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Presentation transcript:

Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Engineering 43 Chp 8 [1-4] Sinusoids Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

Outline – AC Steady State SINUSOIDS Review basic facts about sinusoidal signals SINUSOIDAL AND COMPLEX FORCING FUNCTIONS Behavior of circuits with sinusoidal independent sources Modeling of sinusoids in terms of complex exponentials

Outline – AC SS cont. phasEr PHASORS IMPEDANCE AND ADMITANCE Representation of complex exponentials as vectors Facilitates steady-state analysis of circuits Has NOTHING to do with StarTrek IMPEDANCE AND ADMITANCE Generalization of the familiar concepts of RESISTANCE and CONDUCTANCE to describe AC steady-state circuit operation

Outline – AC SS cont.2 PHASOR DIAGRAMS Representation of AC voltages and currents as COMPLEX VECTORS BASIC AC ANALYSIS USING KIRCHHOFF’S LAWS ANALYSIS TECHNIQUES Extension of node, loop, SuperPosition, Thevenin and other techniques

Sinusoids Recall From Trig the Sine Function Where XM  “Amplitude” or Maximum Value Typical Units = A or V ω  Radian, or Angular, Frequency in rads/sec t  Sinusoid argument in radians (a pure no.) The function Repeats every 2π; mathematically For the RADIAN Plot Above, The Functional Relationship

Sinusoids cont. Now Define the “Period”, T Such That From Above Observe How often does the Cycle Repeat? Define Next the CYCLIC FREQUENCY Now Can Construct a DIMENSIONAL Plot

Sinusoids cont.2 Now Define the Cyclic “Frequency”, f Quick Example USA Residential Electrical Power Delivered as a 115Vrms, 60Hz, AC sine wave Describes the Signal Repetition-Rate in Units of Cycles-Per-Second, or HERTZ (Hz) Hz is a Derived SI Unit Will Figure Out the 2 term During Chp9 Study RMS  “Root (of the) Mean Square”

Sinusoids cont.3 Now Consider the GENERAL Expression for a Sinusoid Where θ  “Phase Angle” in Radians Graphically, for POSITIVE θ

Leading, Lagging, In-Phase Consider Two Sinusoids with the SAME Angular Frequency If  = , Then The Signals are IN-PHASE If   , Then The Signals are OUT-of-PHASE Phase Angle Typically Stated in DEGREES, but Radians are acceptable Both These Forms OK Now if  >  x1 LEADS x2 by (−) rads or Degrees x2 LAGS x1 by ( −) rads or Degrees

Useful Trig Identities To Convert sin↔cos Additional Relations To Make a Valid Phase-Angle Difference Measurement BOTH Sinusoids MUST have the SAME Frequency & Trig-Fcn (sin OR cos) Useful Phase-Difference ID’s  SUM & DIFFERENCE formulas

Example  Phase Angles Given Signals To find phase angle must express BOTH sinusoids using The SAME trigonometric function; either sine or cosine A POSITIVE amplitude Find Frequency in Standard Units of Hz Phase Difference Frequency in radians per second is the PreFactor for the time variable Thus

Example – Phase Angles cont. Convert –6V Amplitude to Positive Value Using It’s Poor Form to Express phase shifts in Angles >180° in Absolute Value Then Next Convert cosine to sine using So Finally Thus v1 LEADS v2 by 120°

Sinusoidal Forcing Functions Consider the Arbitrary LINEAR Ckt at Right. If the independent source is a sinusoid of constant frequency then for ANY variable in the LINEAR circuit the STEADY-STATE Response will be SINUSOIDAL and of the SAME FREQUENCY Thus to Find iss(t), Need ONLY to Determine Parameters A &  Mathematically

Example  RL Single Loop Given Simple Ckt Find i(t) in Steady State Write KVL for Single Loop In Steady State Expect Sub Into ODE and Rearrange

