Switching Theory

1. Definition and Scope of Switching Theory

Definition and Scope of Switching Theory

Switching theory is a mathematical framework for analyzing and designing systems where discrete state changes govern behavior. It underpins digital logic design, telecommunications, and control systems by formalizing the conditions under which a system transitions between distinct states. Unlike continuous systems, switching systems operate on quantized inputs and outputs, making Boolean algebra and finite automata theory fundamental tools.

Core Principles

The theory models switching phenomena using binary variables, where a signal x(t) takes values from a finite set, typically {0, 1}. A combinational switching system’s output y(t) depends solely on current inputs, expressed as:

$$ y(t) = f(x_1(t), x_2(t), \dots, x_n(t)) $$

Sequential systems incorporate memory, with outputs depending on input history. Their behavior is captured by state equations:

$$ \begin{aligned} s(t+1) &= g(s(t), x(t)) \\ y(t) &= h(s(t), x(t)) \end{aligned} $$

Historical Context

Claude Shannon’s 1937 thesis established the link between Boolean algebra and relay circuits, enabling systematic design of digital systems. This work extended George Boole’s symbolic logic to physical implementations, forming the basis for modern computer architecture.

Practical Applications

Modern Extensions

Contemporary research addresses:

Inputs Outputs Switching function f(x)
Combinational vs Sequential Switching Systems A side-by-side comparison of combinational and sequential switching systems, showing inputs, switching functions, outputs, and state memory in sequential systems. Combinational vs Sequential Switching Systems Inputs x(t) Switching Function y(t) = f(x) Outputs y(t) Inputs x(t) Switching Function y(t) = h(s,x) Outputs y(t) State Memory s(t+1) = g(s,x) Combinational Sequential
Diagram Description: The diagram would physically show the relationship between inputs, switching function, and outputs in a combinational system, contrasting it with a sequential system's state memory.

Binary Logic and Boolean Algebra Basics

Fundamentals of Binary Logic

Binary logic operates on two discrete values: 0 (false) and 1 (true). These values correspond to voltage levels in digital circuits, where 0 typically represents a low voltage (e.g., 0V) and 1 represents a high voltage (e.g., 5V or 3.3V in modern systems). The behavior of binary logic is governed by Boolean algebra, a mathematical framework introduced by George Boole in 1854.

The three basic operations in Boolean algebra are:

These operations form the foundation for constructing more complex logic functions, such as NAND, NOR, XOR, and XNOR, which are widely used in digital circuit design.

Boolean Algebraic Identities

Boolean algebra follows a set of axioms and identities that simplify logical expressions. Some fundamental identities include:

$$ \text{Commutative Law: } A + B = B + A \quad \text{and} \quad A \cdot B = B \cdot A $$
$$ \text{Associative Law: } (A + B) + C = A + (B + C) \quad \text{and} \quad (A \cdot B) \cdot C = A \cdot (B \cdot C) $$
$$ \text{Distributive Law: } A \cdot (B + C) = (A \cdot B) + (A \cdot C) \quad \text{and} \quad A + (B \cdot C) = (A + B) \cdot (A + C) $$
$$ \text{De Morgan's Theorems: } \overline{A + B} = \overline{A} \cdot \overline{B} \quad \text{and} \quad \overline{A \cdot B} = \overline{A} + \overline{B} $$

These identities are crucial for optimizing digital circuits, reducing gate counts, and minimizing power consumption in VLSI design.

Logic Gates and Truth Tables

Logic gates are physical implementations of Boolean operations. Each gate has a corresponding truth table that enumerates all possible input combinations and their outputs. For example, the AND gate's truth table is:

A B A · B
0 0 0
0 1 0
1 0 0
1 1 1

Universal gates like NAND and NOR are particularly significant because any Boolean function can be implemented using only one type of these gates, a principle leveraged in transistor-level design.

Canonical Forms: SOP and POS

Boolean functions can be expressed in two canonical forms:

  1. Sum of Products (SOP): A logical OR of multiple AND terms (minterms).
  2. Product of Sums (POS): A logical AND of multiple OR terms (maxterms).

For example, the function \( F(A,B,C) = \overline{A}BC + A\overline{B}C + AB\overline{C} \) is in SOP form. These forms are essential for algorithmic simplification techniques like the Quine-McCluskey method and Karnaugh maps.

Practical Applications

Boolean algebra is the backbone of digital systems, from simple combinational circuits to complex microprocessors. In hardware description languages (HDLs) like VHDL or Verilog, Boolean expressions directly translate to gate-level implementations. Modern EDA tools use Boolean optimization to enhance performance and reduce chip area in ASIC and FPGA designs.

Error-correcting codes, cryptographic algorithms, and even quantum computing frameworks extend these principles to higher abstraction levels, demonstrating the enduring relevance of Boolean logic in cutting-edge technologies.

Logic Gates and Truth Tables A diagram showing IEEE standard symbols for AND, OR, and NOT gates alongside their corresponding truth tables. AND Gate A B Q AND Truth Table A B Q 0 0 0 1 0 0 0 1 1 1 OR Gate A B Q OR Truth Table A B Q 0 0 0 1 0 1 0 1 1 1 NOT Gate A Q NOT Truth Table A Q 0 1 1 0
Diagram Description: The section covers logic gates and truth tables, which are inherently visual concepts. A diagram would show the physical symbols of basic logic gates (AND, OR, NOT) alongside their truth tables.

1.3 Logic Gates and Their Functions

Fundamental Logic Gates

Logic gates are the building blocks of digital circuits, implementing Boolean algebra operations. The seven fundamental gates are:

Mathematical Representation

Each gate corresponds to a Boolean function. For two inputs A and B:

$$ \text{AND: } Y = A \cdot B $$
$$ \text{OR: } Y = A + B $$
$$ \text{XOR: } Y = A \oplus B = A\overline{B} + \overline{A}B $$

Truth Tables and Universality

The NAND and NOR gates are universal, meaning any Boolean function can be implemented using only NAND or only NOR gates. This property is crucial for CMOS technology, where NAND/NOR implementations are more efficient.

A B NAND NOR
0 0 1 1
0 1 1 0
1 0 1 0
1 1 0 0

Propagation Delay and Power Dissipation

The switching characteristics of gates are defined by:

$$ t_{pd} = \frac{t_{PHL} + t_{PLH}}{2} $$

where tPHL is the high-to-low delay and tPLH is the low-to-high delay. Dynamic power consumption is given by:

$$ P_{dyn} = \alpha C_L V_{DD}^2 f $$

where α is the activity factor, CL is load capacitance, and f is switching frequency.

CMOS Implementation

Modern logic gates are primarily implemented using complementary MOSFET pairs. A CMOS NAND gate consists of:

The noise margins are calculated as:

$$ NM_H = V_{OH} - V_{IH} $$
$$ NM_L = V_{IL} - V_{OL} $$

Advanced Logic Families

Beyond static CMOS, several logic families exist for specialized applications:

CMOS NAND Gate Implementation and Truth Tables A schematic diagram of a CMOS NAND gate with labeled PMOS and NMOS transistors, input/output signals, and truth tables for fundamental logic gates. VDD PMOS1 (A) PMOS2 (B) NMOS1 (A) NMOS2 (B) GND A B Y NAND Gate A B Y 0 0 1 0 1 1 1 0 1 1 1 0 Other Fundamental Gates AND: 0/0/0/1 OR: 0/1/1/1 NOR: 1/0/0/0 XOR: 0/1/1/0 XNOR: 1/0/0/1 NOT: 1/0 CMOS NAND Gate Implementation and Truth Tables
Diagram Description: The section covers CMOS implementation and logic gate operations, which are highly visual concepts requiring transistor-level schematics and truth table visualizations.

