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Circuit Analysis - Basic Concepts -
Engineering 43 Circuit Analysis - Basic Concepts - Bruce Mayer, PE Licensed Electrical & Mechanical Engineer
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Circuit Analysis DEVELOP TOOLS FOR THE ANALYSIS AND DESIGN OF BASIC ELECTRIC CIRCUITS
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BASIC ANALYSIS STRATEGY
Note: Some Devices are NONlinear
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Potential↔Flow Circuit
A “Circuit” describes the Relationship between Indestructible-Entity (Mass, Heat, Charge) Movement or FLOW Energy-Producing POTENTIAL (Gravity, Pressure, Voltage, Temperature)
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Electro-Fluidic Analogy
Analogous Fluidic and Electrical Circuits Examine Individual Circuit Elements Fluidic Electrical
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Potential Source Voltage is the Electrical PRESSURE
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Flow (Mass Movement) Source
Current is the Electrical MASS-FLOW
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Conductors (move mass)
Perfect Conductors transport “Flow” with NO LOSS of Potential
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Resistor (hinders flow)
Resistors CAUSE a Loss of Potential with Flow thru Them
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Flow On/Off Control Valves & Switches Turn Flow On/Off
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Capacitor Accumulates Flow
Capacitors Produce a Flow-Change when Pressure Changes
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Inductor Resists FLOW Chg
Inductors Produce a Pressure-Change when Flow Changes
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Another Circuit Analogy
Resistors in “Series Note that “Flow” moves from HI-potential to LO-potential in BOTH Cases
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Electric Circuit – What is it?
ELECTRIC CIRCUIT IS AN INTERCONNECTION OF ELECTRICAL COMPONENTS a b 2 TERMINAL COMPONENT characterized by the current through it and the voltage difference between terminals NODE TYPICAL LINEAR CIRCUIT VI relationship can take linear, Differential, or integral forms
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Basic Concepts Learning Goals
System Of Units: The SI Standard System; Prefixes Basic Electrical Quantities: Charge, Current, Voltage, Power, And Energy Circuit Elements: Active and Passive Passive elements typically have TWO connections: INput & OUTput, and can only DISSIPATE POWER Active elements have at LEAST 3 conntions: Input, OUTput, and CONTROL
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International System of Units (SI)
SI base units The SI system of Units Is Founded On Units For Seven Base Quantities Assumed To Be Mutually Independent, As Given Below SI Base Units Base quantity Name Symbol length meter m mass kilogram kg time second s electric current ampere A thermodynamic temperature kelvin K amount of substance mole mol luminous intensity candela cd Only One “Electrical” Unit
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SI DERIVED ELECTRICAL UNITS
Other quantities, called derived quantities, are defined in terms of the seven base quantities via a system of quantity equations. The SI units for these derived quantities are obtained from these equations and the 7 SI base units Derived quantity Name Symbol Expression in terms of other SI units Expression in terms of SI base units Frequency hertz Hz - s-1 energy, work, quantity of heat joule J N·m m2·kg·s-2 power, radiant flux watt W J/s m2·kg·s-3 electric charge, quantity of electricity coulomb C s·A electric potential difference, electromotive force volt V W/A m2·kg·s-3·A-1 capacitance farad F C/V m-2·kg-1·s4·A2 electric resistance ohm Ω V/A m2·kg·s-3·A-2 electric conductance siemens S A/V m-2·kg-1·s3·A2 inductance henry H Wb/A m2·kg·s-2·A-2
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SI prefixes Factor Name Symbol 1024 yotta Y 10-1 Deci d 1021 zetta Z
10-2 Centi c 1018 exa E 10-3 milli m 1015 peta P 10-6 micro 1012 tera T 10-9 nano n 109 giga G 10-12 pico p 106 mega M 10-15 femto f 103 kilo k 10-18 atto a 102 hecto h 10-21 zepto z 101 deka da 10-24 yocto y
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Derived Units One Ampere Of Current Carries One Coulomb Of Charge Every Second 1 Coulomb = 6.242x1018 e- e- The Charge On An Electron A Volt Is A Measure Of Energy Per Charge (Electrical Potential) Two Points Have A Potential Difference Of One Volt If One Coulomb Of Charge Gains One Joule Of Energy When It Is Moved From One Point To The Other.
