1. Fundamentals of Analog and Digital Design
ET-IDA-134
Lecture-1
Circuit Analysis Fundamentals
Ch-1 and Ch-2
, v2
Prof. W. Adi
Source: Analog Devices, Digilent course material

2. Recommended Textbook:
Course Contents
Recommended Textbook:
Agarwal, Anant, and Jeffrey H. Lang. Foundations of Analog and Digital Electronic Circuits. San Mateo, CA: Morgan Kaufmann Publishers, Elsevier, July ISBN: View e-book versionElsevier companion site: supplementary sections and examples
Lecture Material:
- Provided in the class. Source: Digilent/Analog Devices Course material
- Suplimentary advanced analog and digital design topics with laboratory
Laboratory:
- Digilent Analog Discovery with lab‘s kit.

3. Lecture 1 Course Overview Basic Circuit Parameters
System modeling, analysis and design
Basic Circuit Parameters
Passive Sign Convention
Related educational materials:
Chapter 1.1

4. Pre-requisite and Co-requisite requirements
Pre-requisites (recommended)
Basic exposure to electricity and magnetism
Two semesters of Calculus
Co-requisites (recommended)
Differential equations
Pre- and Co-requisite requirements are rather weak
Superficial introductions to necessary topics provided at the appropriate points during this course
Note that a series of background lectures is available to provide some useful material.

5. Course Goals
Introduction to modeling, analysis and design of electrical circuits
We will often use a systems-level approach:
Note that we may start thinking of the system strictly in terms of the equation(s) relating the input and output
It is entirely possible to lose track of the physics of the system, to the extent that it doesn’t matter whether the system is electrical, mechanical, fluidic, or thermal
It is very common (especially in thermal and fluid systems) to represent the system as an “equivalent” electrical system

6. What are modeling, analysis and design?
We model the system by determining the mathematical relationship between the input and the output
System analysis often refers to determining the output from a system, for some given input
System design involves creating a system to provide some desired output

7. Modeling, analysis, and design – overview

8. General Modeling Approaches

9. Circuits I modeling approach
We will restrict our attention to lumped parameter models of linear, time-invariant systems
Governing equations will be linear, constant-coefficient, ordinary differential equations

10. Slinky demo
Linear
Nonlinear
Lumped
Distributed

11. Basic Circuit Parameters
Charge (q) is the basic quantity in circuit analysis
Units are Coulombs (C)  1 Coulomb  -6.241018 electrons
Current (i) is the rate of change of charge with time:
Units are Amperes (A) 

12. Basic Circuit Parameters – continued
Voltage (v) is the change in energy of a unit charge at two different points:
Units are Volts (V) 

13. Basic Circuit Parameters – continued
Power (P) is the time rate of change of energy:
Units are Watts (W)

14. Passive Circuit Elements
For a passive circuit element, the total energy delivered to the circuit element by the rest of the circuit is non-negative
The element can store energy, but it cannot create energy
Active circuit elements can supply energy to the circuit from external sources

15. Passive Sign Convention
We will assume the sign
of the current relative to
voltage for passive circuit
elements
Positive current enters the
node at the higher voltage
Sign must be known for active circuit elements
Note on active elements:
We haven’t really talked about these, yet. However, if you’ve watched the background lectures, we can mention that:
You must know the voltage polarity on a voltage source (though you know nothing about current direction)
You must know the current direction for a current source (though you don’t know what the voltage polarity is)
More on this later!

16. Passive Sign Convention – continued
You can assume (arbitrarily) either the voltage polarity or the current direction
This assumption dictates the assumed direction of the other parameter
These assumptions provide reference voltage polarities and current directions
Subsequent analysis is performed based on this assumption; a negative result simply means that the assumed voltage polarity or current direction was incorrect
Compare to choosing a “posititve” direction on a free-body diagram?

17. Passive Sign Convention – Example 1
Provide the appropriate sign convention for the missing parameter on the passive elements represented by grey boxes.
Point out active elements. Voltage source and current source. Note that:
We will introduce these types of elements more formally later
The necessary signs are provided on these elements. We do not know:
Anything about the voltage across the current source
Anything about the current through the voltage source

18. Passive Sign Conventions – Hints
It is generally counter-productive to attempt to determine the “correct” voltage polarities and current directions before analyzing the circuit
Just arbitrarily choose either the assumed voltage polarity or current direction for each passive circuit element
This choice dictates the sign of the other parameter
Perform analysis using assumed signs
Negative signs mean that the assumption was incorrect

19. Passive Sign Convention – Example 2
Assign reference voltage and current directions for the passive elements represented by shaded boxes in the circuit below:

20. Passive Sign Convention – Example 3
Assign reference voltage and current directions for the passive elements represented by shaded boxes in the circuit below:
Do this problem at least two ways. (choose one set of reference directions, then erase and choose a different set.) Note that it doesn’t matter!