2. Design Principles of Combinational Circuits

Design Principles of Combinational Circuits

Fundamental Concepts

Combinational circuits are digital logic circuits where the output depends solely on the current input values, with no dependence on previous inputs or internal state. These circuits are constructed from basic logic gates—AND, OR, NOT, NAND, NOR, XOR, and XNOR—arranged to implement Boolean functions. The absence of memory elements distinguishes combinational circuits from sequential circuits.

The behavior of a combinational circuit with n inputs and m outputs can be fully described by m Boolean functions of the form:

$$ F_i(x_1, x_2, ..., x_n) \quad \text{for} \quad i = 1, 2, ..., m $$

Canonical Forms

Two standard representations exist for Boolean functions in combinational circuit design:

For example, the Boolean function F(A,B,C) = Σ(1,3,5,7) in SOP form is:

$$ F = \overline{A}\overline{B}C + \overline{A}BC + A\overline{B}C + ABC $$

Logic Minimization

Minimizing Boolean functions reduces circuit complexity, power consumption, and propagation delay. The Karnaugh map (K-map) provides a graphical method for simplification of functions with up to 6 variables. For larger functions, the Quine-McCluskey algorithm offers a systematic tabular approach.

Consider the K-map for a 4-variable function:

Technology Mapping

After minimization, the Boolean expressions must be implemented using available logic gates. Different technologies (TTL, CMOS, FPGA LUTs) have varying cost functions for gate implementations. Common approaches include:

Timing Considerations

The propagation delay tpd of a combinational circuit is the maximum time from when any input changes until all outputs have settled to their final values. For a circuit with k levels of logic gates, each with delay tgate:

$$ t_{pd} = \sum_{i=1}^{k} t_{gate_i} $$

Hazards—temporary output errors due to unequal propagation delays—must be identified and eliminated through hazard covers in the Boolean expression or careful timing analysis.

Practical Design Methodology

The systematic design of combinational circuits follows these steps:

  1. Problem specification and truth table derivation
  2. Boolean function extraction
  3. Logic minimization using K-maps or algorithmic methods
  4. Technology mapping to available gates
  5. Timing analysis and hazard elimination
  6. Verification through simulation or formal methods

Advanced Optimization Techniques

Modern combinational circuit design employs:

$$ \text{Cost function} = \alpha \cdot \text{gate count} + \beta \cdot \text{level count} + \gamma \cdot \text{fan-out} $$
4-Variable Karnaugh Map with Minterm Grouping A 4x4 Karnaugh Map showing minterm groupings with input variables A,B (vertical) and C,D (horizontal) in Gray code order. Highlighted groups show adjacent 1s with corresponding simplified Boolean terms. 4-Variable Karnaugh Map with Minterm Grouping 00 01 11 10 CD → 00 01 11 10 AB ↓ 0 1 3 2 4 5 7 6 12 13 15 14 8 9 11 10 1 1 0 1 0 1 1 0 1 0 1 1 1 0 0 1 Group 1: A'C' Group 2: BD Group 3: AD' Group 4: B'D' Legend A'C' BD AD' B'D'
Diagram Description: The K-map illustration is incomplete in the SVG placeholder and is a highly visual/spatial concept that requires proper demonstration of cell grouping and minterm adjacency.

2.2 Multiplexers and Demultiplexers

Functional Principles

A multiplexer (MUX) is a combinational circuit that selects one of N input lines and routes it to a single output line based on a set of control signals. For an n-bit selector, the MUX can handle 2n inputs. The Boolean function for a 2:1 MUX is derived as:

$$ Y = S \cdot D_0 + \overline{S} \cdot D_1 $$

where S is the selector, and D0, D1 are the data inputs. For a 4:1 MUX, the logic expands to:

$$ Y = \overline{S_0} \overline{S_1} D_0 + \overline{S_0} S_1 D_1 + S_0 \overline{S_1} D_2 + S_0 S_1 D_3 $$

Demultiplexers: Reverse Operation

A demultiplexer (DEMUX) performs the inverse function, directing a single input to one of 2n outputs. Its operation mirrors a decoder with an enable input repurposed as the data line. The output logic for a 1:4 DEMUX is:

$$ Y_i = D \cdot \text{Decoder}(S_0, S_1) $$

Implementation and Applications

MUX/DEMUX ICs (e.g., 74HC151, 74HC154) are widely used in:

Cascading and High-Density Designs

Larger MUXes are constructed by cascading smaller units. A 16:1 MUX can be built using five 4:1 MUXes, with the first level handling input grouping and the final stage performing selection. Propagation delay scales as:

$$ t_{pd,\text{total}} = t_{pd,\text{stage1}} + t_{pd,\text{stage2}} $$
MUX 4:1 MUX 4:1

Time-Division Multiplexing Case Study

In TDM systems, MUXes interleave signals by assigning fixed time slots. For N channels with bandwidth B, the aggregate rate must satisfy:

$$ R \geq N \times B $$

Clock synchronization is critical; a 1% skew can cause a 10-bit drift in 1000 cycles.

Cascaded 4:1 MUXes forming a 16:1 MUX Block diagram showing four 4:1 multiplexers at the input stage feeding into a single 4:1 multiplexer at the output stage, forming a 16:1 multiplexer. Includes selector lines, data paths, and propagation delay annotations. 4:1 MUX 4:1 MUX 4:1 MUX 4:1 MUX 4:1 MUX D0 D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13 D14 D15 S0 S1 S2 S3 t_pd t_pd 2×t_pd
Diagram Description: The section describes cascading MUXes and their propagation delays, which requires visualizing hierarchical connections and timing relationships.

2.3 Encoders and Decoders

Fundamental Concepts

Encoders and decoders are combinational logic circuits that convert between different digital representations. An encoder compresses multiple input lines into a smaller set of output lines, while a decoder expands a coded input into a larger set of output lines. These circuits are fundamental in digital systems for address decoding, data multiplexing, and error correction.

Binary Encoders

A binary encoder converts 2n input lines into n output lines. The simplest form is the 4-to-2 encoder, which has four inputs (only one active at a time) and two outputs representing the binary code of the active input. The truth table for a 4-to-2 encoder is:

$$ \begin{array}{|c|c|c|c|c|c|} \hline D_3 & D_2 & D_1 & D_0 & Y_1 & Y_0 \\ \hline 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 & 1 \\ \hline \end{array} $$

The Boolean expressions for the outputs are:

$$ Y_1 = D_3 + D_2 $$ $$ Y_0 = D_3 + D_1 $$

Priority Encoders

Standard encoders fail when multiple inputs are active simultaneously. A priority encoder resolves this by assigning priority to the highest-order active input. The 8-to-3 priority encoder (74LS148) includes an additional output (GS, Group Select) that indicates when any input is active.

