UNIT 1: ANALOG CIRCUITS - EXAM-FOCUSED SHORT NOTES

1.0 FEEDBACK AMPLIFIERS & OSCILLATORS

1.1 Feedback Concepts

• Definition: Feeding a portion of the output signal back to the input.

• Types:

  • Positive Feedback: Feedback signal is in-phase with input. Used in oscillators. Tends to cause instability.

  • Negative Feedback: Feedback signal is 180° out-of-phase with input. Used in amplifiers. Improves performance.

• Advantages of Negative Feedback:

  • Stabilizes Gain: Reduces sensitivity to component variations.

  • Extends Bandwidth: Increases bandwidth by factor (1 + Aβ).

  • Reduces Nonlinear Distortion & Noise.

  • Controls Impedance: Increases input impedance (series feedback), decreases output impedance (shunt feedback).

• Disadvantages of Negative Feedback:

  • Reduces Overall Gain.

  • Potential for Instability/Oscillation if phase shift is excessive.

  • Increases circuit complexity and cost.

• Barkhausen Criterion for Oscillation:

  For sustained oscillations, the loop gain must satisfy:

  1. Magnitude Condition: $$\displaystyle |A\beta| = 1 $$

  2. Phase Condition: Total phase shift = 0° or 360° (n·2π)

  • A = Open-loop gain, β = Feedback factor.

1.2 Feedback Topologies & Analysis

• Generalized Gain with Feedback:

$$A_f = \frac{A}{1 + A\beta}$$

*   $$\displaystyle A_f $$ = Gain with feedback, $A$ = Open-loop gain, $\beta$ = Feedback factor.

*   **For Negative Feedback:** $$\displaystyle 1 + A\beta > 1 $$ → $$\displaystyle A_f < A $$.

1.3 Oscillator Fundamentals & Types

• RC Oscillators: (Low-frequency, < 1 MHz)

  • RC Phase Shift Oscillator:

    • Circuit: 3-stage RC network (each providing 60° shift at f) + inverting amplifier (180°).

    • Frequency of Oscillation:

$$f = \frac{1}{2\pi RC\sqrt{6}}$$

    *   **Practical:** Requires amplifier gain $$\displaystyle A_v \geq 29 $$ to compensate for network attenuation.

*   **Wien Bridge Oscillator:**

    *   **Circuit:** Series-parallel RC network (Wien bridge) in positive feedback path + negative feedback path with gain control (often using a thermistor or diodes).

    *   **Frequency of Oscillation:** 

$$f = \frac{1}{2\pi RC}$$

    *   **Advantages:** Stable frequency, low distortion, easy frequency tuning.

    *   **Limitations:** Gain must be precisely set to 3. Sensitive to component drift.

• LC Oscillators: (High-frequency, > 1 MHz)

  • Hartley Oscillator:

    • Circuit: Inductor split into two parts (L1, L2) with capacitor C. Feedback from junction of L1 & L2.

    • Frequency:

$$f = \frac{1}{2\pi\sqrt{L_1 C}} \approx \frac{1}{2\pi\sqrt{(L_1+L_2)C}}$$

*   **Colpitt's Oscillator:**

    *   **Circuit:** Capacitor split into two parts (C1, C2) with inductor L. Feedback from junction of C1 & C2.

    *   **Frequency:** 

$$f = \frac{1}{2\pi\sqrt{L C_{eq}}}, \quad C_{eq} = \frac{C_1 C_2}{C_1 + C_2}$$

• Comparison:

  | Feature | RC Oscillators | LC Oscillators | | :--- | :--- | :--- | | Frequency Range | Audio (Hz - kHz) | RF (kHz - GHz) | | Stability | Moderate (temperature sensitive) | Good (Q-factor high) | | Size/Weight | Small (R, C components) | Large (L components) | | Applications | Audio signal generators, function generators | RF transmitters, local oscillators |

[!TIP] Exam Focus: Deriving oscillation frequency for RC Phase Shift (3-stage) and Wien Bridge is very frequent. Remember the gain conditions (29 for RC, 3 for Wien).

2.0 OPERATIONAL AMPLIFIER (OP-AMP) FUNDAMENTALS

2.1 Ideal vs. Practical Op-Amp (IC 741)

• IC 741 Pin Diagram (8-pin DIP):

  
        +Vcc (Pin 7)
  
          |
  
  Pin 1: Offset Null
  
  Pin 2: Inverting (-)
  
  Pin 3: Non-inverting (+)
  
  Pin 4: -Vcc (or GND for single supply)
  
  Pin 5: Offset Null
  
  Pin 6: Output
  
  Pin 7: +Vcc
  
  Pin 8: NC
  
  

2.2 Key Op-Amp Parameters & Their Effects

• Input Offset Voltage (V_io):

  • Definition: Voltage required at input to make output zero.

