1. Registered Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Engineering 43
Impedance KCL & KVL
Bruce Mayer, PE
Registered Electrical & Mechanical Engineer

2. Review → V-I in Phasor Space
No Phase Shift
Resistors
Inductors
i(t) LAGS
Leading & Lagging are usually reference to VOLTAGE signal
Capacitors
i(t) LEADS

3. Impedance
For each of the passive components, the relationship between the voltage phasor and the current phasor is algebraic (previous sld)
Consider now the general case for an arbitrary 2-terminal element
Since the Phasors V & I Have units of Volts and Amps, Z has units of Volts per Amp (V/A), or OHMS
The Frequency Domain Analog to Resistance is IMPEDANCE, Z

4. Impedance cont. Since V & I are COMPLEX, Then Z is also Complex
However, Z IS a COMPLEX NUMBER that can be written in polar or Cartesian form.
In general, its value DOES depend on the Sinusoidal frequency
Impedance is NOT a Phasor
It’s Magnitude and Phase Do Not Change regardless of the Location within The Circuit
Note that the REACTANCE, X, is a function of ω

5. Impedance cont.2 Thus Summary Of Passive-Element Impedance
The Magnitude and Phase
Examine ZC
Where

6. KVL & KCL Hold In Phasor Spc
Similarly for the Sinusoidal Currents ...

7. Series & Parallel Impedances
Impedances (which have units of Ω) Combine as do RESISTANCES
The SERIES Case
The Parallel Case

8. Admittance The Frequency Domain Analog of CONDUCTANCE is ADMITTANCE
Admittance is Thus Inverse Impedance
Multiply Denominator by the Complex Conjugate
G  CONDUCTance
B  SUSCEPTance
Find G & B In terms of Resistance, R, and Reactance, X
Note that G & R and X & B are NOT Reciprocals

9. Series & Parallel Admittance
Admittance Summarized
Admittances (which have units of Siemens) Combine as do CONDUCTANCES
The SERIES Case
The PARALLEL Case

10. Sp17 Game Plan Start on Next slide of this Lecture
ENGR-43_Lec-05b_Sp17_Impedance_KCL_KVL_ pptx
Complete as much as possible on Lectures
ENGR-43_Lec-05c_Sp17_Thevenin_AC_Power.pptx
ENGR-43_Lec-06a_Sp17_Fourier_XferFcn.pptx

11. MATLAB 𝒙,𝒚 ↔ 𝒓,𝜽 Functions
Rectangular to Polar
Polar to Rectangular
Both use RADIANS only

12. Phasor Diagrams Imaginaryb a
Real
Imaginary
As Noted Earlier Phasors can be Considered as VECTORS in the Complex Plane
See Diagram at Right
See Next Slide for Review of Vector Addition
Text Diagrams follow the PARALLELOGRAM Method
Phasors Obey the Rules of Vector Arithmetic
Which were originally Developed for Force Mechanics

13. Vector Addition Parallelogram Rule For Vector Addition
Examine Top & Bottom of The Parallelogram
Triangle Rule For Vector Addition
Vector Addition is Commutative
Vector Subtraction → Reverse Direction of The Subtrahend B C

14. Example  Phasor Diagram
For The Single-Node Ckt at Right, Draw the Phasor Diagrams as a function of Frequency
First Write KCL
That is, we Can Select ONE Phasor to have a ZERO Phase Angle
In this Case Choose V
Next Examine Frequency Sensitivity of the Admittances
Now we can Select ANY Phasor Quantity, I or V, as the BaseLine

15. Example  Phasor Diagram cont
The KCL
This Eqn Shows That as ω increases
YL Decreases (goes to 0)
YC Increases (goes to +∞)
Now ReWrite KCL using Phasor Notation
Examining the Phase Angles Shows that in the Complex Plane
IR Points RIGHT
IL Points DOWN
IC Points UP
As ω Increases, IC begins to dominate IL

16. Example  Phasor Diagram cont.2
Case-I: ω=Med so That
YL  YC
Case-III: ω=Hi so That
YC  2YL
The Circuit is Basically CAPACITIVE
Case-II: ω=Low so That
YL  2YC
The Circuit is Basically INDUCTIVE

17. KCL & KVL for AC Analysis
Simple-Circuit Analysis
AC Version of Ohm’s Law → V = IZ
Rules for Combining Z and/or Y
KCL & KVL
Current and/or Voltage Dividers
More Complex Circuits
Nodal Analysis
Loop or Mesh Analysis
SuperPosition or Source Xform

18. Methods of AC Analysis cont.
More Complex Circuits
Thevenin’s Theorem
Norton’s Theorem
Numerical Techniques
MATLAB
SPICE