Binary Decoders

A binary decoder performs the inverse operation of an encoder. An n-to-2n decoder activates one of 2n output lines based on the n-bit input code. The 3-to-8 decoder (74LS138) is widely used in memory address decoding, where each output line selects a specific memory chip.

$$ \text{Output}_i = \text{Enable} \cdot \overline{A_2} \cdot \overline{A_1} \cdot \overline{A_0} \quad \text{(for } i=0 \text{)} $$

Applications in Digital Systems

Advanced Variants

Modern systems use error-correcting encoders/decoders (e.g., Hamming code circuits) to detect and correct bit errors in communication channels. These circuits add parity bits during encoding and perform syndrome calculation during decoding to identify erroneous bits.

Encoder Decoder Coded Data
4-to-2 Encoder and 3-to-8 Decoder Functional Diagrams Block diagrams showing a 4-to-2 priority encoder with OR gates and a 3-to-8 decoder with binary input selection. 4-to-2 Encoder D0 D1 D2 D3 Y0 Y1 GS 3-to-8 Decoder A0 A1 A2 EN O0 O1 O2 O3 O4 O5 O6 O7 Binary Input: A2 A1 A0 Outputs: O0-O7 (One-hot)
Diagram Description: The section describes spatial relationships between encoder/decoder inputs/outputs and their binary conversions, which are more intuitively understood visually.

Adders and Subtractors

Binary Adders

Binary adders form the backbone of arithmetic logic units (ALUs) in digital systems. The simplest adder, the half-adder, performs addition of two single-bit binary numbers, producing a sum (S) and a carry (C). Its truth table and Boolean expressions are:

$$ S = A \oplus B $$ $$ C = A \cdot B $$

A full-adder extends this by incorporating a carry-in (Cin), enabling multi-bit addition. The sum and carry-out (Cout) are derived as:

$$ S = A \oplus B \oplus C_{in} $$ $$ C_{out} = (A \cdot B) + (C_{in} \cdot (A \oplus B)) $$

Cascading n full-adders creates an n-bit ripple-carry adder, where the carry propagates sequentially. While simple, this design suffers from latency due to carry propagation delays, leading to the development of faster architectures like carry-lookahead adders.

Carry-Lookahead Adders (CLA)

CLAs mitigate ripple-carry delays by computing carry signals in parallel. The key innovation lies in generating propagate (P) and generate (G) signals:

$$ P_i = A_i \oplus B_i $$ $$ G_i = A_i \cdot B_i $$ $$ C_{i+1} = G_i + (P_i \cdot C_i) $$

By expanding these recursively, carries for all bits are computed simultaneously. A 4-bit CLA, for instance, calculates C4 as:

$$ C_4 = G_3 + P_3G_2 + P_3P_2G_1 + P_3P_2P_1G_0 + P_3P_2P_1P_0C_0 $$

This approach reduces delay from O(n) to O(log n) at the cost of increased hardware complexity.

Binary Subtractors

Subtraction is implemented using two’s complement arithmetic, where the subtrahend is inverted and a ‘1’ is added to the least significant bit (LSB). A full-subtractor mirrors the full-adder but computes a borrow (Bout) instead of a carry:

$$ D = A \oplus B \oplus B_{in} $$ $$ B_{out} = \overline{A} \cdot B + B_{in} \cdot (\overline{A \oplus B}) $$

In practice, a single adder-subtractor circuit suffices by toggling between addition and subtraction modes using a control signal (SUB). When SUB=1, the subtrahend is inverted, and Cin=1 triggers the two’s complement operation.

Parallel Adders and Subtractors

Modern processors employ parallel prefix adders (e.g., Kogge-Stone, Brent-Kung) for high-speed arithmetic. These architectures use tree-like structures to compute carry signals in logarithmic time. For example, the Kogge-Stone adder achieves O(log n) delay with O(n log n) area, making it ideal for high-performance designs.

4-bit Carry-Lookahead Adder Block Diagram

Applications range from low-power embedded systems (using ripple-carry) to supercomputers (employing parallel prefix adders). Recent advancements explore hybrid designs balancing speed, power, and area trade-offs.

4-bit Carry-Lookahead Adder Architecture Block diagram of a 4-bit Carry-Lookahead Adder showing full-adders, propagate/generate logic, and carry computation tree with labeled inputs/outputs. FA0 FA1 FA2 FA3 A0 A1 A2 A3 B0 B1 B2 B3 S0 S1 S2 S3 P0,G0 P1,G1 P2,G2 P3,G3 C0 C4 4-bit Carry-Lookahead Adder Architecture
Diagram Description: The section explains complex adder architectures (ripple-carry, CLA) and their carry propagation logic, which would benefit from a visual representation of signal flow and parallel computation paths.

3. Flip-Flops and Latches

Flip-Flops and Latches

Fundamental Concepts

Flip-flops and latches are bistable multivibrators, capable of storing a single bit of data (0 or 1). They form the backbone of sequential logic circuits, where output depends not only on current inputs but also on previous states. While latches are level-sensitive, flip-flops are edge-triggered, making them more suitable for synchronous systems.

SR Latch: The Basic Building Block

The Set-Reset (SR) latch is the simplest form of a bistable memory element. It consists of two cross-coupled NOR or NAND gates. The state equations for an SR latch with NOR gates are:

$$ Q = \overline{R + \overline{Q_{prev}}}} $$ $$ \overline{Q} = \overline{S + Q_{prev}} $$

Where S (Set) and R (Reset) are active-high inputs. The forbidden state (S = R = 1) must be avoided as it leads to metastability.

D Latch: Eliminating the Forbidden State

The D (Data) latch solves the SR latch's ambiguity by using a single data input (D) and an enable (E) signal. When E = 1, the output Q follows D; when E = 0, the last state is retained. The characteristic equation is:

$$ Q_{next} = E \cdot D + \overline{E} \cdot Q_{current} $$

Edge-Triggered Flip-Flops

Flip-flops introduce clocked operation, responding only to transitions (rising or falling edges) rather than levels. The D flip-flop is the most widely used, with its output updating only at the clock edge:

$$ Q(t+1) = D(t) \cdot \text{CLK\_edge} $$

Master-Slave Configuration

Early edge-triggered designs used master-slave pairs to isolate input and output. The master latch captures the input when the clock is high, while the slave updates the output when the clock falls. Modern implementations use transmission gates for glitch-free operation.

JK Flip-Flop: Versatility with Toggle Mode

The JK flip-flop combines SR and T (toggle) functionalities. Its truth table includes:

The characteristic equation is derived using a Karnaugh map:

$$ Q_{next} = J\overline{Q} + \overline{K}Q $$

Metastability and Timing Constraints

When setup or hold times are violated, flip-flops may enter a metastable state, where the output oscillates or settles unpredictably. The mean time between failures (MTBF) is modeled as:

$$ \text{MTBF} = \frac{e^{t_r/\tau}}{f_{clk} \cdot f_{data} \cdot t_0} $$

Where tr is the resolution time, Ï„ is the time constant of the flip-flop, and t0 is a process-dependent parameter. Synchronizers (cascaded flip-flops) mitigate this risk in cross-clock-domain designs.

