Engineering Mathematics III

Cheatsheet Content

### Field Effect Transistors (FETs) - **Introduction:** Voltage-controlled devices used for amplification and switching. - **JFET (Junction FET):** - **Structure:** N-channel or P-channel, controlled by a reverse-biased PN junction (gate). - **Operation:** Gate-source voltage ($V_{GS}$) controls the width of the channel, thereby controlling drain current ($I_D$). - **Characteristics:** High input impedance, negative transconductance. - **Biasing:** Setting the DC operating point ($Q$-point) for desired amplifier performance. Common methods include self-bias, voltage-divider bias. - **MOSFET (Metal-Oxide-Semiconductor FET):** - **Types:** - **Depletion-type (D-MOSFET):** Can operate in both depletion and enhancement modes. - **Enhancement-type (E-MOSFET):** Requires a positive $V_{GS}$ (for N-channel) to create a channel. - **Structure:** Gate insulated from the channel by an oxide layer. - **Operation:** Gate-source voltage ($V_{GS}$) controls the electric field, inducing a channel and controlling drain current ($I_D$). - **Characteristics:** Extremely high input impedance, widely used in digital and analog circuits. - **Biasing:** Similar to JFETs, ensuring stable operating point. - **Small Signal Amplifiers:** - **Purpose:** Amplify small AC signals without distortion. - **JFET/MOSFET Amplifiers:** Common source, common drain (source follower), common gate configurations. - **Key Parameters:** Voltage gain, current gain, input impedance, output impedance. - **Oscillators:** - **Principle:** Circuits that produce repetitive waveforms (e.g., sine wave, square wave) without an external input signal. - **Conditions for Oscillation:** Barkhausen criteria (loop gain = 1, phase shift = $0^\circ$ or $360^\circ$). - **FET Oscillators:** Can be used in various oscillator circuits (e.g., Wien bridge, Colpitts, Hartley). ### Operational Amplifiers (Op-Amps) - **Characteristics:** - **Ideal Op-Amp:** Infinite open-loop gain, infinite input impedance, zero output impedance, infinite bandwidth, zero offset voltage. - **Practical Op-Amp:** High but finite gain, high but finite input impedance, low but non-zero output impedance, finite bandwidth, small input offset voltage/current. - **Differential Amplifiers:** - **Core of Op-Amp:** Amplifies the difference between two input signals. - **CMRR (Common Mode Rejection Ratio):** Ability to reject common-mode signals. High CMRR is desired. - **Offset Voltages and Currents:** - **Input Offset Voltage ($V_{OS}$):** Voltage required between inputs to make output zero. - **Input Bias Current ($I_B$):** Average of the two input currents. - **Input Offset Current ($I_{OS}$):** Difference between the two input currents. - **Linear Applications of Op-Amps:** - **Inverting Amplifier:** Output is inverted and amplified ($A_V = -R_f/R_{in}$). - **Non-Inverting Amplifier:** Output is non-inverted and amplified ($A_V = 1 + R_f/R_{in}$). - **Voltage Follower (Buffer):** Unity gain, high input impedance, low output impedance. - **Summing Amplifier:** Sums multiple input voltages. - **Integrator:** Output proportional to the integral of the input signal. - **Differentiator:** Output proportional to the derivative of the input signal. - **Instrumentation Amplifier:** High gain, high CMRR, high input impedance, used for precision measurements. - **Active Filters:** Circuits that use op-amps to provide gain and frequency selectivity (e.g., low-pass, high-pass, band-pass). - **Non-Linear Applications of Op-Amps:** - **Comparators:** Compares two input voltages and outputs a high or low state. - **Multivibrators:** Circuits that generate square wave or pulse waveforms. - **Astable Multivibrator:** Free-running oscillator, no stable state. - **Monostable Multivibrator:** One stable state, triggered to the quasi-stable state for a set duration. - **Bistable Multivibrator (Flip-Flop):** Two stable states, changes state upon trigger. - **Schmitt Trigger:** A comparator with hysteresis, provides noise immunity. - **Function Generators:** Produce various waveforms (sine, square, triangle). ### The Timer IC 555 - **Introduction:** Versatile integrated circuit used for timing and oscillation. - **Internal Structure:** Comparators, flip-flop, discharge transistor, voltage divider. - **Applications:** - **Astable Multivibrator:** Generates continuous square waves. Frequency and duty cycle controlled by external resistors and capacitors. - **Monostable Multivibrator:** Generates a single pulse of a specific duration when triggered. Used for pulse generation, missing pulse detection. - **Bistable Multivibrator:** Can be configured as a basic flip-flop. - **Voltage-to-Frequency Converters (VFC):** Converts an input voltage into an output frequency. - **Tone Burst Generators:** Produces a short burst of tone, often used in telecommunications. - **Pulse Width Modulation (PWM):** Generating pulses with varying widths. ### IC Voltage Regulators - **Purpose:** Maintain a constant output voltage despite variations in input voltage or load current. - **Fixed Voltage Regulators:** - **Series:** Common fixed voltage regulators are the 78xx series (positive output, e.g., 7805 for +5V) and 79xx series (negative output, e.g., 7905 for -5V). - **Operation:** Internal reference voltage, error amplifier, pass transistor. - **Adjustable Voltage Regulators:** - **LM317 (positive) / LM337 (negative):** Allow the output voltage to be set by external resistors. - **Formula:** $V_{OUT} = V_{REF} (1 + R_2/R_1) + I_{ADJ}R_2$. - **Variable Power Supplies:** Often built using adjustable IC regulators with potentiometers for continuous voltage variation. - **Switching Regulators (Switch-Mode Power Supplies - SMPS):** - **Principle:** Use a switching element (transistor) to rapidly turn the input voltage on and off, followed by a filter to average the pulsed voltage. - **Advantages:** Higher efficiency than linear regulators, smaller size, lower heat dissipation. - **Types:** Buck (step-down), Boost (step-up), Buck-Boost, Flyback. - **Components:** Switching element, inductor, diode, capacitor, control circuit. ### Data Converters - **Analog-to-Digital Converters (ADC):** - **Purpose:** Convert analog signals (continuous in time and amplitude) into digital signals (discrete values). - **Key Parameters:** - **Resolution:** Number of bits (e.g., 8-bit, 12-bit). Determines the number of discrete levels. - **Sampling Rate:** How often the analog signal is sampled. - **Quantization Error:** Error introduced by representing a continuous range with discrete levels. - **Types:** - **Flash ADC:** Fastest, uses a large number of comparators. - **Successive Approximation Register (SAR) ADC:** Common, uses a DAC and comparator to find the digital value bit by bit. - **Delta-Sigma ADC:** High resolution, used for audio and precision measurements. - **Dual Slope ADC:** High accuracy, slow. - **Digital-to-Analog Converters (DAC):** - **Purpose:** Convert digital signals (discrete values) into analog signals (continuous voltage or current). - **Key Parameters:** - **Resolution:** Number of bits. - **Settling Time:** Time taken for the output to settle to within a specified error band. - **Types:** - **Weighted Resistor DAC:** Uses resistors with binary weighted values. - **R-2R Ladder DAC:** Uses only two resistor values ($R$ and $2R$), easier to manufacture. - **Pulse Width Modulation (PWM) DAC:** Generates an analog voltage by varying the duty cycle of a square wave.