Basic Laws

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

• Wye-Delta Transformations

Portland State University ECE 221 Basic Laws Ver. 1.25 1

Ideal Voltage Sources: Series

v1

= v1+v2

v2

• Ideal voltage sources connected in series add


Ideal Voltage Sources: Parallel

=v1 v2Smoke

• Ideal voltage sources cannot be connected in parallel

• Recall: ideal voltage sources guarantee the voltage between twoterminals 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 V1 = V2, but is a bad idea


Ideal Current Sources: Series

= Smoke

i1i2

• Ideal current sources cannot be connected in series

• Recall: ideal current sources guarantee the current flowingthrough source is at specified value

• Recall: the current entering a circuit element must equal thecurrent leaving a circuit element, Iin = Iout

• Could easily cause component failure (smoke)

• Ideal sources do not exist

• Technically allowed if I1 = I2, but is a bad idea


Ideal Current Sources: Parallel

i1 i2 = i1 + i2

• Ideal current sources in parallel add


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 and cross-sectional area A has a resistance:

R = ρ

A

where

R = resistance of an element in ohms (Ω)ρ = resistivity of the material in ohm-meters = length of cylindrical material in metersA = Cross sectional area of material in meters2


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) thatcan usually be ignored (i.e. we will assume wires have zeroresistance)

• Insulators (e.g. air) have very large resistance (> 50 MΩ) thatcan usually be ignored (omitted from circuit for analysis)

• Resistors have a medium range of resistance and must beaccounted for in the circuit analysis

• Conceptually, a light bulb is similar to a resistor

• Properties of the bulb control how much current flows and howmuch power is dissipated (absorbed & emitted as light and heat)


Ohm’s Law

-v+

i Ri

v

i

v

Linear(Ohm's Law Applies)

Nonlinear(Ohm's Law Does Not Apply)

R

• As with all circuit elements, we need to know how the currentthrough 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)


Resistors & Passive Sign Convention

+

v

-

i

R

+

v

-

i

R

-

v

+

i

R

-

v

+

i

R

• Recall that relationships between current and voltage are signsensitive

• Passive Sign Convention: Current enters the positive terminal ofan element

– If PSC satisfied: v = iR

– If PSC not satisfied: v = −iR


Other Equations Derived from Ohm’s Law

-v+

i R

• Ohm’s law implies: i = ± vR

• Recall p = ±vi. Therefore

– p = v vR = v2

R

– p = (iR)i = i2R

• Resistors cannot produce power

– Therefore, the power absorbed by a resistor will always bepositive

• 1 Ω = 1 V/A


Example 1: Ohm’s Law

+0.5882 V-

R410 V 5 mA

-

10.59 V

+v6- +v2-

-

13.53 V

+ +

1.61 mA 3.38 mAi2

i8

2 kΩ2 kΩ 6 kΩ

8 kΩ

i2 =v6 =R4 =v2 =i8 =


Short Circuit as Zero Resistance

Circuit

-V

+i

Circuit-

0 V

+i

=0 Ω

• 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 addsclutter


Short Circuit as Voltage Source (0 V)

Circuit

i

=0 V Circuit-

0 V

+i

= SmokeVs

• An ideal voltage source Vs = 0 V is also equivalent to a shortcircuit

• Since v = iR and R = 0, v = 0 regardless of i

• Could draw a source with Vs = 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 (ifnot protected)


Open Circuit

Circuit

-V

+i

Circuit-V

+0 A

=∞ Ω

• An element with R = ∞ is called a open circuit

• Often just omitted

• Could draw a resistor with R = ∞, but is unnecessary and wouldadd clutter


Open Circuit as Current Source (0 A)

Circuit Circuit-V

+0 A

=0 A

-V

+I = Smoke

• An ideal current source I = 0 A is also equivalent to an opencircuit

• 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.