21. Passive Sign Convention – Example 4
For the circuit below, the sign convention shown is chosen
After analyzing the circuit, it is determined that I1 = -3mA, I2 = 3mA, V1 = -1.5V, and V2 = 2.5V. Re-draw the circuit showing the actual voltages and currents and their directions

22. Lecture 2 Review Passive Sign Convention Power Generation, Absorption
Power Sources
Resistance
Related educational materials:
Chapter 1.1, 1.2, 1.3

23. Passive sign convention – review
For passive circuit elements, we assume that the current enters the node with the higher voltage potential
Your analyses will not be reliable unless you do this correctly
Examples:
Note that a series of background lectures is available to provide some useful material.

24. Subscript notation can denote voltage polarity
Voltage polarity is sometimes indicated by subscript notation
The order of the subscripts indicates the polarity
The first subscript indicates assumed higher-voltage node
The second subscript is the assumed lower-voltage node

25. “Ground” Voltages are often represented as relative to “ground”:
Ground (symbol: ) is a reference voltage; often 0V
Voltages relative to ground generally not called a voltage difference; they are a difference relative to zero volts
Voltages relative to ground often represented with a single subscript

26. Power Generation and Dissipation
Circuit elements can either dissipate or generate power
Power is dissipated (or absorbed) if current enters the positive voltage node
Power is generated (or supplied) if current enters the negative voltage node

27. Power Generation and Dissipation
Power = voltagecurrent (p= vi)
Power is absorbed if the power is positive (voltage and current are consistent with the passive sign convention)
Power is generated if the power is negative (voltage and current not consistent with the passive sign convention)

28. Examples Determine the power absorbed by the circuit element below.
The circuit element absorbs 10W. Determine the current in the element.

29. Power Supplies Power supplies provide a source of electrical power
Conceptual types of power supplies (models of physical supplies):
Voltage, current sources
Independent, dependent sources
Ideal and non-ideal sources

30. Independent voltage sources
Common symbols:
Independent voltage sources maintain specified voltage, regardless of the current

31. Independent voltage sources – continued
Voltage-current characteristic for constant voltage source:

32. Independent current sources
Common symbol:
Independent current sources maintain specified current, regardless of the voltage

33. Independent current sources – continued
Voltage-current characteristic for constant current source:

34. Ideal power sources – limitations
Ideal sources can provide infinite power
Voltage sources provide specified voltage, regardless of the current  current can be infinite  power can be infinite
Current sources provide specified current, regardless of the voltage  voltage can be infinite  power can be infinite
These models can be unrealistic
We will examine more realistic power source models later

35. Dependent Power Supplies
Some active circuit elements can be modeled as dependent power sources
The current or voltage delivered by the source is controlled by a current or voltage somewhere else in the circuit
Four possible combinations
Voltage controlled voltage source (VCVS)
Current controlled voltage source (CCVS)
Voltage controlled current source (VCCS)
Current controlled current source (CCCS)

36. Dependent Power Supplies – continued
Examples:

37. Resistors Circuit symbol: R is the resistance
Units are ohms ()
Voltage-current relation (Ohm’s Law):

38. Resistors – continued Notes: Resistors can only dissipate energy
The voltage-current relation is algebraic

39. Resistor Power Dissipation
Ohm’s Law:
Power:
Combining:

40. Example
Determine the power (generated or absorbed) by the resistor below:

41. Conservation of energy
In an electrical circuit, the power generated is the same as the power absorbed
Slightly more mathematically,
Recall that power absorbed is positive and power generated is negative

42. Conservation of power – example
Determine the power (absorbed or generated) by the voltage source VS

43. Lecture 3 Review: Kirchoff’s Current Law Kirchoff’s Voltage Law
Ohm’s Law, Power, Power Conservation
Kirchoff’s Current Law
Kirchoff’s Voltage Law
Related educational materials:
Chapter 1.4

44. Review: Ohm’s Law Ohm’s Law Voltage-current characteristic
of ideal resistor:

45. Review: Power
Power:
Power is positive if i, v agree with passive sign convention (power absorbed)
Power is negative if i, v contrary to passive sign convention (power generated)

46. Review: Conservation of energy
Power conservation:
In an electrical circuit, the power generated is the same as the power absorbed.
Power absorbed is positive and power generated is negative

47. Two new laws today: Kirchoff’s Current Law Kirchoff’s Voltage Law
These will be defined in terms of nodes and loops

48. Basic Definition – Node
A Node is a point of connection between two or more circuit elements
Nodes can be “spread out” by perfect conductors

49. Basic Definition – Loop
A Loop is any closed path through the circuit which encounters no node more than once

50. Kirchoff’s Current Law (KCL)
The algebraic sum of all currents entering (or leaving) a node is zero
Equivalently: The sum of the currents entering a node equals the sum of the currents leaving a node
Mathematically:
We can’t accumulate
charge at a node

51. Kirchoff’s Current Law – continued
When applying KCL, the current directions (entering or leaving a node) are based on the assumed directions of the currents
Also need to decide whether currents entering the node are positive or negative; this dictates the sign of the currents leaving the node
As long all assumptions are consistent, the final result will reflect the actual current directions in the circuit

