Fundamentals of Analog and Digital Design ET-IDA-134 Lecture-1 Review: Fundamentals of Circuit Analysis Ch-1 and Ch-2 (L1 to L6) 20.11.2017, v9 Prof. W. Adi Source: Analog Devices, Digilent course material

Fundamentals of Analog and Digital Design Recommended Textbook (MIT): Agarwal, Anant, and Jeffrey H. Lang. Foundations of Analog and Digital Electronic Circuits. San Mateo, CA: Morgan Kaufmann Publishers, Elsevier, July 2005. ISBN: 9781558607354. 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“ unit with lab‘s kit. http://www.digilentinc.com/Classroom/RealAnal/

Laboratory Equipment: „Analog Discovery-1“ Capabilities of Analog Discovery unit: • 2-Channel Oscilloscope • 2-Channel Waveform Generator • 16-Channel Logic Analyzer • 16-Channel Digital Pattern Generator • ±5VDC Power Supplies • Spectrum Analyzer • Network Analyzer • Voltmeter • Digital I/O USB connected with user interface The Analog Parts Kit contains a large selection of components perfect for creating a wide variety of useful circuits & devices. Featuring Analog Devices components, the kit includes transistors, resistors, capacitors, diodes, sensors, and variety of useful ICs including Op Amps, convertors, and regulators. Finally, the kit also comes with an assortment of lead wires, a solderless breadboard, and a screwdriver.

Laboratory Equipment: „Analog Discovery-2“ Capabilities of Analog Discovery unit: Two-channel USB digital oscilloscope (1MΩ, ±25V, differential, 14-bit, 100MS/s, 30MHz+ bandwidth - with the Analog Discovery BNC Adapter Board) Two-channel arbitrary function generator (±5V, 14-bit, 100MS/s, 12MHz+ bandwidth - with the Analog Discovery BNC Adapter Board) Stereo audio amplifier to drive external headphones or speakers with replicated AWG signals 16-channel digital logic analyzer (3.3V CMOS and 1.8V or 5V tolerant, 100MS/s) 16-channel pattern generator (3.3V CMOS, 100MS/s) 16-channel virtual digital I/O including buttons, switches, and LEDs – perfect for logic training applications Two input/output digital trigger signals for linking multiple instruments (3.3V CMOS) Single channel voltmeter (AC, DC, ±25V) Network analyzer – Bode, Nyquist, Nichols transfer diagrams of a circuit. Range: 1Hz to 10MHz Spectrum Analyzer – power spectrum and spectral measurements (noise floor, SFDR, SNR, THD, etc.) Digital Bus Analyzers (SPI, I²C, UART, Parallel) Two programmable power supplies (0…+5V , 0…-5V). The maximum available output current and power depend on the Analog Discovery 2 powering choice: 500mW total when powered through USB. (Each supply can provide between 0mW and 500mW so long as the total does not exceed 500mW.) 2.1W max for each supply when powered by an auxiliary supply. 700mA maximum current for each supply. The Analog Parts Kit contains a large selection of components perfect for creating a wide variety of useful circuits & devices. Featuring Analog Devices components, the kit includes transistors, resistors, capacitors, diodes, sensors, and variety of useful ICs including Op Amps, convertors, and regulators. Finally, the kit also comes with an assortment of lead wires, a solderless breadboard, and a screwdriver.

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

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 Quick introductions to necessary topics provided at the appropriate points on-demand during this course Note that a series of background lectures is available to provide some useful material.

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 Laboratory Objectives: Practical Introduction to analysis and design of electronic circuits

Modeling, analysis, and design – overview

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

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) 

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

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

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

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! !! You can assume (arbitrarily) either the voltage polarity or the current direction !! !Negative results tells that the assumption has to be reversed!

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

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

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

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!

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

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

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.

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

“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

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

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)

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

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

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

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

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

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

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

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)

Dependent Power Supplies – continued Examples:

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

KCL – Example 1 Write KCL at the node below:

KCL – Example 2 Use KCL to determine the current i

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

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

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

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

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.