  • Cause: Transistor mismatch in input differential pair.

  • Effect: Output offset = $$\displaystyle A_{OL} \times V_{io} $$. Can be nullified using external potentiometer on offset null pins (1 & 5).

• Input Bias Current (I_B) & Input Offset Current (I_io):

  • I_B: Average of currents flowing into both inputs ($$\displaystyle I_{B+}, I_{B-} $$).

  • I_io: Difference between the two input bias currents ($$\displaystyle |I_{B+} - I_{B-}| $$).

  • Effect: In high-impedance sources, causes output offset voltage: $$\displaystyle V_{os} \approx I_B \times R_{eq} $$, where $$\displaystyle R_{eq} $$ is equivalent resistance seen by inputs.

• Common-Mode Rejection Ratio (CMRR):

  • Definition: Ratio of differential gain ($$\displaystyle A_d $$) to common-mode gain ($$\displaystyle A_{cm} $$).

  • Formula:

$$\text{CMRR} = \frac{A_d}{A_{cm}} \quad \text{(unitless)} \quad \text{or} \quad \text{CMRR}_{dB} = 20 \log_{10}\left(\frac{A_d}{A_{cm}}\right)$$

*   **Importance:** Measures ability to reject noise/voltage fluctuations common to both inputs (e.g., power supply ripple).

• Slew Rate (SR):

  • Definition: Maximum rate of change of output voltage.

$$\text{SR} = \left| \frac{dV_{out}}{dt} \right|_{max} \quad \text{(V/μs)}$$

*   **Cause:** Limited current available to charge internal compensation capacitor.

*   **Effect:** **Limits large-signal, high-frequency performance.** For a sinusoidal output $$\displaystyle V_{out} = V_m \sin(2\pi f t) $$, the requirement is $$\displaystyle 2\pi f V_m \leq \text{SR} $$.

• Gain-Bandwidth Product (GBW):

  • Definition: Constant product of open-loop gain and frequency for a given op-amp.

$$\text{GBW} = A_{OL} \times f$$

*   **Significance:** For a closed-loop gain $$\displaystyle A_{CL} $$, the -3dB bandwidth is approximately $$\displaystyle \text{GBW} / A_{CL} $$.

• Power Supply Rejection Ratio (PSRR):

  • Definition: Ratio of change in supply voltage to resulting change in input offset voltage. High PSRR means output is insensitive to power supply variations.

[!TIP] Common Pitfall: Do not confuse Slew Rate (large-signal, rate limit) with Bandwidth (small-signal, frequency limit). SR limits the maximum slope of output, while GBW defines the trade-off between gain and bandwidth.

3.0 DIFFERENTIAL AMPLIFIER (DA)

3.1 Basic Differential Amplifier Stage

• Structure: Two identical transistors (Q1, Q2) with common emitter resistor (R_E) or current source (for better CMRR). Often uses an active load (current mirror) for high differential gain.

• Modes:

  • Differential Input: $$\displaystyle V_{in1} = +V_d/2 $$, $$\displaystyle V_{in2} = -V_d/2 $$.

  • Common-Mode Input: $$\displaystyle V_{in1} = V_{in2} = V_{cm} $$.

3.2 Configurations & Gain Expressions

• Single Input, Unbalanced Output (Most Common):

  • One input grounded (say $$\displaystyle V_{i2} = 0 $$), signal applied to other ($$\displaystyle V_{i1} = V_{in} $$). This is equivalent to a differential input with $$\displaystyle V_{in2} = 0 $$.

  • Differential Voltage Gain ($$\displaystyle A_d $$):

$$A_d = \frac{V_{out}}{V_{id}} = \frac{R_C}{2r_e'} \quad \text{(for BJT, without emitter degeneration)}$$

    *   $$\displaystyle r_e' \approx 26\text{mV}/I_E $$ (dynamic emitter resistance).

*   **Common-Mode Voltage Gain ($$\displaystyle A_{cm} $$):** 

$$A_{cm} = \frac{V_{out}}{V_{icm}} = -\frac{R_C}{2R_E + r_e'} \quad \text{(approx.)}$$

    *   Large $$\displaystyle R_E $$ (or current source) minimizes $$\displaystyle A_{cm} $$.

