Semiconductors and Semiconductor devices

The three basic transistor configuration modes are:

1. Common Emitter:

   • Emitter terminal is common to both input and output circuits
   • Input applied to base, output taken from collector
   • Provides high current gain and voltage gain (typically 50-1000)
   • Phase inversion occurs between input and output signals
   • Most widely used configuration for amplification

2. Common Base:

   • Base terminal is common to both input and output circuits
   • Input applied to emitter, output taken from collector
   • Current gain less than 1, but good voltage gain
   • No phase inversion between input and output
   • Good for high-frequency applications due to low input impedance

3. Common Collector (Emitter Follower):

   • Collector terminal is common to both input and output circuits
   • Input applied to base, output taken from emitter
   • Current gain high, but voltage gain less than 1
   • No phase inversion between input and output
   • Features high input impedance and low output impedance

OR Gate (parallel switches):

• Truth table:
  • A=0, B=0 → C=0
  • A=0, B=1 → C=1
  • A=1, B=0 → C=1
  • A=1, B=1 → C=1
• Implementation: Switches in parallel
• Output is 1 if ANY input is 1

AND Gate (series switches):

• Truth table:
  • A=0, B=0 → C=0
  • A=0, B=1 → C=0
  • A=1, B=0 → C=0
  • A=1, B=1 → C=1
• Implementation: Switches in series
• Output is 1 only if ALL inputs are 1

Practical Applications:

• OR gates (parallel):

  • Safety systems where any trigger should activate the system
  • Lighting controlled from multiple locations
  • Alarm systems with multiple sensors
  • Redundant systems where backup is needed

• AND gates (series):

  • Security systems requiring multiple conditions
  • Machine safety interlocks requiring multiple checks
  • Authorization systems requiring multiple approvals
  • Control systems where all conditions must be met

Inclusive OR (OR) vs. Exclusive OR (XOR):

1. Definition:

   • OR (inclusive OR): Output is TRUE if at least one input is TRUE
   • XOR (exclusive OR): Output is TRUE if exactly one input is TRUE (inputs differ)

2. Truth Tables:

   OR Gate: | A | B | A OR B | |---|---|--------| | 0 | 0 | 0 | | 0 | 1 | 1 | | 1 | 0 | 1 | | 1 | 1 | 1 |

   XOR Gate: | A | B | A XOR B | |---|---|---------| | 0 | 0 | 0 | | 0 | 1 | 1 | | 1 | 0 | 1 | | 1 | 1 | 0 |

3. Boolean expressions:

   • OR: Y = A + B
   • XOR: Y = A⊕B = AB̄ + ĀB (can be read as "A or B, but not both")

4. Implementation:

   • XOR can be constructed using AND, OR, and NOT gates as: (A AND (NOT B)) OR ((NOT A) AND B)
   • XOR symbol is similar to OR but with an additional curved line on the input side

5. Applications:

   • XOR is extensively used in arithmetic circuits, especially for binary addition and parity checking
   • XOR is used in encryption algorithms due to its reversible properties

The diode-based OR gate works through these mechanisms:

• Two diodes (D₁ and D₂) connect inputs A and B to the output node X
• When either input is HIGH (logic 1/+5V):
  • The corresponding diode becomes forward-biased
  • Current flows through the diode to the output
  • The output X becomes HIGH (+5V)
• When both inputs are LOW (logic 0/0V):
  • Neither diode conducts
  • No current flows
  • The output X remains LOW (0V)
• The resistor R connects to ground, ensuring the output is pulled LOW when no input is active

Truth table verification:

• A=0, B=0 → X=0 (neither diode conducts)
• A=1, B=0 → X=1 (D₁ conducts)
• A=0, B=1 → X=1 (D₂ conducts)
• A=1, B=1 → X=1 (both diodes conduct)

Advantages:

• Simplicity: Minimal components needed (just diodes and resistors)
• Speed: Faster than relay-based logic
• No power consumption in steady state (except through pull-down resistors)
• Passive implementation requiring no power supply for the switching elements

Limitations:

