J. McNames Portland State University ECE 221 Basic Laws Ver

Transcription

1 Basic Laws Overview Ideal sources: series & parallel Resistance & Ohm s Law Definitions: open circuit, short circuit, conductance Definitions: nodes, branches, & loops Kirchhoff s Laws Voltage dividers & series resistors Current dividers & parallel resistors WyeDelta Transformations J. McNames Portland State University ECE 221 Basic Laws Ver

2 Ideal Voltage Sources: Series v 1 = v 1 v 2 v 2 Ideal voltage sources connected in series add J. McNames Portland State University ECE 221 Basic Laws Ver

3 Ideal Voltage Sources: Parallel v 1 v 2 = Smoke Ideal voltage sources cannot be connected in parallel Recall: ideal voltage sources guarantee the voltage between two terminals is at the specified potential (voltage) Immovable object meets unstoppable force In practice, the stronger source would win Could easily cause component failure (smoke) Ideal sources do not exist Technically allowed if V 1 = V 2, but is a bad idea J. McNames Portland State University ECE 221 Basic Laws Ver

4 Ideal Current Sources: Series i 2 i 1 = Smoke Ideal current sources cannot be connected in series Recall: ideal current sources guarantee the current flowing through source is at specified value Recall: the current entering a circuit element must equal the current leaving a circuit element, I in = I out Could easily cause component failure (smoke) Ideal sources do not exist Technically allowed if I 1 = I 2, but is a bad idea J. McNames Portland State University ECE 221 Basic Laws Ver

5 Ideal Current Sources: Parallel i 1 i 2 = i 1 i 2 Ideal current sources in parallel add J. McNames Portland State University ECE 221 Basic Laws Ver

6 Resistance: Defined All materials resist the flow of current Resistance is usually represented by the variable R Depends on geometry and resistivity of the material A cylinder of length l and crosssectional area A has a resistance: R = ρ l A where R = resistance of an element in ohms (Ω) ρ = resistivity of the material in ohmmeters l = length of cylindrical material in meters A = Cross sectional area of material in meters 2 J. McNames Portland State University ECE 221 Basic Laws Ver

7 Resistance: Basic Concepts & Assumptions We will always measure resistance in Ohms Ohms are denoted by the greek letter Omega: Ω Examples: 50 Ω, 1 kω, 2.5 MΩ Conductors (e.g. wires) have very low resistance (< 0.1 Ω) that can usually be ignored (i.e. we will assume wires have zero resistance) Insulators (e.g. air) have very large resistance (> 50 MΩ) that can usually be ignored (omitted from circuit for analysis) Resistors have a medium range of resistance and must be accounted for in the circuit analysis Conceptually, a light bulb is similar to a resistor Properties of the bulb control how much current flows and how much power is dissipated (absorbed & emitted as light and heat) J. McNames Portland State University ECE 221 Basic Laws Ver

8 Ohm s Law v v i R v R Linear (Ohm's Law Applies) i Nonlinear (Ohm's Law Does Not Apply) As with all circuit elements, we need to know how the current through and voltage across the device are related Many materials have a complicated nonlinear relationship (including light bulbs): v = ±f(i) Materials with a linear relationship satisfy Ohm s law: v = ±mi The slope, m, is equal to the resistance of the element Ohm s Law: v = ±ir Sign, ±, is determined by the passive sign convention (PSC) i J. McNames Portland State University ECE 221 Basic Laws Ver

9 Resistors & Passive Sign Convention i i i i v R v R v R v R Recall that relationships between current and voltage are sign sensitive Passive Sign Convention: Current enters the positive terminal of an element If PSC satisfied: v = ir If PSC not satisfied: v = ir J. McNames Portland State University ECE 221 Basic Laws Ver

10 Other Equations Derived from Ohm s Law i R v Ohm s law implies: i = ± v R Recall p = ±vi. Therefore p = v v R = v2 R p = (ir)i = i 2 R Resistors cannot produce power Therefore, the power absorbed by a resistor will always be positive 1 Ω = 1 V/A J. McNames Portland State University ECE 221 Basic Laws Ver