*   **CMRR:** 

$$\text{CMRR} = \left| \frac{A_d}{A_{cm}} \right| \approx \frac{R_E}{r_e'} \quad \text{(for BJT)}$$

• Dual Input, Balanced Output:

  • Outputs taken from both collectors ($$\displaystyle V_{c1} $$ and $$\displaystyle V_{c2} $$). Differential output $$\displaystyle V_{od} = V_{c1} - V_{c2} $$.

  • $$\displaystyle A_d = R_C / r_e' $$ (twice the single-ended unbalanced gain).

3.3 Applications & Advantages

• Principle: Amplifies difference between two signals, rejects common signal.

• Advantages:

  • Excellent common-mode rejection (noise, DC drift).

  • High differential gain.

• Instrumentation Amplifier (3-Op-Amp IA):

  • Structure: First stage = two unity-gain buffers feeding a single-input unbalanced DA. Second stage = difference amplifier.

  • Output Voltage:

$$V_{out} = \left(1 + \frac{2R}{R_G}\right) (V_2 - V_1)$$

*   **Advantages:** Very high CMRR, high input impedance, gain set by single resistor $$\displaystyle R_G $$.

[!TIP] Exam Derivation: Be prepared to derive $$\displaystyle A_d $$ and $$\displaystyle A_{cm} $$ for a single-input unbalanced output DA using small-signal analysis. Key is applying superposition: differential signal sees $$\displaystyle R_E $$ as AC ground (if bypassed), common-mode signal sees full $$\displaystyle R_E $$.

4.0 OP-AMP APPLICATIONS I: LINEAR CIRCUITS

4.1 Basic Amplifier Configurations

$$A_v = -\frac{R_f}{R_{in}}$$

| ≈ $$\displaystyle R_{in} $$ (due to virtual ground) | | Non-Inverting Amplifier |

$$A_v = 1 + \frac{R_f}{R_{in}}$$

| Very High (≈ op-amp's Z_in) | | Voltage Follower (Buffer) | Non-inverting with $$\displaystyle R_f = 0 $$, $$\displaystyle R_{in} = \infty $$ |

$$A_v = 1$$

| Very High (≈ op-amp's Z_in) |

4.2 Voltage Regulators

• Fixed Regulators (78XX/79XX):

  • Pins: Input, Ground, Output.

  • Operation: Internal reference, error amplifier, pass transistor.

  • Advantages: Simple, built-in protection (thermal shutdown, current limiting).

  • Limitations: Fixed output, dropout voltage (~2V), poor efficiency for large voltage drops.

• Adjustable Regulator (LM317/LM337):

  • Pins: Adjust (Adj), Output, Input.

  • Working: Maintains 1.25V between Output and Adj pins. External resistors set output.

  • Output Voltage Formula:

$$V_{out} = 1.25\text{V} \left(1 + \frac{R_2}{R_1}\right) + I_{adj} R_2 \approx 1.25\text{V} \left(1 + \frac{R_2}{R_1}\right)$$

    *   $$\displaystyle I_{adj} $$ (~50 μA) is usually negligible.

*   **Advantages over Fixed:** Variable output, better regulation, higher current capability.

• Linear Regulator Specs:

  • Line Regulation: $$\displaystyle \Delta V_{out} / \Delta V_{in} $$ (mV/V or %).

  • Load Regulation: $$\displaystyle \Delta V_{out} / \Delta I_{load} $$ (mV/A or %).

4.3 Active Filters

• Filter Response: Plot of Magnitude (dB) vs. Frequency (log scale).

  • Passband: Frequency range with little attenuation.

  • Stopband: Frequency range with high attenuation.

  • Cutoff Frequency ($$\displaystyle f_c $$): Frequency at which gain drops to $$\displaystyle 0.707 A_{max} $$ (-3dB).

  • Roll-off: Rate of attenuation in stopband (dB/octave or dB/decade).

• 1st-Order Butterworth Low-Pass Filter (LPF):

  • Circuit: RC low-pass filter followed by op-amp voltage follower (to isolate load).

  • Transfer Function:

$$H(s) = \frac{A_{max}}{1 + sRC}$$

*   **Cutoff Frequency:** 

$$f_c = \frac{1}{2\pi RC}$$

*   **Roll-off:** **-20 dB/decade** after $$\displaystyle f_c $$.

