5.6 Number Representation

• Value of binary number AnAn-1...A2A1A0 :

5.5 Simplifying Logic Expressions

• Rules

• De Morgan's law:

5.4 Standard Sum-of-Products Representation

• Logic expressions in the sum-of-products form are represented using two levels of operations as a set of product (AND) terms, each comprising one or more variables in their true forms (for example, A) or complement forms (e.g., A), combined using the OR function

• We can write a sum-of-products expression from a truth table representation by first writing a product term for each row in the truth table with a 1 in its output column, and then summing these product terms. Each product term comprises an AND function of all the input variables. A variable will appear in its true or complement form in a product term corresponding to a given row in the truth table depending on whether it appears as a 1 or a 0 in that row

5.3 Combinational Gates

• A combinational gate is an abstract representation of a circuit that satisfies two properties:

1. Its outputs are a function of its inputs alone.

2. It satisfies the static discipline.

5.2 Boolean Logic

5.1 Voltage Levels and Static Discipline

• To send a logical 0, the sender must produce an output voltage value that is less than VOL. Correspondingly, the receiver must interpret input voltages below VIL as a logical 0.
• Similarly, to send a logical 1, the sender must produce an output voltage value that is greater than VOH. Further, the receiver must interpret voltages above VIH as a logical 1.
• Noise Margin: The absolute value of the difference between the prescribed output voltage for a given logical value and the corresponding forbidden region voltage threshold for the receiver is called the noise margin for that logical value.
• NM0 = VIL − VOL
• NM1 = VOH − VIH
• The static discipline is a specification for digital devices. The static discipline requires devices to interpret correctly voltages that fall within the input thresholds (VIL and VIH). As long as valid inputs are provided to the devices, the discipline also requires the devices to produce valid output voltages that satisfy the output thresholds (VOL and VOH).

5.0 The Digital Abstraction

• Value discretization forms the basis of the digital abstraction
• Although the digital approach seems wasteful of signal dynamic range, it has a significant advantage over analog transmission in the presence of noise

4.5 Incremental Analysis

• Process of linearizing device models over a very narrow operating range is called incremental analysis
• Systematic procedure for finding incremental voltages and currents for a circuit with a nonlinear device characterized by the v-i relation iD = f(vD) :

4.4 Piecewise Linear Analysis

• Piecewise linear analysis represents the nonlinear v-i characteristics of each nonlinear element by a succession of straight-line segments, then make calculations within eachstraight-line segment using the linear analysis tools already developed

4.3 Graphical Analysis

• There are many nonlinear circuits that cannot be solved analytically. Usually we must resort to trial-and-error solutions on a computer. Such solutions provide answers, but usually give little insight about circuit performance and design. Graphical solutions, on the other hand, provide insight at the expense of accuracy

• For circuits with two nonlinear elements, the method is less useful, as it involves sketching one nonlinear characteristic on another. Nonetheless, crude sketches can still provide much insight
• Assuming E = 3 V, R = 500 ohm

4.2 Analytical Solutions

• Node method and its foundational Kirchhoff’s voltage and current laws are derived from Maxwell’s Equations with no assumptions about linearity.
• The superposition method, the Thévenin method, and the Norton method however do require a linearity assumption

4.1 Introduction to Nonlinear Elements

• Diode is a nonlinear device

• An analytical expression of diode

3.6 The Norton Equivalent Network

The Norton equivalent circuit for any linear network at a given pair of terminals consists of a current source iN in parallel with a resistor RN. The current iN and resistance RN can be obtained as follows:

1. iN can be found by applying a short at the designated terminal pair on the original network and calculating or measuring the current through the short circuit.

2. RN can be found in the same manner as RTH, that is, by calculating or measuring the resistance of the open-circuit network seen from the designated terminal pair with all independent sources internal to the network set to zero; that is, with voltage sources replaced with short circuits, and current sources replaced with open circuits.

3.5 Thevenin's Theorem

The Thévenin equivalent circuit for any linear network at a given pair of terminals consists of a voltage source vTH in series with a resistor RTH. The voltage vTH and resistance RTH can be obtained as follows:

1. vTH can be found by calculating or measuring the open-circuit voltage at the designated terminal pair on the original network.