Practical Applications

Flip-flops are ubiquitous in:

Modern FPGAs and ASICs use flip-flops as the primary storage element, with optimized cell libraries for power-performance tradeoffs.

SR Latch and D Flip-Flop Timing Diagrams A diagram showing the SR latch schematic with NOR gates and feedback paths on the left, and D flip-flop timing waveforms on the right, including clock, D input, Q output, and setup/hold time markers. NOR NOR S R Q QÌ… CLK D Q setup hold propagation delay SR Latch D Flip-Flop Timing
Diagram Description: The SR latch's cross-coupled gate structure and flip-flop timing diagrams are inherently spatial concepts that require visual representation.

3.2 Counters and Registers

Synchronous vs. Asynchronous Counters

Counters are sequential circuits that progress through a predefined sequence of states upon each clock pulse. Synchronous counters update all flip-flops simultaneously using a common clock signal, minimizing propagation delay skew. The state transition logic for an n-bit synchronous binary counter follows:

$$ Q_i^{t+1} = Q_i^t \oplus \left( \bigwedge_{j=0}^{i-1} Q_j^t \right) $$

where Qi represents the i-th bit. In contrast, asynchronous (ripple) counters cascade flip-flops with each output triggering the next stage, introducing cumulative propagation delays. Their maximum operating frequency is constrained by:

$$ f_{max} = \frac{1}{n \cdot t_{pd}} $$

with tpd as the flip-flop delay. Synchronous designs dominate high-speed applications like DDR memory interfaces, while ripple counters persist in low-power scenarios such as frequency dividers.

Ring and Johnson Counters

Ring counters form circular shift registers where only one bit is active (e.g., 0001 → 0010 → 0100 → 1000). They provide natural n-state decoding without additional logic. A Johnson counter (twisted ring counter) feeds the inverted output back to the input, generating 2n states for an n-stage register. The state sequence for a 3-bit Johnson counter is:

These find use in quadrature phase generation and industrial sequence controllers.

Shift Register Architectures

Shift registers serialize/deserialize data through cascaded D-flip-flops. Key configurations include:

Metastability risks in high-speed serial interfaces necessitate synchronizer chains with MTBF given by:

$$ MTBF = \frac{e^{t_r/\tau}}{f_{clk} \cdot f_{data}} $$

where tr is resolution time and Ï„ the flip-flop time constant.

Applications in Modern Systems

Linear Feedback Shift Registers (LFSRs) generate pseudo-random sequences for cryptography and built-in self-test (BIST). A maximal-length n-stage LFSR has period 2n - 1, achieved when feedback taps correspond to primitive polynomials. For example, the polynomial:

$$ x^4 + x^3 + 1 $$

yields a 15-cycle sequence. In processor design, circular buffers built from registers enable zero-overhead looping in digital signal processors, while hardware FIFOs use dual-clock registers for asynchronous domain crossing.

D0 D1 D2 CLK
Counter Timing and State Transition Diagrams A side-by-side comparison of synchronous and asynchronous counter timing diagrams, with state transition graphs for ring and Johnson counters below. Counter Timing and State Transition Diagrams Asynchronous Counter Synchronous Counter CLK Q0 Q1 Q2 tpd Q0 Q1 Q2 Ring Counter State Transitions Johnson Counter State Transitions 000 100 110 111 000 100 110 111
Diagram Description: The section covers sequential circuit behaviors and timing relationships that are inherently visual, particularly the difference between synchronous and asynchronous counters and the state transitions in ring/Johnson counters.

3.3 Finite State Machines

Definition and Mathematical Representation

A Finite State Machine (FSM) is a computational model defined by a finite set of states, transitions between those states, and actions triggered by inputs. Formally, an FSM is a 5-tuple:

$$ M = (Q, \Sigma, \delta, q_0, F) $$

For Mealy machines, the output depends on both the state and input, whereas Moore machines produce outputs based solely on the current state.

State Transition Diagrams and Tables

FSMs are often visualized using state transition diagrams, where nodes represent states and edges denote transitions labeled with input/output pairs. Alternatively, a state transition table provides a matrix representation of \( \delta \).

Consider a simple vending machine FSM:

Idle Dispense Coin Inserted Item Selected

Deterministic vs. Non-Deterministic FSMs

A Deterministic FSM (DFSM) has exactly one transition per input-state pair. In contrast, a Non-Deterministic FSM (NFSM) may have multiple possible transitions for a single input, requiring backtracking or parallel exploration.

$$ \text{DFSM: } \delta(q, a) = q' $$ $$ \text{NFSM: } \delta(q, a) = \{q_1, q_2, \dots\} $$

Practical Applications

Optimization Techniques

State minimization reduces redundant states while preserving behavior. The Hopcroft algorithm partitions states into equivalence classes for optimal representation:

$$ P_{k+1} = \{ B \in P_k | \forall a \in \Sigma, B \text{ splits w.r.t. } a \} $$

Hardware Implementation: FSMs are realized using flip-flops (state storage) and combinational logic (transition function). The number of flip-flops \( n \) required is:

$$ n = \lceil \log_2 |Q| \rceil $$
Vending Machine State Transition Diagram A state transition diagram for a vending machine FSM, showing states as circular nodes and transitions as labeled directional arrows. Idle Dispense Coin Inserted Item Selected
Diagram Description: The section includes a state transition diagram for a vending machine FSM, which visually shows states as nodes and transitions as labeled edges.

4. Switching Devices: Diodes and Transistors

4.1 Switching Devices: Diodes and Transistors

Semiconductor Diodes as Switching Elements

The diode is the simplest semiconductor switching device, operating based on the principles of p-n junction behavior. Under forward bias (VF > Vth), the diode conducts with minimal resistance, while under reverse bias, it blocks current flow until breakdown occurs. The switching characteristics are governed by:

$$ I_D = I_S \left( e^{\frac{V_D}{nV_T}} - 1 \right) $$

where IS is the reverse saturation current, n is the ideality factor (1-2), and VT = kT/q is the thermal voltage. Practical switching diodes exhibit non-ideal behaviors:

Bipolar Junction Transistors (BJTs) in Switching Applications

BJTs operate as current-controlled switches through three distinct regions:

  1. Cutoff: Both junctions reverse-biased (IB = 0, IC ≈ 0)
  2. Active: Base-emitter forward biased, base-collector reverse biased (IC = βIB)
  3. Saturation: Both junctions forward-biased (VCE ≈ 0.2V)

The switching speed is limited by:

$$ t_{on} = t_d + t_r $$ $$ t_{off} = t_s + t_f $$

where td is delay time, tr rise time, ts storage time, and tf fall time. The Ebers-Moll model describes large-signal behavior:

$$ I_C = I_S \left( e^{\frac{V_{BE}}{V_T}} - e^{\frac{V_{BC}}{V_T}} \right) - \frac{I_S}{\beta_R} \left( e^{\frac{V_{BC}}{V_T}} - 1 \right) $$