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Derived Units cont. The Ohm Is A Measure Of The Resistance To The Flow Of Charge There Is One Ohm Of Resistance If It Requires One Volt Of Electrical-Pressure To Drive Through One Ampere Of Current One Watt Of Power Is Required To Drive One Ampere Of Current Against An Electromotive Potential Difference Of One Volt
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Current And Voltage Ranges
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Current = Charge-Flow Strictly Speaking Current (i) is a Basic Quantity And Charge (q) Is Derived However, Physically The Electric Current is Created By Charged-Particle Movement An Electric Circuit Acts a Pipeline That Moves Charge from One Pt to Another The Time-Rate-of-Change for Charge Constitutes an Electric Current; Mathematically
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Water-Flow Analogy Water Particles in a Pipeline Move
From HIGH-Pressure to LOW-Pressure In Current-Flow Electrical Charges move From HIGH-Potential (Voltage) to LOW-Potential Direction Convention Almost all Solid-State Current Flow is Transported by ELECTRONS, Which Move from LOW Potential to HIGH
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Fluid↔Current Flow Analogs
Think of the Voltage as the Electrical “Pressure” Current as the Electrical Fluid Wire as the Electrical “Pipe”
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Current & Electron Flow
These Alternatives Constitute Conventional POSITIVE Current Flow POSITIVE Charges Moving: Hi-v → Lo-v NEGATIVE Charges Moving: Lo-v → Hi-v + q(t) - Vhi Vlo
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Current/Charge Calculation
If The Charge is Given, Determine The Current By Differentiation If The Current Is Known, Determine The Charge By Integration EXAMPLE + q(t) Vhi Vlo
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Find The Charge That Passes During In The Interval 0<t<1s
Another Example Find The Charge That Passes During In The Interval 0<t<1s By MATLAB >> t=[0:0.001:1]; % in Sec >> i_of_t = exp(-2*t); % in mA >> q = trapz(t, i_of_t) % in mCoul q = 0.4323 >> syms x t >> q_of_t = int(exp(-2*x), 0, t) q_of_t = -1/2*exp(-2*t)+1/2 Units? Find The Charge As A Function Of Time And the units for the charge?...
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Graphical Example Zero currents DETERMINE THE CURRENT
Here we are given the charge flow as a function of time. To determine current we must take derivatives. PAY ATTENTION TO UNITS Zero currents Graphical Example
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Convention For Currents
It Is Absolutely Necessary To Indicate The Direction Of Movement Of Charged Particles The Universally Accepted Convention In Electrical Engineering is That Current is the Flow Of POSITIVE Charges We Indicate Our ASSUMED Direction Of Flow For Positive Charges THE REFERENCE DIRECTION A Positive Value For The Current Indicates Flow In The Direction Of The Arrow (The Reference Direction) A Negative Value For The Current Indicates Flow In The OPPOSITE Direction Than The Reference Direction
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Double Index Notation If The Initial And Terminal Node Are Labeled One Can Indicate Them As SubIndices For The Current Name POSITIVE CHARGES FLOW LEFT-RIGHT POSITIVE CHARGES FLOW RIGHT-LEFT
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Example This Example Illustrates The Various Ways In Which The Current Notation Can Be Used To Find Iab Assuming that Charge is CONSERVED More on this later 3A 7A out = 7A in • Draw arrows for Iab, label I with -2A
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Voltage Conventions One Definition For the Volt unit
Two Points Have A Voltage Differential Of One Volt If One Coulomb Of Charge Gains (Or Loses) One Joule Of Energy When It Moves From One Point To The Other If The Charge GAINS Energy Moving From a To b, Then b Has HIGHER Voltage Than a If It Loses Energy Then b Has LOWER Voltage Than a
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Voltage Conventions cont.