2. RTH can be found by calculating or measuring the resistance of the open-circuit network seen from the designated terminal pair with all independent sources internal to the network set to zero. That is, with independent voltage sources replaced with short circuits, and independent current sources replaced with open circuits. (Dependent sources must be left intact, however.)

3.4 Superposition

Superposition principle states that for all linear systems:

The net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually.

Superposition method:

1. For each independent source, form a subcircuit with all other independent sources set to zero. Setting a voltage source to zero implies replacing the voltage source with a short circuit, and setting a current source to zero implies replacing the current source with an open circuit.

2. From each subcircuit corresponding to a given independent source, find the response to that independent source acting alone. This step results in a set of individual responses.

3. Obtain the total response by summing together each of the individual responses.

3.3 Node Method

Steps of the node method can be written as:

1. Select a reference node, called ground, from which all other voltages will be measured. Define its potential to be 0 V.

2. Label the potentials of the remaining nodes with respect to the ground node. Any node connected to the ground node through either an independent or a dependent voltage source should be labeled with the voltage of that source. The voltages of the remaining nodes are the primary unknowns and should be labeled accordingly. Since there are generally far fewer nodes than branches in a circuit, there will be far fewer primary unknowns to determine in a node analysis.

3. Write KCL for each of the nodes that has an unknown node voltage (in other words, the ground node and nodes with voltage sources connected to ground are excluded), using KVL and element laws to obtain the currents directly in terms of the node voltage differences and element parameters. Thus, one equation is written for each unknown node voltage.

4. Solve the equations resulting from Step 3 for the unknown node voltages. This is the most difficult step in the analysis.

5. Back-solve for the branch voltages and currents. More specifically, use node voltages and KVL to determine branch voltages as desired. Then, use the branch voltages, the element laws, and KCL to determine the branch currents, again as desired.

3.2 The Node Voltage

• A node voltage is the potential difference between the given node and some other node that has been chosen as a reference node. The reference node is called the ground.

3.1 Network Theorems

• The basic network analysis method is fundamental but unfortunately often insufficient. The problem is that frequently we deal with complicated circuits in which we are interested in relating only one output variable to one input variable.
• There exist better approaches to the organization of circuit analysis which greatly simplify circuit analysis and provide substantial insight about how circuits behave, ie Node Method

2.6 Dependent Sources and The Control Concept

• Voltage-controlled current source:

• Linear voltage-controlled current source: iout = g*vin
• g is a constant coefficient called transconductance with unit of conductance
• Current-controlled current source:

• Linear current-controlled current source: (coefficient = current transfer ratio)
• Linear voltage-controlled voltage source: (coefficient = voltage transfer ratio)
• Linear current-controlled voltage source: vout = r*iIN (r = transresistance)
• VCCS, CCCS, VCVS, CCVS:

2.4 Series and Parallel Simplication

• Voltage Divider: v2 = (R2/(R1 + R2))*V
• Resistors in series: Rs = R1 + R2
• Current Divider: i2 = (G2/(G1 + G2))*I
• Resistors in parallel: Rp = R1*R2/(R1+R2)

2.3 Circuit Analysis: Basic Method

Basic method of circuit analysis:

1. Define each branch current and voltage in the circuit (follow associated variables convention)
2. Assemble the element laws for the elements
3. Apply Kirchhoff’s current and voltage laws
4. Jointly solve the equations assembled in Steps 2 and 3 for the branch variables defined in Step 1

Summary:

1. Define voltages and currents for each element
2. Write KVL
3. Write KCL
4. Write constituent relations
5. Solve 

2.2 Kirchhoff's Laws

• Kirchhoff’s current law (KCL) - The current flowing out of any node in a circuit must equal the current flowing in. That is, the algebraic sum of all branch currents flowing into any node must be zero.
• Kirchhoff’s voltage law (KVL) - The algebraic sum of the branch voltages around any closed path in a network must be zero.
• Voltages across two parallel connected elements must be the same.

