Nodal Analysis and Circuit Simplification using Series and Parallel Combinations, Study notes of Circuit Theory

Download Nodal Analysis and Circuit Simplification using Series and Parallel Combinations and more Study notes Circuit Theory in PDF only on Docsity! M. Horowitz, J. Plummer, R. Howe 1 E40M Solving Circuits using Nodal Analysis, Part II and EveryCircuitTM M. Horowitz, J. Plummer, R. Howe 2 The Key Idea from Last Lecture Systematic Nodal Analysis 1. Label all the nodes (VA, VB, or V1, V2, etc.), after selecting the node you choose to be Gnd. 2. Label all the branch currents (i1, i2, etc.) and choose directions for each of them 3. Write the KCL equations for every node except the reference (Gnd) • Sum of the device currents at each node must be zero 4. Substitute the equations for each device’s current as a function of the node voltages, when possible 5. Solve the resulting set of equations M. Horowitz, J. Plummer, R. Howe 5 Parallel Combinations • Two resistors in parallel – The total current through parallel resistors is the sum of the currents through the two resistors – The voltage across each resistor is the same … they “share a voltage” – So the effective resistance of parallel resistors is: 1/R = 1/R1 + 1/R2 R = (R1·R2) / (R1+R2) R1 R2 M. Horowitz, J. Plummer, R. Howe 6 Using Series and Parallel Combinations to Simplify Circuits Example: Find the resistance between node a and node b M. Horowitz, J. Plummer, R. Howe 7 But What About This Circuit? • R3 and R4 are in series – But I need to find the voltage at the node I will eliminate – “collapse and then expand” • First eliminate the node to simplify the circuit M. Horowitz, J. Plummer, R. Howe 10 Voltage Divider • First simplify circuit to a single resistor and find the current I • Then use the current to find the voltage Va +- aV Va = R1 R2 I I = M. Horowitz, J. Plummer, R. Howe 11 Current Divider • In this case simplify the circuit to a single resistor, then find voltage across each resistor and use it to find the current through each resistor I R1 R2 I1 I2 a M. Horowitz, J. Plummer, R. Howe 12 Intuition on Dividers • Voltage divider: R2 = 10 R1 … what is Va = VR2 in terms of V? • Current divider: R2 = 10 R1 … what is I2 in terms of I? M. Horowitz, J. Plummer, R. Howe 15 Superposition For Linear Circuits • Reason: – Resistors, voltage, and current sources are linear – Resulting equations are linear • What’s the benefit? – Superposition enables the analysis of several simpler circuits in place of one complicated circuit M. Horowitz, J. Plummer, R. Howe 16 Applying Superposition • Calculate the response of the circuit for each independent source at a time, with the other’s turned off • What happens when we turn off a source? – Voltage sources: have 0 V (are shorted … replace by a wire) – Current sources: have 0 current (are opened … replace by a broken wire) VI + - VI + - VI + -= +X X open-circuited so I = 0 short-circuited so V = 0 M. Horowitz, J. Plummer, R. Howe 17 • We need to “zero-out” sources into order to find the sub-circuits (one per source) • Find the current I Applying Superposition F. T. Ulaby and M. M. Maharbiz, Circuits, NSTP, 2009, p. 97. M. Horowitz, J. Plummer, R. Howe 20 Circuit Debugging • For future labs you will be building more complex circuits – You will build these circuits using breadboards • These circuits will contain many different components – Including transistors with three connections • Sometimes these circuits won’t work the way you expect – Perhaps your circuit is wrong – Or perhaps you just connected it up wrong – How do you debug it in either case? M. Horowitz, J. Plummer, R. Howe 21 Circuit Simulator • We create a program to estimate how our circuit will behave • The program shows the wiring in a nice way – and makes it easy to probe the voltage and current – It has built-in voltage and current meters • It also makes it easy to change component values – So you can tune/play with your circuit • You are going to use an easy-to-use simulator: EveryCircuit M. Horowitz, J. Plummer, R. Howe 22 Every Circuit http://everycircuit.com/app/ M. Horowitz, J. Plummer, R. Howe 25 Every Circuit’s Keyboard Shortcuts • R : Rotate selected device • F : Flip selected device • A : Adjust parameter of a selected device • T : Toggle selected switch • W : Add / remove voltage of selected node or current of selected device to / from oscilloscope • S : Adjust simulation speed • Esc : deselect all • Arrows : move selected component or workspace • Plus / Minus : zoom in / out • Space : start or pause simulation • Delete : delete selected device or cut selected wire • Ctrl + Z : Undo • Ctrl + Y : Redo M. Horowitz, J. Plummer, R. Howe 26 Activate Your License • http://everycircuit.com/licensekeyactivation 259825287547 (Good during spring quarter) M. Horowitz, J. Plummer, R. Howe 27 Learning Objectives • Understand how to solve for device voltage and currents – First label node voltages (KVL) – Solve current equations at each node (KCL) – Called nodal analysis • Be able to break a large circuit into smaller circuits – This is standard “divide and conquer” approach • Recognize some common circuit patterns – Which reduce the complexity of the circuit you need to solve – Start with series and parallel resistors • Superposition is a powerful tool for handling multiple sources • EveryCircuit can solve your circuits, so you can be sure your homework and prelab answers are correct!