Example  RL Single Loop cont Recall the Expanded ODE Equating the sin & cos PreFactors Yields Solving for Constants A1 and A2 Recognize as an ALGEBRAIC Relation for 2 Eqns in 2 Unkwns Found Solution using ONLY Algebra This is Good

Example  RL Loop cont.2 Using A1 & A2 State the Soln Also the Source-V If in the ID =x, and ωt = y, then Would Like Soln in Form Comparing Soln to Desired form → use sum-formula Trig ID

Example  RL Loop cont.3 Dividing These Eqns Find Now Find A to be Trig ID: tan(-x) = -tan(x) Elegant Final Result, But VERY Tedious Calc for a SIMPLE Ckt  Not Good Subbing for A &  in Solution Eqn

Complex Exponential Form Solving a Simple, One-Loop Circuit Can Be Very Tedious for Sinusoidal Excitations To make the analysis simpler relate sinusoidal signals to COMPLEX NUMBERS. The Analysis Of the Steady State Will Be Converted To Solving Systems Of Algebraic Equations ... Start with Euler’s Identity Where Note: The Euler Relation can Be Proved Using Taylor’s Series (Power Series) Expansion of ej

Complex Exponential cont Now in the Euler Identity, Let Notice That if So Next Multiply by a Constant Amplitude, VM Now Recall that LINEAR Circuits Obey SUPERPOSITION Separate Function into Real and Imaginary Parts

Complex Exponential cont.2 Consider at Right The Linear Ckt with Two Driving Sources General Linear Circuit By KVL The Total V-src Applied to the Circuit Now by SuperPosition The Current Response to the Applied Sources This Suggests That the…

Complex Exponential cont.3 Application of the Complex SOURCE will Result in a Complex RESPONSE From Which The REAL (desired) Response Can be RECOVERED; That is General Linear Circuit Then The Desired Response can Be RECOVERED By Taking the REAL Part of the COMPLEX Response Thus the To find the Response a COMPLEX Source Can Be applied.

General Linear Circuit Realizability General Linear Circuit We can NOT Build Physical (REAL) Sources that Include IMAGINARY Outputs We can also NOT invalidate Superposition if we multiply a REAL Source by ANY CONSTANT Including “j” Thus Superposition Holds, mathematically, for We CAN, However, BUILD These

Complex Numbers Reviewed A b a Real Imaginary Consider a General Complex Number This Can Be thought of as a VECTOR in the Complex Plane This Vector Can be Expressed in Polar (exponential) Form Thru the Euler Identity Where Then from the Vector Plot

Complex Number Arithmetic Consider Two Complex Numbers The PRODUCT n•m The SUM, Σ, and DIFFERENCE, , for these numbers Complex DIVISION is Painfully Tedious See Next Slide

Complex Number Division For the Quotient n/m in Rectangular Form Use the Complex CONJUGATE to Clear the Complex Denominator The Generally accepted Form of a Complex Quotient Does NOT contain Complex or Imaginary DENOMINATORS The Exponential Form is Cleaner See Next Slide

Complex Number Division cont. For the Quotient n/m in Exponential Form However Must Still Calculate the Magnitudes A & D...

Example  RL Single Loop This Time, Start with a COMPLEX forcing Function, and Recover the REAL Response at The End of the Analysis Let Thus Assume Current Response of the Form In a Linear Ckt, No Circuit Element Can Change The Driving Frequency, but They May induce a Phase Shift Relative to the Driving Sinusoid Then The KVL Eqn

Example – RL Single Loop cont. Taking the 1st Time Derivative for the Assumed Solution Then the KVL Eqn Then the Left-Hand-Side (LHS) of the KVL Canceling ejt and Solving for IMej

Example – RL Single Loop cont.2 Clear Denominator of The Imaginary Component By Multiplying by the Complex Conjugate Next A & θ The Response in Rectangular Form: a+jb