Example 1 – continued

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

Example 2 – Alternate approach

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

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

Circuit Analysis Example 3 – continued

Lecture 4 Review: Circuit analysis examples KVL, KCL Circuit analysis examples Series, parallel circuit elements Related educational materials: Chapter 1.4, 1.5

Review: KVL & KCL KVL: algebraic sum of all voltage differences around any closed loop is zero KCL: algebraic sum of all currents entering a node is zero

Review: Circuit analysis General circuit analysis approach: Assign element voltages, currents according to passive sign convention Apply KVL, KCL, and voltage-current relations as necessary to solve for desired circuit parameters The general idea is to write as many equations as you have unknowns, and solve for the desired unknowns

Circuit analysis – example 1 For the circuit below, determine: vAC, vX, vDE, RX, and the power absorbed by the 2 resistor

Example 1 – continued

Circuit analysis – example 2 Determine the voltages across both resistors.

Example 2 – continued

Circuit analysis – example 3 We have a “dead” battery, which only provides 2V Second battery used to “charge” the dead battery – what is the current to the dead battery?

Non-ideal voltage source models Add a “source resistance” in series with an ideal voltage source We will define the term series formally later

Non-ideal current source models Add a “source resistance” in parallel with an ideal current source We will define the term parallel formally later

Example 3 – revisited Our battery charging example can now make sense Include internal (source resistances) in our model

Ideal sources can provide infinite power Connect a “load” to an ideal voltage source:

Non-ideal sources limit power delivery “Loaded” non-ideal voltage source

Non-ideal sources limit power delivery “Loaded” non-ideal current source

When are ideal source models “good enough”? Ideal and non-ideal voltage sources are the “same” if RLoad >> RS Ideal and non-ideal current sources are the “same” if RLoad << RS

Series and parallel circuit elements Circuit elements are in series if all elements carry the same current KCL at node “a” provides i1 = i2

Series and parallel circuit elements Circuit elements are in parallel if all elements have the same voltage difference KVL provides v1 = v2

Circuit reduction In some cases, series and parallel combinations of circuit elements can be combined into a single “equivalent” element This process reduces the overall number of unknowns in the circuit, thus simplifying the circuit analysis Fewer elements  fewer related voltages, currents The process is called circuit reduction

Lecture 5 Review: Circuit reduction Related educational materials: Series, parallel circuit elements Circuit reduction Related educational materials: Chapter 2.1, 2.2, 2.3

Review: series and parallel circuit elements Elements in series if they have the same current Elements in parallel have the same voltage

Circuit reduction Some circuit problems can be simplified by combining elements to reduce the number of elements Reducing the number of elements reduces the number of unknowns and thus the number of equations which must be written to determine these unknowns

Series circuit elements – example 1 Apply KCL at any node  all elements have the same current All of the above circuit elements are in series

Series element circuit reduction – example 1 KVL around the loop: -V1 + i·R1 + V2 + i·R2 + i·R3 – V3 + i·R4 = 0 (-V1 + V2– V3) + i(R1 + R2 + R3 + R4) = 0

Series circuit reduction Notes: Voltage sources in series add directly to form an equivalent voltage source Resistances in series add directly to form an equivalent resistance

Series circuit reduction – Example 2 Determine the power delivered by the 20V source

Voltage Division Series combination of N resistors:

Voltage Divider Formula Ratio of VK to the total voltage is the same as the ratio of RK to the total series resistance

Voltage Dividers – special case Voltage source in series with two resistors:

Voltage division – example 1 Determine the power dissipated by the 2 resistor

Voltage division – example 2 Determine the voltage V1 in the circuit below.

Parallel circuit elements – example 1 Apply KVL around any loop  all elements have the same voltage All of the above circuit elements are in parallel

Parallel element circuit reduction – example 1 KCL at upper node:

Parallel circuit reduction Notes: Current sources in parallel add directly to form an equivalent current source Definition: Conductance is the inverse of resistance Units are siemens or mhos (abbreviated S or ) Conductances in parallel add directly to form an equivalent conductance

Parallel element circuit example 1 – revisted

Parallel circuit reduction – Example 2 Determine the power delivered by the 2A source

Current Division Parallel combination of N resistors:

Current Divider Formula Ratio of iK to the total current is the same as the ratio of GK to the total parallel conductance

Current Divider – special case Current source in parallel with two resistors

Current division – example 1 Determine the current in the 2 resistor

Current division – example 2 Determine the value of R which makes i = 2mA

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

Circuit Reduction – example 1 Determine the current in the 2 resistor. (Note: we wrote the governing equations for this example in lecture 3.)

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

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

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

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

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

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

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

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

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

Example 3 – continued

Example 3 – continued

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

Example 4 – continued