• 2nd-Order Filters (Sallen-Key):

  • Topology: Uses op-amp with RC feedback network.

  • Butterworth Response: Maximally flat passband. Roll-off = -40 dB/decade.

[!TIP] Design Tip: For a 1st-order LPF, choose $R$ and $C$ such that $$\displaystyle f_c = 1/(2\pi RC) $$. The op-amp buffer ensures the filter's characteristics are not loaded by the next stage.

5.0 OP-AMP APPLICATIONS II: NON-LINEAR & SPECIAL FUNCTION CIRCUITS

5.1 Integrator & Differentiator

5.2 Comparators & Detectors

• Zero-Crossing Detector:

  • Circuit: Op-amp without feedback. $$\displaystyle V_{in} $$ to (+) input, (-) input grounded.

  • Operation: Output switches from $$\displaystyle +V_{sat} $$ to $$\displaystyle -V_{sat} $$ when $$\displaystyle V_{in} $$ crosses 0V.

  • Applications: AC to digital conversion, phase detection.

• Schmitt Trigger:

  • Circuit: Op-amp with positive feedback (resistors from output to (+) input).

  • Key Concept: Hysteresis. Two distinct threshold voltages:

    • Upper Threshold ($$\displaystyle V_{UT} $$): When output = $$\displaystyle -V_{sat} $$, $$\displaystyle V_{in} $$ must rise to $$\displaystyle V_{UT} $$ to switch high.

    • Lower Threshold ($$\displaystyle V_{LT} $$): When output = $$\displaystyle +V_{sat} $$, $$\displaystyle V_{in} $$ must fall to $$\displaystyle V_{LT} $$ to switch low.

  • Hysteresis Width: $$\displaystyle V_{HYS} = V_{UT} - V_{LT} $$.

  • Applications: Noise immunity, debouncing, waveform shaping (sine to square).

5.3 Waveform Generators & Timers (555 IC)

• Pin Configuration (8-pin DIP):

  | Pin | Name | Function | | :--- | :--- | :--- | | 1 | GND | Ground | | 2 | TRIG | Trigger (low pulse to start timing) | | 3 | OUT | Output | | 4 | RESET | Master reset (active low) | | 5 | CTRL | Control voltage (modulates threshold) | | 6 | THR | Threshold (ends timing when > 2/3 Vcc) | | 7 | DIS | Discharge (open collector to ground) | | 8 | Vcc | Supply voltage (5-15V) |

• Astable Multivibrator (Oscillator):

  • Circuit: $$\displaystyle R_A $$, $$\displaystyle R_B $$, $C$ connected between Vcc, DIS (pin 7), THR (pin 6), TRIG (pin 2).

  • Operation: Capacitor $C$ charges through $$\displaystyle R_A+R_B $$ to $$\displaystyle 2/3 V_{cc} $$ (THR), discharges through $$\displaystyle R_B $$ to $$\displaystyle 1/3 V_{cc} $$ (TRIG).

  • Frequency & Duty Cycle:

$$f = \frac{1.44}{(R_A + 2R_B)C}$$

$$\text{Duty Cycle \%} = \frac{R_A + R_B}{R_A + 2R_B} \times 100\%$$

    *   **Note:** Duty cycle is always >50% because $$\displaystyle t_{charge} > t_{discharge} $$.

• Monostable Multivibrator (One-Shot):

  • Circuit: $R$, $C$ between Vcc, DIS, and THR. Trigger pulse to pin 2.

  • Operation: Output goes high for fixed time $T$ on negative trigger pulse, then returns low.

  • Pulse Width:

$$T = 1.1 \times R \times C$$

• Schmitt Trigger using 555:

  • Circuit: Use 555 as two comparators with positive feedback. Connect TRIG and THR together as input. Output from pin 3.

  • Thresholds: $$\displaystyle V_{UT} = 2/3 V_{cc} $$, $$\displaystyle V_{LT} = 1/3 V_{cc} $$.

5.4 Other Special Circuits

• Peak Detector:

  • Circuit: Diode in feedback path of op-amp, capacitor across load.

  • Operation: When $$\displaystyle V_{in} > V_{cap} $$, diode conducts, capacitor charges to peak. When $$\displaystyle V_{in} < V_{cap} $$, diode reverse-biased, capacitor holds peak.

• Sample and Hold (S/H):

  • Concept: Captures (samples) input voltage at a specific instant and holds (stores) it for a period.

  • Circuit: Analog switch (FET) in series with capacitor. Control logic opens/closes switch.