Field-Effect Transistors (FETs) as Switches

FETs provide voltage-controlled switching with superior high-frequency performance compared to BJTs. The drain current in saturation (for MOSFETs) follows:

$$ I_D = \frac{\mu_n C_{ox}}{2} \frac{W}{L} (V_{GS} - V_{th})^2 (1 + \lambda V_{DS}) $$

Key switching parameters include:

  • Threshold voltage (Vth): Minimum gate voltage for channel formation
  • Transconductance (gm):
    $$ g_m = \frac{\partial I_D}{\partial V_{GS}} = \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th}) $$
  • Figure of Merit (FOM):
    $$ \text{FOM} = R_{DS(on)} \times Q_g $$

Comparative Switching Characteristics

Parameter BJT MOSFET
Control Mechanism Current Voltage
Switching Speed Moderate (ns) Fast (ps-ns)
Input Impedance Low Very High
Conduction Loss VCE(sat) I2RDS(on)

Practical Considerations in Switching Circuits

Real-world implementations must account for:

  • Parasitic elements: Package inductances (Ls, Ld), junction capacitances
  • Thermal effects: RθJC and derating curves
  • Safe Operating Area (SOA): Boundaries defined by current, voltage, and power limits

For power applications, the switching loss is calculated as:

$$ P_{sw} = \frac{1}{2} V_{DS} I_D (t_r + t_f) f_{sw} $$

where fsw is the switching frequency. Modern devices like SiC MOSFETs and GaN HEMTs achieve superior performance through wider bandgap materials:

$$ R_{on,sp} \propto \frac{1}{\mu_n \epsilon_s E_c^3} $$

where Ec is the critical electric field and μn is electron mobility.

Switching Characteristics of Diodes, BJTs, and FETs Comparative schematic showing I-V curves of diodes, operating regions of BJTs, transfer characteristics of FETs, and switching timing diagrams. I V Forward Bias Reverse Bias Diode I-V Curve Ic Vce BJT Operating Regions Cutoff Active Saturation Id Vgs FET Transfer Curve Vth RDS(on) V/I Time td tr ts tf trr Switching Waveforms Input Output Switching Characteristics of Diodes, BJTs, and FETs
Diagram Description: The section covers complex switching behaviors and comparisons between devices that would benefit from visual representation of operating regions, switching waveforms, and device characteristics.

4.2 Switching Characteristics and Timing

Dynamic Behavior of Switching Devices

The transient response of semiconductor switches, such as MOSFETs and IGBTs, is governed by charge carrier dynamics and parasitic capacitances. The switching process is divided into four distinct phases:

  • Turn-on delay (td(on)) - Time between gate drive activation and initial drain current rise
  • Current rise time (tr) - Interval for drain current to reach 90% of final value
  • Turn-off delay (td(off)) - Time between gate drive removal and current beginning to fall
  • Current fall time (tf) - Duration for current to decay to 10% of initial value
$$ t_{sw} = t_{d(on)} + t_r + t_{d(off)} + t_f $$

Switching Loss Mechanisms

Power dissipation during switching transitions occurs through two primary mechanisms:

$$ P_{sw} = \frac{1}{2}V_{DS}I_D(f_{sw})(t_r + t_f) + Q_{rr}V_{DS}f_{sw} $$

Where the first term represents capacitive switching losses and the second accounts for reverse recovery losses in body diodes. Modern wide-bandgap devices (GaN, SiC) achieve faster switching times (ns range) compared to silicon (10s-100s ns).

Gate Drive Considerations

The gate drive circuit critically impacts switching performance through:

  • Drive current capability (IG = VDR/RG)
  • Miller plateau duration
  • Common source inductance effects

Optimal gate resistance balances switching speed against EMI generation:

$$ R_{G(opt)} = \sqrt{\frac{L_{loop}}{C_{iss}}} $$

Timing Diagrams and Dead-Time

In bridge configurations, precise dead-time (tdead) management prevents shoot-through while minimizing body diode conduction:

$$ t_{dead} > t_{d(off)}^{HS} - t_{d(on)}^{LS} + \Delta t_{margin} $$

Package and Layout Parasitics

Parasitic inductances (Ls, Ld, Lg) and capacitances (Cgd, Cds) create:

  • Voltage overshoot during turn-off (Lsdi/dt)
  • False turn-on from dV/dt coupling
  • Oscillations during switching transitions

Kelvin source connections and low-inductance package designs (e.g., DirectFET) mitigate these effects in high-speed switching applications (>1MHz).

Thermal Considerations

Switching losses generate localized heating at the die level, creating thermal impedance challenges:

$$ Z_{th(j-c)} = \sum_{i=1}^{n}R_{th,i}C_{th,i}\left(1-e^{-t/\tau_i}\right) $$

Where time constants (Ï„i = Rth,iCth,i) range from microseconds (bond wires) to seconds (heat sink).

MOSFET Switching Timing Diagram Oscilloscope-style voltage and current waveforms showing MOSFET switching characteristics, including gate voltage (VGS), drain current (ID), drain-source voltage (VDS), and key timing parameters (td(on), tr, td(off), tf, tdead, Miller plateau). Time VGS ID VDS Miller plateau td(on) tr td(off) tf tdead Legend VGS ID VDS
Diagram Description: The section describes time-domain switching characteristics and timing relationships that would be clearer with visual waveforms.

4.3 Power Dissipation and Heat Management

Fundamentals of Power Dissipation in Switching Devices

Power dissipation in switching devices arises primarily from two sources: static (leakage) losses and dynamic (switching) losses. Static losses occur due to finite off-state leakage current (Ileak) and on-state resistance (RDS(on) in MOSFETs or VCE(sat) in BJTs). Dynamic losses result from transient currents and voltages during switching events, governed by the overlap of I and V waveforms.

$$ P_{\text{total}} = P_{\text{static}} + P_{\text{dynamic}} $$

Dynamic Power Dissipation

For a CMOS gate switching at frequency f, dynamic power is derived from the energy required to charge/discharge load capacitance CL:

$$ P_{\text{dynamic}} = \alpha C_L V_{DD}^2 f $$
where α is the activity factor (probability of a transition). High-speed switching circuits exacerbate this loss due to quadratic dependence on supply voltage VDD.

Short-Circuit Currents

During switching transitions, a brief shoot-through current flows when both NMOS and PMOS transistors are partially on. This loss component is approximated as:

$$ P_{\text{short-circuit}} = \frac{\beta}{12} (V_{DD} - 2V_T)^3 \tau f $$
where β is the transistor gain, VT the threshold voltage, and τ the transition time.

Thermal Resistance and Junction Temperature

Heat removal is quantified via thermal resistance θJA (junction-to-ambient). The steady-state junction temperature is:

$$ T_J = T_A + \theta_{JA} P_{\text{total}} $$
Exceeding the maximum TJ (typically 125–150°C for silicon) accelerates failure mechanisms like electromigration.

Heat Management Techniques

  • Package Selection: Low-θJC packages (e.g., copper slugs, flip-chip) enhance heat transfer to heatsinks.
  • Active Cooling: Forced air or liquid cooling systems for high-power applications (>100W).
  • Switching Frequency Optimization: Balancing speed and dynamic losses through adaptive clocking.
  • Layout Strategies: Thermal vias, distributed power devices, and avoiding hot-spot concentration.

