1. Circuit Analysis - Basic Concepts -
Engineering 43
Circuit Analysis - Basic Concepts -
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. Circuit Analysis
DEVELOP TOOLS FOR THE ANALYSIS AND DESIGN OF BASIC LINEAR ELECTRIC CIRCUITS

3. BASIC ANALYSIS STRATEGY

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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.

14. 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 Electromotive Force 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

15. Current And Voltage Ranges

16. 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

17. 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

18. Fluid↔Current Flow Analogs
Think of the
Voltage as the Electrical “Pressure”
Current as the Electrical Fluid
Wire as the Electrical “Pipe”

19. 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

20. 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

21. 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?...

22. 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

23. 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 The 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

24. 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

25. 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

26. 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

27. 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)

28. 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

29. 2-Index Notation for Voltages
Instead Of Showing The Reference Polarity We Agree That The FIRST SubIndex Denotes The Point With POSITIVE Reference Polarity

30. 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

31. 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

32. Example Recall Again Potential = Work/Charge Work = 5 J Charge = 1C
The Voltage Difference Is 5V
Which Point Has Higher Voltage?
VAB = 5V

33. 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)

34. 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

35. Passive Sign Convention
Power Received (Absorbed) 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

36. 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

37. 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

38. 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 ←

39. Understanding the Passive Sign Convention
Examine The Voltage Across The Component or System, and The Current Through It S1 S2 A A’ B B’

40. Passive Sign Convention
When In Doubt Label The Terminals Of The Component
Element SUPPLIES Power
Element RECEIVES Power

41. Passive Sign Convention
Select Voltage Reference Polarity Based On CURRENT Reference Direction
In These Examples the Unknown V & I have Polarity & Direction OPPOSITE to the Passive Assumption

42. 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 B B

43. 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

44. Circuit Elements Passive Elements Active, Independent Sources R, C, L
(a)  V-source
(b)  Battery
Basically a V-Source
(c)  I-Source

45. Dependent Sources Dependent on V Dependent on I Units for µ, g?
Units for r, β?

46. Dependent Source Exercises
Determine Power Supplied by Sources A B B A
Take VOLTAGE Polarity as Reference
Take CURRENT Direction as Reference

47. Power Example Determine Power Absorbed Or Supplied By Each Element
Note That
Power-Supplied = Power-Absorbed
In both Cases = 152W

48. Done for First Day Please Be Ready for a LAB during the Next Meeting.
We’ll Learn About
Resistors
The Analog Volt-Ohm-Amp-Meter (VOAM)
The Digital MultiMeter (DMM)
See you then… 
See Me to Add

49. Example Use Power Balance To Compute I0
Thus a 1A Current Flows in the Direction Assumed in the Figure