Example – RL Single Loop cont.3 First A A &  Correspondence with Assumed Soln A → IM θ →  Cast Solution into Assumed Form And θ

Example – RL Single Loop cont.4 The Complex Exponential Soln Where Recall Assumed Soln Finally RECOVER the DESIRED Soln By Taking the REAL Part of the Response

Example – RL Single Loop cont.5 By Superposition Explicitly SAME as Before 

Phasor Notation Imaginary b a Real Imaginary If ALL dependent Quantities In a Circuit (ALL i’s & v’s) Have The SAME FREQUENCY, Then They differ only by Magnitude and Phase That is, With Reference to the Complex-Plane Diagram at Right, The dependent Variable Takes the form Borrowing Notation from Vector Mechanics The Frequency PreFactor Can Be Written in the Shorthand “Phasor” Form

Phasor Notation cont. Because of source superposition one can consider as a SINGLE source, a System That contains REAL and IMAGINARY Components Or by SuperPosition Since ejt is COMMON To all Terms we can work with ONLY the PreFactor that contains Magnitude and Phase info; so The Real Steady State Response Of Any Circuit Variable Will Be Of The Form

Phasor Characteristics Since in the Euler Reln The REAL part of the expression is a COSINE, Need to express any SINE Function as an Equivalent CoSine Turn into Cos Phasor Examples Phasors Combine As the Complex Polar Exponentials that they are

Example  RL Single Loop As Before Sub a Complex Source for the Real Source The Form of the Responding Current For the Complex Quantities Phasor Variables Denoted as BOLDFACE CAPITAL Letters Then The KVL Eqn in the Phasor Domain Recall The KVL

Example  RL Single Loop cont. To recover the desired Time Domain Solution Substitute This is A LOT Easier Than Previous Methods The Solution Process in the Frequency Domain Entailed Only Simple Algebraic Operations on the Phasors Then by Superposition Take

Resistors in Frequency Domain The v-i Reln for R Thus the Frequency Domain Relationship for Resistors

Resistors in -Land cont. Phasors are complex numbers. The Resistor Model Has A Geometric Interpretation In the Complex-Plane The Current & Voltage Are CoLineal i.e., Resistors induce NO Phase Shift Between the Source and the Response Thus voltage and current sinusoids are said to be “IN PHASE” R → IN Phase

Inductors in Frequency Domain The v-i Reln for L Thus the Frequency Domain Relationship for Inductors

Inductors in -Land cont. The relationship between phasors is algebraic. To Examine This Reln Note That Therefore the Current and Voltage are OUT of PHASE by 90° Plotting the Current and Voltage Vectors in the Complex Plane Thus

Inductors in -Land cont.2 In the Time Domain Short Example L → current LAGS Phase Relationship Descriptions The VOLTAGE LEADS the current by 90° The CURRENT LAGS the voltage by 90° In the Time Domain

Capacitors in Frequency Domain The v-i Reln for C Thus the Frequency Domain Relationship for Capacitors

Capacitors in -Land cont. The relationship between phasors is algebraic. Recall Therefore the Voltage and Current are OUT of PHASE by 90° Plotting the Current and Voltage Vectors in the Complex Plane Thus

Capacitors in -Land cont.2 In the Time Domain Short Example C → current LEADS Phase Relationship Descriptions The CURRENT LEADS the voltage by 90° The VOLTAGE LAGS the current by 90° In the Time Domain

WhiteBoard Work Let’s Work Text Problem 8.5

Sp2010 MTE2 Review - P4 The 2.7 kΩ Resistors are NOT in Parallel Parallel  SAME Potential Across Elements

Sp2010 MTE2 Review – P5 The 3.6&1.2 kΩ OR the 1.2&3 kΩ OR 3.6&3 kΩ OR 3&1.2&3.6 kΩ Resistors are NOT in Parallel Parallel  SAME Potential Across Elements Particularly important for Find RTH for time const RTHC