52. KCL – Example 1
Write KCL at the node below:

53. KCL – Example 2
Use KCL to determine the current i

54. Kirchoff’s Voltage Law (KVL)
The algebraic sum of all voltage differences around any closed loop is zero
Equivalently: The sum of the voltage rises around a closed loop is equal to the sum of the voltage drops around the loop
Mathematically:
If we traverse a loop, we end up
at the same voltage we started with

55. Kirchoff’s Voltage Law – continued
Voltage polarities are based on assumed polarities
If assumptions are consistent, the final results will reflect the actual polarities
To ensure consistency, I recommend:
Indicate assumed polarities on circuit diagram
Indicate loop and direction we are traversing loop
Follow the loop and sum the voltage differences:
If encounter a “+” first, treat the difference as positive
If encounter a “-” first, treat the difference as negative

56. KVL – Example
Apply KVL to the three loops in the circuit below. Use the provided assumed voltage polarities

57. Circuit analysis – applying KVL and KCL
In circuit analysis, we generally need to determine voltages and/or currents in one or more elements
We can determine voltages, currents in all elements by:
Writing a voltage-current relation for each element (Ohm’s law, for resistors)
Applying KVL around all but one loop in the circuit
Applying KCL at all but one node in the circuit

58. Circuit Analysis – Example 1
For the circuit below, determine the power absorbed by each resistor and the power generated by the source. Use conservation of energy to check your results.

59. Example 1 – continued

60. Circuit Analysis – Example 2
For the circuit below, write equations to determine the current through the 2 resistor

61. Example 2 – Alternate approach

62. Circuit Analysis
The above circuit analysis approach (defining all “N” unknown circuit parameters and writing N equations in N unknowns) is called the exhaustive method
We are often interested in some subset of the possible circuit parameters
We can often write and solve fewer equations in order to determine the desired parameters

63. Circuit analysis – Example 3
For the circuit below, determine:
(a) The current through the 2 resistor
(b) The current through the 1 resistor
(c) The power (absorbed or generated) by the source

64. Circuit Analysis Example 3 – continued

65. Lecture 6 Review: Circuit reduction examples Practical application
Temperature measurement
Related educational materials:
Chapter 2.3

66. Review: series resistors and voltage division
Equivalent resistance: Voltage divider formula:

67. Review: parallel resistance and current division
Equivalent resistance: Current divider formula:

68. Checking parallel resistance results
The equivalent resistance of a parallel combination of resistors is less than the smallest resistance in the combination
Resistance decreases as resistors are added in parallel
Range of equivalent resistance:
Rmin is the lowest resistance; N is the number of resistors

69. Examples: Non-ideal “loaded” power sources
Loaded voltage source:
Loaded current source:

70. Circuit Reduction
Series and parallel combinations of circuit elements can be combined into a “equivalent” elements
The resulting simplified circuit can often be analyzed more easily than the original circuit

71. Circuit reduction – example 1
Determine the equivalent resistance of the circuit below

72. Circuit reduction – example 2
Determine Vout in the circuit below.

73. Circuit reduction – example 3
In the circuit below, find i1, VS, and VO.

74. Example 3 – continued

75. Example 3 – continued

76. Circuit reduction – example 4
In the circuit below, determine
(a) the equivalent resistance seem by the source,
(b) the currents i1 and i2

77. Example 4 – continued

78. Practical application – temperature measurement
Design a temperature measurement system whose output voltage increases as temperature increases
In general, we will typically have other design objectives
For example, power and sensitivity requirements
We neglect these for now; lab 2 will provide a more rigorous treatment of this problem

79. Temperature sensors: thermistors
Thermistors are sensors whose resistance changes as a function of temperature
Thermistors are classified as either NTC (negative temperature coefficient) or PTC (positive temperature coefficient)
Resistance increases with temperature for PTCs; Resistance decreases with temperature for NTCs
A resistance variation is generally not directly useful; information is generally relayed with voltage
We need to convert the resistance change to a voltage change

80. Example thermistor characteristics
Response:
NTC 25C
Negative temperature coefficient thermistor with (nominal) resistance of 10k at 25C

81. Initial Design Concept
Use voltage divider to convert resistance variation to voltage variation
Design problem: choose Vs and R to obtain desired variation in Vout for a given variation in temperature

82. Potential Design Issues
Sensitivity
Our design requirements may specify a minimum voltage change per degree of temperature change (the sensitivity of the instrumentation system)
We can affect the sensitivity with our choice of R
Power requirements
We can increase the sensitivity by increasing VS
Increasing VS increases the power required by the system; increasing power (generally) increases cost
The above can cause us to modify or discard our initial design concept!

83. Effect of resistance change on voltage

84. Demo: Change of thermistor resistance with temperature (DMM)
Change of output voltage from voltage divider
R<<RTH
R>>RTH
Intermediate R