2.1 Terminology

• The junction points at which the terminals of two or more elements are connected are referred to as the nodes of a circuit
• The connections between the nodes are referred to as the edges or branches of a circuit
• Circuit loops are defined to be closed paths through a circuit along its branches

• Branch current is the current along a branch of the circuit
• Branch voltage is the potential difference measured across a branch

2.0 Resistive Networks

• Solving or analyzing a circuit generally involves finding the voltage across, and current through, each of the circuit elements
• When circuit obeys the lumped matter discipline, Maxwell's Equations can be simplified into two algebraic relationships stated as Kirchhoff's voltage & current law (KVL & KCL)

1.8 Signal Representation

• Signals in the physical world are most commonly analog
• Value discretization forms the basis of the digital abstraction, which yields a number of advantages such as better noise immunity compared to an analog signal representation.

1.7 Modeling Physical Elements

• Resistor self-heating, with the associated change in value, prompts manufacturers to provide power ratings for resistors, to indicate maximum power dissipation (pmax) without significant value change or burnout.

• Battery terminal voltage expression: vt = V + iR

1.6 Ideal Two-Terminal Elements

• An ideal voltage source is a device that maintains a constant voltage at its terminals regardless of the amount of current drawn from those terminals.
• Two type of voltage source: independent & dependent
• An ideal conductor is a element in which any amount of current can flow without loss of voltage or power.
• An ideal linear resistor obeys Ohm's law
• Conductance (G) is the reciprocal resistance:

G = 1 / R

• The element law for a independent voltage source supplying a voltage V:

v = V

• The element law for ideal wire (short circuit):

v = 0

• The element law for open circuit:

i = 0

• The element law for a current source supply a current I:

i = I

1.5 Practical Two-Terminal Elements

• Power delivered by battery is the product of voltage and current : p = VI
• If a constant amount of power p is delivered over an interval T, the energy w supplied is : w = pT
• Ohm's law : v = iR
• An open circuit is an element through which no current flows, regardless of its terminal voltage
• A short circuit is an element across which no voltage can appear regardless of the current through it
• A two-terminal resistor is any two-terminal element that has an algebraic relation between its instantaneous terminal current and its instantaneous terminal voltage
• Associated Variables Convention define current to flow in at the device terminal assigned to be positive in voltage

1.4 Limitations of the lumped circuit abstraction

• Third postulate of lumped matter discipline requires signal speed to be significantly lower than the speed of electromagnetic waves
• Electromagnetic waves travel at 15 cm per nanosecond in insulator with 4x dielectric constant of vacuum
• Electromagnetic waves propagation delay across a 1-cm chip is about 1/15 ns
• When signal speeds are comparable to speed of electromagnetic waves, lumped matter discipline is violated, and therefore cannot use lumped circuit abstraction. ( resolve through distributed circuit model)
• Capacitive & inductive effects on lumped elements (resistors, wires) resulting from electric fields and magnetic fluxes generated by high frequency oscillator will violate lumped matter discipline ( resolve through separate these effects into new lumped element-capacitor & inductor)

1.3 The Lumped Matter Discipline

• Lumped matter discipline (constraints) provides the foundation for lumped circuit abstraction.
• Lumped matter discipline imposes three constraints on how we choose lumped circuit elements:

1. The rate of change of magnetic flux linked with any portion of the circuit must be zero at all time (allowed unique voltage across the terminals of an element)
2. The rate of change of the charge at any node in the circuit must be zero for all time. A node is any point in the circuit at which two or more element terminals are connected using wires. (allowed unique current across the terminals of an element)
3. The signal timescales must be much larger than the propagation delay of electromagnetic waves through the circuit

1.2 The Lumped Circuit Abstraction

• Capped a set of lumped elements that obey the lumped matter discipline using ideal wires to form an assembly that performs a specific function results in the Lumped Circuit Abstraction

1.1 The Power of Abstraction

• Engineering is purposeful use of Science 
• The set of abstraction are derived through the discretization (lumping) discipline