• Signal degradation: Voltage drop across forward-biased diodes (0.6-0.7V for silicon)
• No signal amplification (unlike transistor-based gates)
• Limited fan-out: Cannot drive multiple subsequent gates efficiently
• Cannot implement all logic functions (NOT gates cannot be implemented with diodes alone)
• Level restoration required for cascading multiple gates
• Susceptible to noise in complex circuits
• Higher power dissipation in pull-down resistors compared to CMOS

The difference arises from the default output state and how the diodes control it:

OR gate with pull-down resistor:

• Default state: Output pulled to ground (logic 0) through resistor R
• Active operation: Any HIGH input forward-biases its diode, pulling output HIGH
• The resistor creates a path to ground when all inputs are LOW
• Without the pull-down resistor, the output would float when all inputs are LOW

AND gate with pull-up resistor:

• Default state: Output pulled to +V (logic 1) through pull-up resistor
• Active operation: Any LOW input forward-biases its diode, pulling output LOW
• The resistor creates a path to +V when all inputs are HIGH
• Without the pull-up resistor, the output would float when all inputs are HIGH

This complementary design reflects the logical duality between OR and AND functions, where OR activates with any HIGH input, while AND deactivates with any LOW input.

An XOR gate can be constructed using basic logic gates in the following way:

1. Create two parallel paths from the inputs A and B:

   • First path: Generate (A AND NOT B) using an AND gate with input A and the inverted input B
   • Second path: Generate (NOT A AND B) using an AND gate with the inverted input A and input B

2. Combine these two results with an OR gate to produce the final output: (A AND NOT B) OR (NOT A AND B)

Step-by-step implementation:

1. Pass input B through a NOT gate to get B̄
2. Pass input A through a NOT gate to get Ā
3. Use an AND gate with inputs A and B̄ to produce AB̄
4. Use an AND gate with inputs Ā and B to produce ĀB
5. Use an OR gate with inputs AB̄ and ĀB to produce the final output AB̄ + ĀB

This implementation follows directly from the Boolean expression X = AB̄ + ĀB.

NAND gates are universal because:

1. Any digital logic function can be constructed using only NAND gates
2. The three fundamental operations (NOT, AND, OR) can all be built from only NAND gates:
   • NOT: Connect both inputs together
   • AND: NAND followed by NOT
   • OR: NOT both inputs first, then NAND them

Advantages in circuit implementation:

• Simplified manufacturing by requiring only one type of gate
• Reduced component inventory and standardization
• More efficient IC design with uniform gate structures
• Lower production costs through mass production of a single gate type
• Improved reliability through component standardization
• Simplified troubleshooting by limiting gate variety

Engineers can create any complex logic function using this single versatile building block, making NAND gates the foundation of modern digital circuit design.

De Morgan's theorem states that the negation of a conjunction is the disjunction of the negations, and vice versa:

• (A·B)̅ = Ā+B̄
• (A+B)̅ = Ā·B̄

To create an OR gate using only NAND gates:

1. First apply the identity: A+B = (Ā·B̄)̅
2. Recognize that (Ā·B̄)̅ is precisely the NAND operation on the complements of inputs
3. Implementation requires:
   • Two NOT gates to generate Ā and B̄ (can be implemented as NAND gates with tied inputs)
   • One NAND gate taking Ā and B̄ as inputs

The output of this arrangement will give A+B, effectively creating an OR gate using only NAND gates. This principle is fundamental in digital circuit design when working with standardized gate arrays.

To verify functional equivalence between logic gate configurations:

1. Construct a complete truth table for both configurations
2. Compare outputs for all possible input combinations
3. If outputs match for every input combination, the configurations are equivalent

For NAND(Ā,B̄) = OR(A,B):

| A | B | Ā | B̄ | NAND(Ā,B̄) | OR(A,B) | |---|---|---|---|-----------|---------| | 0 | 0 | 1 | 1 | 0 | 0 | | 0 | 1 | 1 | 0 | 1 | 1 | | 1 | 0 | 0 | 1 | 1 | 1 | | 1 | 1 | 0 | 0 | 1 | 1 |

The outputs match for all four input combinations, confirming the equivalence. This technique is called functional verification through exhaustive testing, and it's a fundamental method in digital logic design for proving circuit equivalence without relying on algebraic manipulation.