11 Example 1: Ohm s Law i ma 3.38 ma 2 kω 6 kω 2 kω V v 6 v 2 10 V V 8 kω 5 ma i V R 4 i 2 = v 6 = R 4 = v 2 = i 8 = J. McNames Portland State University ECE 221 Basic Laws Ver

12 Short Circuit as Zero Resistance Circuit i V 0 Ω = Circuit i 0V An element (or wire) with R = 0 is called a short circuit Often just drawn as a wire (line) Could draw a resistor with R = 0, but is unnecessary and adds clutter J. McNames Portland State University ECE 221 Basic Laws Ver

13 Short Circuit as Voltage Source (0 V) Circuit i 0V = Circuit i 0V V s = Smoke An ideal voltage source V s = 0 V is also equivalent to a short circuit Since v = ir and R = 0, v = 0 regardless of i Could draw a source with V s = 0 V, but is not done in practice Cannot connect a voltage source to a short circuit Irresistible force meets immovable object In practice, the wire usually wins and the voltage source melts (if not protected) J. McNames Portland State University ECE 221 Basic Laws Ver

14 Open Circuit Circuit i V Ω = Circuit 0 A V An element with R = is called a open circuit Often just omitted Could draw a resistor with R =, but is unnecessary and would add clutter J. McNames Portland State University ECE 221 Basic Laws Ver

15 Open Circuit as Current Source (0 A) Circuit V 0 A = Circuit 0 A V I = Smoke An ideal current source I = 0 A is also equivalent to an open circuit Could draw a source with I = 0 A, but is not done in practice Cannot connect a current source to an open circuit Irresistible force meets immovable object In practice, you blow the current source (if not protected) The insulator (air) usually wins. Else, sparks fly. J. McNames Portland State University ECE 221 Basic Laws Ver

16 Conductance Sometimes conductance is specified instead of resistance Conductance is a measure of the ability of an element to conduct electric current Inverse of resistance G = 1 R = i v Units: siemens (S) or mhos ( ) 1 S = 1 = 1 A/V v = ±Ri i = ±Gv p = vi = i 2 R = v2 R p = vi = v 2 G = i2 G J. McNames Portland State University ECE 221 Basic Laws Ver

17 Circuit Building Blocks Before we can begin analysis, we need a common language and framework for describing circuits For this course, networks and circuits are the same Networks are composed of nodes, branches, and loops J. McNames Portland State University ECE 221 Basic Laws Ver

18 Branches Defined 2 kω 6 kω 2 kω 10 V 8 kω 5 ma 5 kω Example: How many branches? Branch: a single twoterminal element in a circuit Segments of wire are not counted as elements (or branches) Examples: voltage source, resistor, current source J. McNames Portland State University ECE 221 Basic Laws Ver

19 Nodes Defined 2 kω 6 kω 2 kω 10 V 8 kω 5 ma 5 kω Example: How many nodes? How many essential nodes? Node: the point of connection between two or more branches May include a portion of the circuit (more than a single point) Essential Node: the point of connection between three or more branches J. McNames Portland State University ECE 221 Basic Laws Ver

20 Loops Defined 2 kω 6 kω 2 kω 10 V 8 kω 5 ma 5 kω Example: How many loops? Loop: any closed path in a circuit J. McNames Portland State University ECE 221 Basic Laws Ver

21 Overview of Kirchhoff s Laws The foundation of circuit analysis is The defining equations for circuit elements (e.g. Ohm s law) Kirchhoff s current law (KCL) Kirchhoff s voltage law (KVL) The defining equations tell us how the voltage and current within a circuit element are related Kirchhoff s laws tell us how the voltages and currents in different branches are related They govern how elements within a circuit are related J. McNames Portland State University ECE 221 Basic Laws Ver