Dimensionally, the Volt is a Derived Unit Voltage Is Always Measured In a RELATIVE Form As The Potential DIFFERENCE Between Two Points It Is Essential That Our Notation Allows Us To Determine Which Point Has The Higher Electrical Potential (Voltage)
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POINT A HAS 2V MORE THAN POINT B
THE + AND - SIGNS DEFINE THE REFERENCE POLARITY If The Number V Is Positive, Then Point A Has V Volts More Than Point B, If The Number V Is Negative, Then Point A Has |V| Less Than Point B POINT A HAS 2V MORE THAN POINT B POINT A HAS 5V LESS THAN POINT B
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2-Index Notation for Voltages
Instead Of Showing The Reference Polarity We Agree That The FIRST SubIndex Denotes The Point With POSITIVE Reference Polarity
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Energy = Potential/Charge
Voltage Is A Measure Of ENERGY Per Unit CHARGE Charges Moving Between Points With Different Voltage ABSORB or RELEASE Energy – They May Transfer Energy From One Point To Another Converts chemical energy stored in the battery to thermal energy in the lamp filament which turns incandescent and glows Battery supplies energy to charges. Lamp absorbs energy from charges. The net effect is an energy TRANSFER EQUIVALENT CIRCUIT Charges gain energy here Charges supply Energy here
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Example What Energy Is Required To Move 120 [C] From Point-B To Point-A? Recall Potential = Work/Charge 120 [C] Since The Charges Move To a Point With Higher Voltage –The Charges Gained (Or Absorbed) Energy
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Example Recall Again Potential = Work/Charge Work = 5 J Charge = 1C
The Voltage Difference Is 5V Which Point Has Higher Voltage? VAB = 5V
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Energy and Power Each Coulomb Of Charge Loses 3[J] Or Supplies 3[J] Of Energy To The Element The Element Receives Energy At A Rate Of 6[J/s] The Electric Power Received By The Element Is 6[W] How Do We Recognize If An Element SUPPLIES Or RECEIVES Power? 2[C/s] PASS THROUGH THE ELEMENT In General Element dissipates power if current moves: Vhi→Vlo
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Passive Sign Convention
Power Received (Absorbed/Dissipated) is Positive While Power SUPPLIED (Generated) is Considered NEGATIVE If Voltage And Current Are Both Positive The Charges Move From High To Low Voltage, then The Component Receives, or Dissipates Energy – It Is A Passive Element
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Passive Element Assumption
We Assume PASSIVE Elements A Consequence Of This Convention Is That The Reference Directions For Current And Voltage Are Not Independent Given This Reference Potential-Polarity Then the Reference Direction For Current
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Passive Element Convention
The Voltage Polarity Given That The Reference Current Flows From Pt-A to Pt-B for the assumed Passive Element + − VA POSITIVE Relative to VB VAB > 0 For the PASSIVE Element
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Example The Element Receives 20W of Power
What is the Current? Select Reference Direction Based On Passive Sign Convention Vab = -10V Given 2A of Current Flows RIGHT-to-LEFT ←
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Passive Sign Convention
When In Doubt Label The Terminals Of The Component Element SUPPLIES Power Element RECEIVES Power
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Passive Sign Convention
Select Voltage Reference Polarity Based On CURRENT Reference Direction These elements SUPPLY Power In These Examples the Unknown V & I have Polarity & Direction OPPOSITE to the Passive Assumption
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Assigning The Reference
If V Given Across the Element Iref Flows: Vhi→Vlo IF I Given Thru the Element Vref DROPS in The Current Direction A A LEFT element DISSIPATES Power. RIGHT element SUPPLIES Power. B B
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Example Determine Power Absorbed By Each Circuit Element; 1, 2, and 3
In this Passive Ckt, a single V-Source is an Active, Power-SUPPLYING Element Current Runs VLo→VHi; Sets I Direction Assume that Voltage DROPS Across the Two Passive Elements is In the DIRECTION of Current Flow Note The Supplied-Received Power-Balance
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-APPENDIX- More Examples
Engineering 43 -APPENDIX- More Examples Applied Math Bruce Mayer, PE Licensed Electrical & Mechanical Engineer
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Done for Lecture/Discussion
Please Be Ready for a LAB during the Next Meeting. Mr. Phillips will have something fun & interesting to do See you then… See Me to Add
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Example Use Power Balance To Compute I0
Thus a 1A Current Flows in the Direction Assumed in the Figure
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Mathematical Analysis
Develop A Set Of Mathematical Equations That Represent The Circuit A Mathematical Model Learn How To Solve The Model To Determine Circuit Behavior (ENGR25) This Course Teaches The Basic Techniques Used To Develop Mathematical Models For Electric Circuits
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Linear Models The Models That Will Be Developed Have “Nice” Mathematical Properties. In Particular They Will Be Linear Which Means That They Satisfy The Principle Of Superposition: Model: Principle of Superposition Given 1, 2, are CONSTANTS
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Sophisticated Math Tools
This Course Applies the Tools Learned in MTH1 & ENGR25 to Solve Complex Physics/Math Circuit Models For The First Part We Will Be Expected To Solve Systems Of Algebraic Equations Watch for These in MTH-6
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Math Tools cont. Later The Models Will Be Differential Equations Of The Form Technically These Are Linear, Nonhomogeneous, Ordinary, Constant Coefficient Differential Equations Of The 1st & 2nd Order Watch for them in MTH-4
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Example A Camcorder Battery Plate Claims That The Unit Stores 2700 mAHr At 7.2V What Is The Total Charge And Energy Stored? Charge → The 2700 mAhr Specification Indicates That The Unit Can Deliver 2700 mA for ONE Hour Or 27 mA for 100 hrs; or 270 mA for 10 hrs Stored Energy → The Charges Are Moved Thru a 7.2V Potential Difference (0.02 kWhr)
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