored energy is independent of the voltage waveform between t0 & t1 4. If voltage waveform is periodic for some constant T > 0 Over any time interval of length T Net stored energy in capacitor = 0 5. Instantaneous stored energy in capacitor integral of power over the interval (-, t]. a. Assuming all voltages & currents = 0 @ t = - Capacitor Combinations Series Capacitors in series behave like resistors in parallel. Parallel Capacitors in parallel behave like resistors in series. Source Free / Zero Input Response 1st Order Circuits Theorems, Requirements, Conditions, Assumptions, and Definitions 1. 1st Order Differential Equation Only presence of the first order derivative of some function x(t) appears 2. Source Free 1st Order DE No sources. 3. *In Source Free 1st Order DEs, one assumes the presence of an initial inductor current or initial* capacitor voltage. 4. 1st Order RL & RC Circuits circuits with only one resistor and either one inductor (RL) or one capacitor (RC). 5. For DC Sources - 1st Order RL & RC Circuits have voltages and currents of the form: 6. Must use Thevenin Equivalent in circuits for value of R in . 7. Time Constant () time it takes a source free system to decay by e-1 = .37 (37%) 8. Solution to 1st Order DE is a waveform satisfying eq : 9. The above eq works for all continuous and piecewise continuous time functions f(t) Unit Step Function (u(t)) Source Free / Zero Input Response 1st Order Capacitor-Resistor (RC) Circuits Theorems, Requirements, Conditions, Assumptions, and Definitions 1. In RC circuit driven by an independent source for a long time Capacitor behaves as an open. How to Derive DE model for RC Circuits 1. Apply KVL, which implies 2. However 3. Substitute and Multiply by RC yields the DE model: 4. is given initial condition Source Free / Zero Input Response 1st Order Inductor-Resistor (RL) Circuits Theorems, Requirements, Conditions, Assumptions, and Definitions 1. In RL circuit driven by an independent source for a long time Inductor behaves as a short. How to Derive DE model for RL Circuits 1. At top of node apply KCL, which implies 2. However 3. Substitute and Multiply by R/L yields the DE model: 4. is given initial condition DC / Step Response 1st Order Inductor-Resistor (RL) Circuits How to Derive DE model for Step Response RL Circuits 1. Circuit Model for an Inductor: 2. By KVL and Ohms: r 3. Substitute for leads to the DE Model 4. is given initial condition DC / Step Response 1st Order Capacitor-Resistor (RC) Circuits How to Derive DE model for RC Circuits 1. Apply KVL, which implies 2. However 3. Substitute and Multiply by RC yields the DE model: 4. is given initial condition ... View Full Document

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