Conductance

• Sometimes conductance is specified instead of resistance

• Conductance is a measure of the ability of an element to conductelectric current

• Inverse of resistance

• G = 1R = i

v

• Units: siemens (S) or mhos ()

• 1 S = 1 = 1 A/V

v = ±Ri p = vi = i2R =v2

R

i = ±Gv p = vi = v2G =i2

G


Circuit Building Blocks

• Before we can begin analysis, we need a common language andframework for describing circuits

• For this course, networks and circuits are the same

• Networks are composed of nodes, branches, and loops


Branches Defined

10 V 5 mA

2 kΩ2 kΩ

5 kΩ

6 kΩ

8 kΩ

Example: How many branches?

• Branch: a single two-terminal element in a circuit

• Segments of wire are not counted as elements (or branches)

• Examples: voltage source, resistor, current source


Nodes Defined

10 V 5 mA

2 kΩ2 kΩ

5 kΩ

6 kΩ

8 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 morebranches


Loops Defined

10 V 5 mA

2 kΩ2 kΩ

5 kΩ

6 kΩ

8 kΩ

Example: How many loops?

• Loop: any closed path in a circuit


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 withina circuit element are related

• Kirchhoff’s laws tell us how the voltages and currents in differentbranches are related

• They govern how elements within a circuit are related


Kirchhoff’s Current Law

i1

i2

i3

i4i5

i1 + i2 − i3 − i4 + i5 = 0i1 + i2 + i5 = i3 + i4

• Kirchhoff’s Current Law (KCL): the algebraic sum of currentsentering a node (or a closed boundary) is zero

• The sum of currents entering a node is equal to the sum of thecurrents 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


Kirchhoff’s Current Law for Boundaries

10 V 5 mA

i1

i4

i2

i3

2 kΩ2 kΩ

5 kΩ

6 kΩ

8 kΩ

i1 − i2 + i3 − i4 = 0i1 + i3 = i2 + i4

• KCL also applies to closed boundaries for all circuits


Example 2: Kirchhoff’s Current Law

5 mA

i1 i3

10 V

i6

i8

2 kΩ 3 kΩ

5 kΩ

6 kΩ

8 kΩ

Apply KCL to each essential node in the circuit.

Essential Node 1:

Essential Node 2:

Essential Node 3:


Kirchhoff’s Voltage Law

M∑

m=1

Vm = 0

• Kirchhoff’s Voltage Law (KVL): the algebraic sum of voltagesaround a closed path (or loop) is zero

• Based on the conservation of energy

• Analogous idea in hydraulic systems: sum of pressure drops andrises in any closed path must be equal


Example 3: Kirchhoff’s Voltage Law

+-

10 V 5 mA-

+v6- +v3-

-

+ +v2

v8 v4

-

+VI

2 kΩ 3 kΩ

5 kΩ

6 kΩ

8 kΩ

Apply KVL to each loop in the circuit.

Loop 1:Loop 2:Loop 3:Loop 4:Loop 5:Loop 6:


Comments on Ohm’s Law, KCL, and KVL

Ohm’s Law: v = ±iRKCL:

∑In = 0

KVL:∑

Vm = 0

• Much of the circuit analysis that we will do is based on these threelaws

• These laws alone are sufficient to analyze many circuits


Ohm’s Law for Fluids

• Ohm’s law applies in fluid mechanics

• For turbulent flow, the pressure is related to the rate of flowsquared - not analogous

• For laminar flow,

Q =πr4∆P

8µL∆P =

8µL

πr4Q

where

– Q = flow rate (m3/s)

– r = pipe radius (m)

– L = pipe length (m)

– ∆P = pressure drop (kN/m2)

– µ = dynamic viscosity of fluid


Ohm’s Law for Fluids Continued

• If we define R = 8µLπr4 , then

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


Example 4: Applying the Basic Laws

10 V 5 mA

+- v2 +- v6

-

+vI

2 kΩ 6 kΩ

Find v2, v6 and vI .



2vo

12 V

io

4 V

+ -vo

4 kΩ

6 kΩ

Find io and vo.


Example 5: Workspace



5 mA

i7

10 Vi3

i2

-

+v

I

-

+v3

20 kΩ

30 kΩ

70 kΩ

Find i7, i3, i2, v3, and vI .


Example 6: Workspace


Basic Laws

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