  • Modes: Sample: Switch closed, capacitor follows input. Hold: Switch open, capacitor holds voltage.

6.0 INTEGRATED CIRCUITS (ICs) & SUPPORTING TOPICS

6.1 IC Fundamentals

• Definition: Circuit in which all components are fabricated on a single semiconductor substrate (chip).

• Advantages:

  • Miniaturization, high reliability, low cost (mass production).

  • Reduced parasitic capacitance/inductance, low power consumption.

  • High switching speed, consistent performance.

• Disadvantages:

  • High initial design/mask cost, limited flexibility (cannot modify internally).

  • Thermal management challenges, entire chip fails if one component fails.

• Basic Building Components:

  • Transistors: BJTs, MOSFETs (most common).

  • Diodes: PN junction diodes.

  • Resistors: Diffused (poor tolerance, ~20%), pinch resistors (better).

  • Capacitors: MOS capacitors (gate oxide), junction capacitors (reverse-biased PN).

  • Interconnects: Multiple metal layers (Al, Cu).

6.2 Datasheets

• Importance: The definitive guide for selecting, using, and troubleshooting an IC.

• Typical Information:

  • Absolute Maximum Ratings: Voltage, current, power, temperature limits.

  • Electrical Characteristics: $$\displaystyle V_{io} $$, $$\displaystyle I_B $$, $$\displaystyle A_{OL} $$, GBW, SR, CMRR, PSRR, input voltage range, output swing.

  • Timing Diagrams & Waveforms.

  • Pin Configuration/Diagram.

  • Package Information & Thermal Data.

  • Typical Application Circuits.

6.3 Power Supply Considerations

• Dual Power Supply (±Vcc):

  • Need: Allows op-amp to handle bipolar (AC) output signals symmetrically.

  • Generation: Center-tapped transformer + rectifier/filter, or charge pump circuit (using capacitors and diodes).

• Effect of Power Supply Variation:

  • Causes output offset voltage shift.

  • PSRR specification quantifies this rejection.

• Switching Regulator (Brief):

  • Principle: Switch (transistor) turns on/off rapidly, stores energy in inductor, releases to load. Uses PWM.

  • Advantages vs. Linear: High efficiency (80-90%), less heat, can step-up (boost) or step-down (buck).

  • Disadvantages: More complex, generates EMI/RF noise, larger output ripple.

  • Topologies: Buck (step-down), Boost (step-up), Buck-Boost.

7.0 CALCULATION-BASED PROBLEMS (KEY FORMULAS)

• RC Phase Shift Oscillator Frequency:

$$f = \frac{1}{2\pi RC\sqrt{6}}$$

• Wien Bridge Oscillator Frequency:

$$f = \frac{1}{2\pi RC}$$

• Inverting Amplifier Gain:

$$A_v = -\frac{R_f}{R_{in}}$$

• Non-Inverting Amplifier Gain:

$$A_v = 1 + \frac{R_f}{R_{in}}$$

• Differential Amplifier Output (Single-Ended):

$$V_{out} \approx -\frac{R_f}{2R_1}(V_2 - V_1) \quad \text{(for $$\displaystyle R_1=R_2 $$, $$\displaystyle R_f=R_3 $$)}$$

• Input Offset Voltage due to Bias Current:

$$V_{os} \approx I_B \times R_{eq} \quad \text{where } R_{eq} \text{ is parallel combination of input resistors.}$$

• LM317 Output Voltage:

$$V_{out} = 1.25\left(1 + \frac{R_2}{R_1}\right) + I_{adj}R_2$$

• 1st-Order LPF Cutoff:

$$f_c = \frac{1}{2\pi RC}$$

• 555 Astable Frequency & Duty Cycle:

$$f = \frac{1.44}{(R_A + 2R_B)C}, \quad \text{Duty} = \frac{R_A + R_B}{R_A + 2R_B} \times 100\%$$

• 555 Monostable Pulse Width:

$$T = 1.1 \times R \times C$$

• Slew Rate Limit for Sine Wave:

$$f_{max} = \frac{\text{SR}}{2\pi V_m} \quad \text{where } V_m \text{ is peak output amplitude.}$$

[!TIP] Final Exam Strategy: When solving problems, first identify the circuit type (feedback topology, oscillator, op-amp config). Then apply the correct formula from the boxed section above. Always check units (μF, kΩ, Hz) and remember that in 741, $$\displaystyle V_{sat} \approx \pm 13V $$ for ±15V supplies.