22 Kirchhoff s Current Law i 1 i 2 i3 i 1 i 2 i 3 i 4 i 5 = 0 i 1 i 2 i 5 = i 3 i 4 i 5 i 4 Kirchhoff s Current Law (KCL): the algebraic sum of currents entering a node (or a closed boundary) is zero The sum of currents entering a node is equal to the sum of the currents leaving a node Common sense: All of the electrons have to go somewhere The current that goes in, has to come out some place Based on law of conservation of charge J. McNames Portland State University ECE 221 Basic Laws Ver

23 Kirchhoff s Current Law for Boundaries 2 kω 6 kω 2 kω i 1 i 2 10 V 8 kω 5 ma 5 kω i 4 i 3 i 1 i 2 i 3 i 4 = 0 i 1 i 3 = i 2 i 4 KCL also applies to closed boundaries for all circuits J. McNames Portland State University ECE 221 Basic Laws Ver

24 Example 2: Kirchhoff s Current Law 2 kω 6 kω 3 kω i 1 i 6 i 3 10 V 8 kω 5 ma 5 kω i 8 Apply KCL to each essential node in the circuit. Essential Node 1: Essential Node 2: Essential Node 3: J. McNames Portland State University ECE 221 Basic Laws Ver

25 Kirchhoff s Voltage Law M m=1 V m = 0 Kirchhoff s Voltage Law (KVL): the algebraic sum of voltages around a closed path (or loop) is zero Based on the conservation of energy Analogous idea in hydraulic systems: sum of pressure drops and rises in any closed path must be equal J. McNames Portland State University ECE 221 Basic Laws Ver

26 Example 3: Kirchhoff s Voltage Law 2 kω 6 kω 3 kω v 2 v 6 v 3 10 V v 8 8 kω 5 ma V I v 4 5 kω Apply KVL to each loop in the circuit. Loop 1: Loop 2: Loop 3: Loop 4: Loop 5: Loop 6: J. McNames Portland State University ECE 221 Basic Laws Ver

27 Comments on Ohm s Law, KCL, and KVL Ohm s Law: KCL: KVL: v = ±ir In = 0 Vm = 0 Much of the circuit analysis that we will do is based on these three laws These laws alone are sufficient to analyze many circuits J. McNames Portland State University ECE 221 Basic Laws Ver

28 Ohm s Law for Fluids Ohm s law applies in fluid mechanics For turbulent flow, the pressure is related to the rate of flow squared not analogous For laminar flow, Q = πr4 P 8µL P = 8µL πr 4 Q where Q = flow rate (m 3 /s) r = pipe radius (m) L = pipe length (m) P = pressure drop (kn/m 2 ) µ = dynamic viscosity of fluid J. McNames Portland State University ECE 221 Basic Laws Ver

29 If we define R = 8µL πr 4, then Ohm s Law for Fluids Continued Q = P R P = R Q This is Ohm s law for laminar fluid flow in a pipe Kirchhoff s laws also apply to fluid networks Analogs Resistor Pipe Voltage source Pressure source Current source Flow rate source Capacitor Fluid capacitance (tanks) Inductor Fluid inductance (inertia) Transformers Fluid transformers (change in pipe diameter) But there are no fluid analogs to transistors or op amps J. McNames Portland State University ECE 221 Basic Laws Ver

30 Example 4: Applying the Basic Laws 2 kω 6 kω v 2 v 6 10 V 5 ma v I Find v 2, v 6 and v I. J. McNames Portland State University ECE 221 Basic Laws Ver

31 Example 5: Applying the Basic Laws 4 kω 2v o i o 12 V 4 V 6 kω v o Find i o and v o. J. McNames Portland State University ECE 221 Basic Laws Ver

32 Example 5: Workspace J. McNames Portland State University ECE 221 Basic Laws Ver

33 Example 6: Applying the Basic Laws 70 kω 20 kω 10 V i 7 v 3 30 kω i 3 i 2 5 ma v I Find i 7, i 3, i 2, v 3, and v I. J. McNames Portland State University ECE 221 Basic Laws Ver

34 Example 6: Workspace J. McNames Portland State University ECE 221 Basic Laws Ver