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

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

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

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

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

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

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

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

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

Sp17 Game Plan Start on Next slide of this Lecture ENGR-43_Lec-05b_Sp17_Impedance_KCL_KVL_160320.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

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

Phasor Diagrams Imaginary b 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

Charles Proteus Steinmetz All Done for Today Charles Proteus Steinmetz http://www.enzim.hu/~szia/cddemo/edemo4.htm Delveloper of Phasor Analysis

Appendix HP48G+ Complex No.s Engineering 43 Appendix HP48G+ Complex No.s Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

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

From: HP-48_Complex_Numbers_1605.pptx

From: HP-48_Complex_Numbers_1605.pptx

From: HP-48_Complex_Numbers_1605.pptx

Appendix White Board Problems Engineering 43 Appendix White Board Problems Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

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

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

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

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