Analog Circuits (EC-405) - Unit 4 Short Notes

UNIT 4: Analog Circuits - Short Notes

1. Operational Amplifier (Op-Amp) Fundamentals

Ideal Op-Amp Characteristics:

• Infinite open-loop voltage gain ($$\displaystyle A_{OL} \to \infty $$)

• Infinite input impedance ($$\displaystyle Z_{in} \to \infty $$), zero input current

• Zero output impedance ($$\displaystyle Z_{out} = 0 $$)

• Infinite bandwidth (gain independent of frequency)

• Zero input offset voltage ($$\displaystyle V_{io} = 0 $$)

• Infinite Common-Mode Rejection Ratio (CMRR)

• Infinite Power Supply Rejection Ratio (PSRR)

Practical Op-Amp Key Parameters:

IC 741 Op-Amp:

• Pin Configuration (8-pin DIP):

  • Pin 2: Inverting input (−)

  • Pin 3: Non-inverting input (+)

  • Pin 6: Output

  • Pin 7: V− (negative supply)

  • Pin 4: V+ (positive supply, often grounded for single supply)

  • Pin 1 & 5: Offset null (optional)

  • Pin 8: NC (no connection)

• Equivalent Circuit Model: Input stage (differential pair), gain stage, output stage (push-pull).

[!TIP]

For 741, typical $$\displaystyle V_{io} \approx 1\,\text{mV} $$, $$\displaystyle I_B \approx 80\,\text{nA} $$, $SR \approx 0.5\,\text{V/μs}$, GBW $\approx 1\,\text{MHz}$.

2. Differential Amplifiers

Basic Structure: Two transistors with common emitter resistor; two inputs ($$\displaystyle V_1 $$, $$\displaystyle V_2 $$) and two outputs ($$\displaystyle V_{out1} $$, $$\displaystyle V_{out2} $$).

Configurations:

• Single-Input Balanced Output: One input grounded; outputs taken from both collectors (equal magnitude, opposite phase).

• Single-Input Unbalanced Output: One input grounded; output taken from one collector.

• Dual-Input Balanced/Unbalanced Output: Both inputs active; output may be single-ended or differential.

Analysis:

• Differential Mode Voltage Gain:

  $$\displaystyle A_d = \frac{v_{od}}{v_{id}} = \frac{v_{o1} - v_{o2}}{v_{i1} - v_{i2}} $$

  For dual-input balanced with $$\displaystyle R_E $$ fully bypassed: $$\displaystyle A_d = -g_m R_C $$ (single-ended) or $$\displaystyle -2g_m R_C $$ (differential).

• Common-Mode Voltage Gain:

  $$\displaystyle A_c = \frac{v_{oc}}{v_{ic}} = \frac{v_{o1} + v_{o2}}{v_{i1} + v_{i2}} $$

  Ideally $$\displaystyle A_c = 0 $$ with infinite $$\displaystyle R_E $$; practically small.

• CMRR:

  $$\displaystyle \text{CMRR} = \left| \frac{A_d}{A_c} \right| $$ (linear) or $$\displaystyle 20\log_{10}\left(\frac{A_d}{A_c}\right)\,\text{dB} $$.

Application: Instrumentation Amplifier

• Uses three op-amps: two buffers for high input impedance, one differential amp.

• Gain set by single resistor: $$\displaystyle A_v = 1 + \frac{2R_f}{R_{gain}} $$.

• High CMRR, high input impedance.

3. Feedback in Amplifiers

Concept: Portion of output fed back to input.

• Negative Feedback: Feedback signal opposes input; stabilizes gain, reduces distortion.

• Positive Feedback: Feedback aids input; used in oscillators.

Feedback Topologies:

Advantages of Negative Feedback:

• Gain Stability: $$\displaystyle A_{f} = \frac{A}{1 + A\beta} \approx \frac{1}{\beta} $$ if $A\beta \gg 1$.

• Bandwidth Extension: $$\displaystyle BW_{f} = BW \times (1 + A\beta) $$ (Gain-Bandwidth Trade-off).

• Distortion Reduction: Distortion reduced by factor $(1 + A\beta)$.

• Impedance Modification: As per table above.

Barkhausen Criterion for Oscillations:

1. Loop gain magnitude: $$\displaystyle |A\beta| = 1 $$

2. Phase shift: $$\displaystyle \angle A\beta = 0^\circ $$ (or $$\displaystyle 360^\circ n $$)

4. Oscillators

Barkhausen Criterion: Conditions for sustained oscillations (see above).

RC Oscillators:

• RC Phase Shift Oscillator:

  • Circuit: 3 identical RC sections in feedback network + inverting amp (op-amp or transistor).

  • Working: Each RC gives $$\displaystyle 60^\circ $$ phase shift at $$\displaystyle f_0 $$; total $$\displaystyle 180^\circ $$ + amp $$\displaystyle 180^\circ $$ = $$\displaystyle 360^\circ $$.

  • Frequency of Oscillation:

$$ f_0 = \frac{1}{2\pi RC\sqrt{6}} \quad \text{(for 3-stage, equal R, C)} $$

• Design: Given $$\displaystyle f_0 $$, $$\displaystyle R = \frac{1}{2\pi f_0 C\sqrt{6}} $$.

• Wien Bridge Oscillator:

  • Circuit: Series RC and parallel RC in positive feedback; negative feedback via amplitude control (e.g., thermistor, diodes).