Case Study: IGBT Module Cooling

A 1.2kV/100A IGBT module dissipating 300W requires a heatsink with θSA < 0.5°C/W to maintain TJ < 125°C in a 40°C ambient. Phase-change materials or direct-bonded copper substrates are often employed in such scenarios.

Junction θJC Case θCS Heatsink θSA Thermal Resistance Network
Thermal Resistance Network and Power Dissipation Components A thermal equivalent circuit showing junction, case, heatsink, and thermal resistances (θJC, θCS, θSA) with annotated power dissipation sources (static, dynamic, short-circuit). TJ (Junction) θJC Case θCS Heatsink θSA TA Pstatic Pdynamic Pshort-circuit Power Dissipation Components Legend Junction (TJ) Case Heatsink
Diagram Description: The section includes a thermal equivalent circuit and power dissipation concepts that are inherently spatial and benefit from visual representation.

5. Digital Signal Processing

5.1 Digital Signal Processing

Sampling and Quantization

The foundation of digital signal processing lies in converting continuous-time signals into discrete-time representations. The Nyquist-Shannon sampling theorem dictates that a signal must be sampled at least twice its highest frequency component to avoid aliasing. Mathematically, if a signal has bandwidth B, the sampling frequency fs must satisfy:

$$ f_s \geq 2B $$

Quantization maps sampled amplitudes to discrete levels, introducing quantization error. For an N-bit ADC, the signal-to-quantization-noise ratio (SQNR) is:

$$ \text{SQNR} = 6.02N + 1.76 \text{ dB} $$

Discrete-Time Systems and Z-Transform

Linear time-invariant (LTI) systems in discrete-time are characterized by difference equations:

$$ y[n] = \sum_{k=0}^{M} b_k x[n-k] - \sum_{k=1}^{N} a_k y[n-k] $$

The Z-transform converts these into algebraic equations in the complex z-domain:

$$ H(z) = \frac{\sum_{k=0}^{M} b_k z^{-k}}{1 + \sum_{k=1}^{N} a_k z^{-k}} $$

Pole-zero analysis in the Z-plane determines stability: a system is stable if all poles lie within the unit circle.

Finite Impulse Response (FIR) Filters

FIR filters exhibit no feedback, yielding an impulse response of finite duration. The output is a weighted sum of past inputs:

$$ y[n] = \sum_{k=0}^{N} h[k] x[n-k] $$

Design methods include windowing (e.g., Hamming, Kaiser) and frequency sampling. FIR filters are inherently stable and can achieve linear phase.

Infinite Impulse Response (IIR) Filters

IIR filters incorporate feedback, enabling sharper cutoffs with fewer coefficients. The transfer function includes poles and zeros:

$$ H(z) = \frac{b_0 + b_1 z^{-1} + \cdots + b_M z^{-M}}{1 + a_1 z^{-1} + \cdots + a_N z^{-N}} $$

Common design techniques include bilinear transform (for analog-to-digital conversion) and Butterworth/Chebyshev approximations.

Fast Fourier Transform (FFT)

The FFT reduces the complexity of computing the Discrete Fourier Transform (DFT) from O(N2) to O(N log N). The DFT of a sequence x[n] is:

$$ X[k] = \sum_{n=0}^{N-1} x[n] e^{-j 2\pi kn/N} $$

Radix-2 FFT algorithms exploit symmetry and periodicity of the twiddle factors e−j2πkn/N.

Multirate Signal Processing

Sample rate conversion involves decimation (downsampling) and interpolation (upsampling). Anti-aliasing filters are critical for decimation to prevent spectral overlap. Polyphase implementations optimize computational efficiency.

Applications in Modern Systems

  • Wireless Communications: OFDM uses FFT for spectral efficiency.
  • Audio Processing: MP3 compression relies on subband coding.
  • Radar/Sonar: Matched filters improve SNR in detection.
Sampling and Quantization Process A diagram illustrating the process of converting a continuous analog signal into discrete quantized levels through sampling and quantization. Time Amplitude Time Amplitude Original Signal with Sampling Instants Quantized Signal Original signal Sampling points Quantized signal Quantization error
Diagram Description: The sampling and quantization process would benefit from a visual representation of how a continuous signal is sampled and quantized into discrete levels.

5.2 Microprocessor Design

Architecture Fundamentals

Modern microprocessors are built upon the Von Neumann or Harvard architectures, differing primarily in memory organization. The Von Neumann architecture employs a unified memory for instructions and data, while Harvard architecture separates them, enabling simultaneous access. The choice impacts performance, with Harvard architectures often used in embedded systems for deterministic execution.

Instruction Set Design

The instruction set architecture (ISA) defines the processor's interface to software. RISC (Reduced Instruction Set Computing) and CISC (Complex Instruction Set Computing) represent two dominant paradigms. RISC architectures, such as ARM, prioritize simple, fixed-length instructions with a load-store model, while CISC (e.g., x86) supports complex, variable-length instructions, reducing code size but increasing hardware complexity.

$$ \text{CPI}_{\text{avg}} = \sum_{i=1}^{n} (\text{CPI}_i \times f_i) $$

where CPIi is the cycles per instruction for the i-th instruction, and fi is its frequency in the program. RISC designs typically achieve lower average CPI due to streamlined execution.

Pipelining and Parallelism

Pipelining divides instruction execution into stages (fetch, decode, execute, memory access, write-back), allowing concurrent processing of multiple instructions. The theoretical speedup is given by:

$$ S = \frac{n \cdot k}{n + k - 1} $$

where n is the number of instructions and k is the pipeline depth. Hazards (structural, data, control) limit efficiency, mitigated via techniques like forwarding, branch prediction, and out-of-order execution.

Cache Hierarchy

Caches reduce memory latency through locality exploitation. A multi-level hierarchy (L1, L2, L3) balances speed and capacity. The average access time (Tavg) is:

$$ T_{\text{avg}} = h \cdot T_{\text{cache}}} + (1 - h) \cdot T_{\text{mem}}} $$

where h is the hit rate. Set-associative caches optimize between direct-mapped and fully associative designs, minimizing conflict misses.

Power-Performance Tradeoffs

Dynamic power dissipation follows:

$$ P_{\text{dynamic}}} = \alpha \cdot C \cdot V^2 \cdot f $$

where α is activity factor, C is load capacitance, V is voltage, and f is frequency. Techniques like clock gating, voltage scaling, and near-threshold computing optimize energy efficiency.

Case Study: Superscalar Processors

Superscalar designs (e.g., Intel Skylake) exploit instruction-level parallelism (ILP) by dispatching multiple instructions per cycle. Key components include:

  • Register renaming to eliminate false dependencies.
  • Tomasulo’s algorithm for dynamic scheduling.
  • Speculative execution with rollback mechanisms.

Modern designs integrate heterogeneous cores (e.g., ARM big.LITTLE) for workload-adaptive power management.

Von Neumann vs Harvard Architecture A side-by-side comparison of Von Neumann and Harvard architectures, highlighting their memory organization and bus structure differences. Von Neumann vs Harvard Architecture Von Neumann Architecture (Unified Bus) CPU Unified Memory Data & Instructions Harvard Architecture (Separate Buses) CPU Instruction Memory Data Memory Instruction Bus Data Bus Key Differences: Von Neumann: Single bus for both data and instructions Harvard: Separate buses for data and instructions
Diagram Description: A diagram would physically show the comparison between Von Neumann and Harvard architectures, highlighting their memory organization differences.

