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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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”

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

9. 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

11. 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

12. 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

13. 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°

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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…

23. 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.

24. 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

25. Complex Numbers ReviewedA 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

26. 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

27. 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

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

29. 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

30. 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

31. 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

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

33. 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

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

35. Phasor Notation Imaginaryb 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

36. 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

37. 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

38. 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

39. 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

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

41. 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

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

43. 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

44. 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

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

46. 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

47. 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

48. WhiteBoard Work
Let’s Work Text Problem 8.5

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

50. 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