19. Example For The Ckt At Right, Find VS if Then I2 by Ohm
𝐕 1
Example
For The Ckt At Right, Find VS if
Then I2 by Ohm
Solution Plan: GND at Bot, then Find in Order
I3 → V1 → I2 → I1 → VS
I3 First by Ohm
Then I1 by KCL
Then V1 by Ohm = ZI

20. Example cont. Then VS by Ohm & KVL Then Zeq
𝐕 1
Then Zeq
Note That in passing we have I1 and VS
Thus can find the Circuit’s Equivalent (BlackBox) Impedance

21. Nodal Analysis for AC Circuits
For The Ckt at Right Find IO
Use Node Analysis
Specifically a SuperNode that Encompasses The V-Src  KCL at SN
The Relation For IO
And the SuperNode Constraint
In SuperNode KCL Sub for V1

22. Nodal Analysis cont. Solving for For V2 The Complex Arithmetic Recall

23. Loop Analysis for AC Circuits
Same Ckt, But Different Approach to Find IO
Note: IO = –I3
Constraint: I1 = –2A0°
The Loop Eqns
Simplify Loop2 & Loop3
Solution is I3 = –IO
Recall I1 = –2A0°
Two Eqns In Two Unknowns: 𝐈 2 & 𝐈 3

24. Loop Analysis cont Isolating I3 Then The Solution
The Next Step is to Solve the 3 Eqns for I2 and I3
So Then Note
Could also use a SuperMesh to Avoid the Current Source

25. Recall Source SuperPosition= +
Circuit With Current Source Set To Zero
OPEN Ckt
Circuit with Voltage Source set to Zero
SHORT Ckt
By Linearity

26. AC Ckt Source SuperPosition
Same Ckt, But Use Source SuperPosition to Find IO
DeActivate V-Source
The Reduced Ckt
Combine The Parallel Impedances

27. AC Source SuperPosition cont.
Find I-Src Contribution to IO by I-Divider
The V-Src Contribution by V-Divider
Now Deactivate the I-Source (open it)

28. AC Source SuperPosition cont.2
Sub for Z”
The Total Response
Finally SuperPose the Response Components

29. Multiple Frequencies
When Sources of Differing FREQUENCIES excite a ckt then we MUST use SuperPosition for every set of sources with NON-EQUAL FREQUENCIES
An Example
We Can Denote the Sources as Phasors
But canNOT COMBINE the Source due to DIFFERING frequencies

30. Multiple Frequencies cont.1
Must Use SuperPosition for EACH Different ω
V1 first (ω = 10 r/s)
V2 next (ω = 20 r/s)
The Frequency-1 Domain Phasor-Diagram

31. Multiple Frequencies cont.2
The Frequency-2 Domain Phasor-Diagram
Recover the Time Domain Currents
Finally SuperPose
Note the MINUS sign from CW-current assumed-Positive

32. Source Transformation
Source transformation is a good tool to reduce complexity in a circuit
WHEN IT CAN BE APPLIED
“ideal sources” are not good models for real behavior of sources
A real battery does not produce infinite current when short-circuited
Resistance → Impedance Analogy

33. Source Transformation
Same Ckt, But Use Source Transformation to Find IO
Start With I-Src
Then the Reduced Circuit
Next Combine the Voltage Sources And Xform

34. Source Transformation cont
The Reduced Ckt
Now Combine the Series-Parallel Impedances
The Reduced Ckt
IO by I-Divider

35. WhiteBoard Work
Let’s Work This Nice Problem to Find VO
7e LE7.14

36. Charles Proteus Steinmetz
All Done for Today
Charles Proteus Steinmetz
Delveloper of Phasor Analysis

37. Appendix HP48G+ Complex No.s
Engineering 43
Appendix HP48G+ Complex No.s
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

38. HP 48G+ : Using Memory Purple LEFT Arrow
From: HP 48g Quick Start Guide

39. From: HP-48_Complex_Numbers_1605.pptx

40. From: HP-48_Complex_Numbers_1605.pptx

41. From: HP-48_Complex_Numbers_1605.pptx

42. Appendix White Board Problems
Engineering 43
Appendix White Board Problems
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

45. WhiteBoard Work Let’s Work this Nice Problem
See Next Slide for Phasor Diagrams

46. P8.29 Phasor Diagrams
Tip-To-Tail Phasor (Vector) Addition
7e P7.33

47. WhiteBoard Work Let’s Work Some Phasor Problems
7e probs 7.13, 7.20, LE 7.10

48. Skip for 22MAr16 Meeting