5.3 Communication Systems

Fundamentals of Switching in Communication Systems

Switching theory in communication systems governs the efficient routing of signals through networks, ensuring optimal bandwidth utilization and minimal latency. At its core, switching involves the dynamic allocation of resources—whether in circuit-switched, packet-switched, or message-switched architectures. The mathematical foundation relies on queuing theory and graph optimization, where the probability of blocking Pb in an M/M/c/c queue is given by:

$$ P_b = \frac{\frac{\rho^c}{c!}}{\sum_{k=0}^c \frac{\rho^k}{k!}} $$

where ρ = λ/μ is the traffic intensity, λ is the arrival rate, and μ is the service rate. This Erlang-B formula is critical for designing telephone networks and data centers.

Circuit Switching vs. Packet Switching

Circuit switching establishes a dedicated path for the entire duration of a session, as seen in traditional PSTN networks. Its efficiency is quantified by the link utilization factor:

$$ U = \frac{T_{\text{active}}}{T_{\text{total}}} $$

where Tactive is the time the channel carries data. In contrast, packet switching (e.g., IP networks) dynamically allocates bandwidth, optimizing resource sharing but introducing variable latency. The end-to-end delay D for a packet traversing N nodes is:

$$ D = \sum_{i=1}^N \left( \frac{L}{R_i} + Q_i + P_i \right) $$

where L is packet length, Ri is transmission rate, Qi is queuing delay, and Pi is processing delay at node i.

Multiplexing Techniques

Time-Division Multiplexing (TDM) and Frequency-Division Multiplexing (FDM) are foundational to modern systems. TDM allocates time slots, while FDM partitions bandwidth into subcarriers. For OFDM (Orthogonal FDM), the subcarrier spacing Δf must satisfy:

$$ \Delta f = \frac{1}{T_u} $$

where Tu is the useful symbol duration. Guard intervals mitigate intersymbol interference (ISI) in high-speed systems like 5G.

Switching in Optical Networks

Wavelength-selective switches (WSS) enable reconfigurable optical add-drop multiplexing (ROADM). The switching speed of a MEMS-based WSS is bounded by the mechanical resonance frequency:

$$ f_r = \frac{1}{2\pi} \sqrt{\frac{k}{m}} $$

where k is the spring constant and m the mirror mass. This limits all-optical networks to microsecond-scale switching.

Practical Applications

  • Data Centers: Non-blocking Clos topologies minimize contention in spine-leaf architectures.
  • 5G: Packet-switched fronthaul leverages eCPRI for flexible bandwidth allocation.
  • Satellite Networks: Onboard processors use fast packet switching to route traffic between beams.
Circuit Switching vs. Packet Switching A side-by-side comparison of circuit switching (left) and packet switching (right) architectures, showing dedicated paths vs. dynamic routing with labeled nodes and data flows. Circuit Switching vs. Packet Switching Circuit Switching A B Time Slot 1 Time Slot 2 Packet Switching X Y R1 R2 P1 P2 Q Q Dedicated Path Packets Routers Dynamic Route
Diagram Description: A diagram would visually contrast circuit switching (dedicated path) and packet switching (dynamic routing) architectures, showing physical paths and resource allocation.

6. Key Textbooks on Switching Theory

6.1 Key Textbooks on Switching Theory

  • Tsutomu Sasao - Switching Theory for Logic Synthesis — Switching Theory for Logic Synthesis Tsutomu Sasao Springer, Feb. 1999/ ISBN: -7923-8456-3 . PREFACE ix 1 Mathematical Foundation 1 1.1 Set 1 1.2 Relation 4 1.3 Equivalence Class 5 1.4 Function 8 1.5 Ordered Set 10 2 Lattice and Boolean Algebra 17 2.1 Algebra 17 2.2 Lattice 17 2.3 Distributive Lattice and Complemented Lattice 18 2.4 Boolean ...
  • PDF Switching and Finite Automata Theory - Cambridge University Press ... — Switching and finite automata theory / Zvi Kohavi. - 3rd ed. / Niraj Jha. p. cm. ISBN 978--521-85748-2 (hardback) 1. Sequential machine theory. 2. Switching theory. I. Jha, Niraj K. II. Title. QA267.5.S4K64 2009 511.3 5 - dc22 2009036353 ISBN 978--521-85748-2 Hardback Additional resources for this publication at www.cambridge.org ...
  • Fundamentals of Switching Theory | PDF | Boolean Algebra - Scribd — Fundamentals of Switching Theory - Free download as PDF File (.pdf), Text File (.txt) or read online for free. This document provides the forward and introduction to a textbook on switching theory and logic circuit design. It discusses the author's experience teaching switching theory and their motivation for developing this textbook. The textbook aims to provide students with fundamental ...
  • PDF Logic Design And Switching Theory — Switching Theory and Logic Design - BIET The course will make them learn the basic theory of switching circuits and ... Electronic digital circuits are also types … Introduction To Switching Theory And Logical ... word. In this review, we shall explore the book is key themes, examine its writing style, and analyze its overall affect readers ...
  • PDF Chapter 6 Switching Theory - From Art and Skills to ... - Springer — 132 6 Switching Theory - From Art and Skills to Scientific Methods 6.1 Switching Theory from Boolean Algebra The first page of the MSC Thesis by C.E. Shannon that converted logic circuit design from arts and skills into a scientific discipline by assuming the Boolean algebra as the underlying algebraic structure to
  • Switching Theory and Logic Design (STLD) - Springer — Electronic digital circuits are also types of switching circuits. In digital systems, the numbers are represented by binary numbers rather than decimal system. The binary ... 274 6 Switching Theory and Logic Design (STLD) 6.3.3 Fractional Decimal Number to Binary It is done by successive multiplication of 2 and carry of the number after decimal is
  • Switching Theory and Logic Design : A. Anand Kumar : Free Download ... — Ask the publishers to restore access to 500,000+ books. An icon used to represent a menu that can be toggled by interacting with this icon. ... Switching Theory and Logic Design ... Rcs_key 26702 Republisher_date 20231002151110 Republisher_operator [email protected] Republisher_time 308 Scandate ...
  • PDF Switching and Finite Automata Theory, Third Edition — Topics in switching and finite automata theory have been an important part of the curriculum in electrical engineering and computer science departments for several decades. The third edition of this book builds on the comprehensive foundation provided by the second edition and adds: significant new material
  • Switching Theory and Logic Design - O'Reilly Media — This book illustrates the usefulness of switching theory and its applications, with examples to acquaint the student with the necessary background. This book has been designed as a prerequisite to many other courses like Digital Integrated Circuits, Computer Organisation, Digital Instrumentation, Digital Control, Digital Communications and ...
  • Switching Theory and Logic Design Book | PDF - Scribd — This comprehensive text fulfills the course requirement on the subject of Switching Theory and Logic Design for B. Tech degree course in Electronics Engineering of different technical Universities. This text is also bound to serve as a useful reference book for various competitive examinations.

6.2 Research Papers and Journals

  • PDF Chapter 6 Switching Theory - From Art and Skills to ... - Springer — 132 6 Switching Theory - From Art and Skills to Scientific Methods 6.1 Switching Theory from Boolean Algebra The first page of the MSC Thesis by C.E. Shannon that converted logic circuit design from arts and skills into a scientific discipline by assuming the Boolean algebra as the underlying algebraic structure to
  • An introduction to the control of switching electronic systems — He received the IEEE Conference on Decision and Control Best Student Paper Award in 2003 and was also a finalist in 2002. In 2009, he was awarded an Australian Research Fellowship. His research interests cover several areas of automatic control, signal processing, communications, and power electronics. ☆
  • Circuit-switching systems and networks | Understanding ... — Having considered in Chapters 1 and 2 how a call is conveyed across one or more networks, this chapter is all about how the exchanges or more precisely, the switching units within the exchanges in a PSTN actually work. Section 6.2 considers the basic components of circuit-switching systems, often referred to as 'Voice switches'.
  • An introduction to the control of switching electronic systems — Also, research in quantum control is inspired by the potential of quantum computing. ... Much of the literature on MPC for switching electronic applications uses, for computational reasons, ... a Best Paper award by Asian Journal of Control, and 2 Best Engineering Text Book awards from the International Federation of Automatic Control in 1984 ...
  • Unified theory on preservation of conditionally non‐blocking switches ... — 1 Some background on switching theory. An N × N switch is a device with N inputs and N outputs. It provides connection paths from inputs to outputs. An I/O connection pattern comprises concurrent I/O connection paths such that every output is connected from at most one input. A switch accommodates a collection of I/O connection patterns and selects one of them at a time.
  • Switching Theory and Logic Design (STLD) | SpringerLink — 6.7.1 Basics. De Morgan related Boolean with algebra.George Boole constructed an algebra known as Boolean algebra. Boolean algebra implements operations of a system of logic required for digital circuits based on on-off/true-false/high-low/ 1-0 bistates. Boolean statements may take the form of algebric equations, logic block diagrams, or truth tables as:
  • Nodal dynamic equation used for electromagnetic transient simulation of ... — 6.1 Modified equation at switching time. In this paper, the switches are ideal switches. If a switch is at on-state, node i and node j will have the same electric potential, we assume it as a 0 V voltage source, or a 0 A Norton current source whose internal resistance tends to zero. If the switch is at off-state, we assume it as a 0 A current ...
  • General topology for asymmetrical multilevel inverter with reduced ... — Different switching operations to get the multilevel output waveform for the structure shown in Fig. 1 are given in Table 1.In Table 1, 1 and 0, respectively, imply the 'ON' and 'OFF' state of the switches. It can be observed from Table 1 that number of levels in output voltage will be more if the magnitudes of both voltage sources are different from each other as in case of equal ...
  • Structural analysis of switching systems modelled by bond graph — a. In the non-ideal approach [Citation 6], switches are modeled as resistive elements what point out the loss of power in the ON mode eventually associated with a modulated transformer.The modulation is done using a Boolean variable. The main advantage of such an approach is that it leads to a unique causal bond graph model, no matters the state that switches can have.

6.3 Online Resources and Tutorials

  • Switching Theory and Logic Design Book | PDF - Scribd — Switching Theory and Logic Design Book - Free download as PDF File (.pdf), Text File (.txt) or read online for free. ... Chapter 2 introduces the basic building blocks of digital electronics. It starts with ... - 459 = 540 Thus, 9's complement of 459 = 540 (ii) By subtracting each digit from 9 9 9 -3 6 6 3 So, 9's complement of 36 is 63 ...
  • PDF Ch2: Boolean Algebra and Logic Gates Ch3: Gate-Level Minimization Ch4 ... — Switching Theory & Logic Design 1403271-4 Ref: M. Morris Mano and Michael D. Ciletti, Digital Design, Prentice Hall Prof. Adnan Gutub 1 Chapter 6 Registers and Counters Switching Theory & Logic Design 1403271-4 Prof. Adnan Gutub Main Ref: M. Morris Mano and Michael D. Ciletti, Digital Design, Prentice Hall Ch1: Digital Systems and Binary Numbers
  • Tsutomu Sasao - Switching Theory for Logic Synthesis — Switching Theory for Logic Synthesis Tsutomu Sasao Springer, Feb. 1999/ ISBN: -7923-8456-3 . PREFACE ix 1 Mathematical Foundation 1 1.1 Set 1 1.2 Relation 4 1.3 Equivalence Class 5 1.4 Function 8 1.5 Ordered Set 10 2 Lattice and Boolean Algebra 17 2.1 Algebra 17 2.2 Lattice 17 2.3 Distributive Lattice and Complemented Lattice 18 2.4 Boolean ...
  • MPG.eBooks - Staff View: Switching Theory for Logic Synthesis — Switching Theory for Logic Synthesis . Switching Theory for Logic Synthesis covers the basic topics of switching theory and logic synthesis in fourteen chapters. Chapters 1 through 5 provide the mathematical foundation. Chapters 6 through 8 include an introduction to sequential circuits, optimization of sequential machines and asynchrono...
  • Switching Theory - Wiley Online Library — Switching Theory Architectures and Performance in Broadband ATM Networks Achille Pattavina Politecnico di Milano, Italy ... -470-84191-5 (Electronic) "..... d'i nostri sensi ch'è del rimanente non vogliate negar l'esperienza, di retro al sol, del mondo senza gente. Considerate la vostra semenza: ... 6.3.2.1. Specific architectures ...
  • Switching Theory and Logic Design (STLD) | SpringerLink — Switching circuits are for the use of binary variables and application of binary logic. Electronic digital circuits are also types of switching circuits. In digital systems, the numbers are represented by binary numbers rather than decimal system. The binary numbers are also used in arithmetic operations.
  • Switching Theory and Logic Design - O'Reilly Media — This book illustrates the usefulness of switching theory and its applications, with examples to acquaint the student with the necessary background. This book has been designed as a prerequisite to many other courses like Digital Integrated Circuits, Computer Organisation, Digital Instrumentation, Digital Control, Digital Communications and ...
  • Introduction to the Theory of Switching Circuits — Addeddate 2017-04-27 04:48:26 Identifier B-001-001-212 Identifier-ark ark:/13960/t9770tw89 Ocr ABBYY FineReader 11.0
  • PDF Lab 3: BJT Digital Switch - The University of Texas at Dallas — EE/CE 3111 Electronic Circuits Laboratory Spring 2015 Professor Y. Chiu 3 There are three possible operation regions for a BJT we can see along the load line. Saturation region: In this region the collector current does not increase for any increase in the base current. The collector essentially acts like a voltage source of 0.2V-0.3V.
  • PDF Fundamentals of Electronic Circuit Design - University of Cambridge — 1.5 Electronic Signals Electronic signals are represented either by voltage or current. The time-dependent characteristics of voltage or current signals can take a number of forms including DC, sinusoidal (also known as AC), square wave, linear ramps, and pulse-width modulated signals. Sinusoidal signals are perhaps the most important signal forms