  • Frequency:

$$ f_0 = \frac{1}{2\pi RC} $$

• Advantages: Low distortion, good frequency stability.

• Limitations: Requires amplitude stabilization; sensitive to component tolerances.

LC Oscillators:

• Colpitt’s Oscillator:

  • Circuit: LC tank with split capacitor ($$\displaystyle C_1 $$, $$\displaystyle C_2 $$) in feedback.

  • Frequency:

$$ f_0 = \frac{1}{2\pi\sqrt{L \cdot \frac{C_1 C_2}{C_1 + C_2}}} $$

• Hartley’s Oscillator:

  • Circuit: LC tank with split inductor ($$\displaystyle L_1 $$, $$\displaystyle L_2 $$) in feedback.

  • Frequency:

$$ f_0 = \frac{1}{2\pi\sqrt{(L_1 + L_2) \cdot C}} \quad (\text{with mutual inductance } M) $$

Generalized LC Analysis:

Oscillation frequency determined by resonant frequency of tank circuit; feedback fraction $\beta$ set by capacitive/inductive divider.

5. Op-Amp Applications

Basic Circuits:

• Voltage Follower (Buffer): Output connected to inverting input; $$\displaystyle A_v = 1 $$, high $$\displaystyle Z_{in} $$, low $$\displaystyle Z_{out} $$.

• Inverting Amplifier:

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

$$\displaystyle Z_{in} = R_{in} $$.

• Non-Inverting Amplifier:

$$ A_v = 1 + \frac{R_f}{R_1} $$

$$\displaystyle Z_{in} \to \infty $$.

Mathematical Operations:

• Integrator:

  $$\displaystyle V_{out} = -\frac{1}{RC} \int V_{in} dt $$

  Uses capacitor in feedback; limited by DC errors (add $$\displaystyle R_f $$ parallel to $C$).

• Differentiator:

  $$\displaystyle V_{out} = -RC \frac{dV_{in}}{dt} $$

  Uses capacitor at input; prone to high-frequency noise (add $R$ in series with $C$).

Active Filters:

• Filter Types:

  • Low Pass (LPF): Passes low frequencies, attenuates high.

  • High Pass (HPF): Passes high frequencies, attenuates low.

  • Band Pass (BPF): Passes band of frequencies.

  • Band Stop (BSF): Attenuates band of frequencies.

• Butterworth Filters: Maximally flat passband.

  • 1st Order LPF:

$$ A_v(s) = \frac{A_0}{1 + sRC} $$

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

• 2nd Order LPF (Sallen-Key):

$$ A_v(s) = \frac{A_0}{1 + \frac{s}{\omega_0 Q} + \left(\frac{s}{\omega_0}\right)^2} $$

Design for $$\displaystyle \omega_0 = \frac{1}{RC} $$, $$\displaystyle Q = \frac{1}{3 - A_0} $$ (for unity gain $$\displaystyle Q=0.5 $$).

Other Applications:

• Voltage Comparator (Zero Crossing Detector): Op-amp without feedback; output saturates positive/negative based on input polarity.

• Peak Detector: Diode in feedback; capacitor holds peak voltage.

• Sample and Hold: Switch (FET) samples input, capacitor holds value when switch opens.

6. 555 Timer IC

Block Diagram & Pin Configuration (8-pin DIP):

Astable Multivibrator:

• Circuit: $$\displaystyle R_1 $$, $$\displaystyle R_2 $$, $C$ connected; pins 2 & 6 tied to capacitor junction.

• Operation: Capacitor charges via $$\displaystyle R_1+R_2 $$, discharges via $$\displaystyle R_2 $$.

• Frequency & Duty Cycle:

$$ T = 0.693 (R_1 + 2R_2) C $$

$$ f = \frac{1.44}{(R_1 + 2R_2)C} $$

Duty Cycle % = $$\displaystyle \frac{R_1 + R_2}{R_1 + 2R_2} \times 100\% $$ (always >50%).

Monostable Multivibrator:

• Circuit: $R$, $C$ from discharge pin to ground; trigger pin 2.

• Operation: Output high for fixed time $T$ on negative trigger.

• Time Period:

$$ T = 1.1 RC $$

Schmitt Trigger using 555:

• Pins 2 & 6 connected to input; output at pin 3.

• Upper threshold = $$\displaystyle \frac{2}{3}V_{cc} $$, lower threshold = $$\displaystyle \frac{1}{3}V_{cc} $$.

• Hysteresis width = $$\displaystyle \frac{1}{3}V_{cc} $$.

7. Voltage Regulators

Types:

• Fixed Voltage Regulators: 78xx (positive), 79xx (negative); e.g., 7805 = +5V.

• Adjustable Voltage Regulators: LM317 (positive), LM337 (negative).

Linear Regulators:

• Characteristics: Low output ripple, simple, inefficient (heat dissipation).

• Specifications: Line regulation, load regulation, temperature coefficient.

• Advantages: Low cost, easy to use, low noise.

• Limitations: Poor efficiency ($$\displaystyle \eta = \frac{V_{out}}{V_{in}} $$), limited current.

LM317 Voltage Regulator:

• Pin Config: Pin 1 = Adjust, Pin 2 = Output, Pin 3 = Input.

• Typical Circuit:

  $$\displaystyle V_{out} = 1.25\,\text{V} + I_{adj}R_2 $$ (since $$\displaystyle I_{adj} \approx 50\,\mu\text{A} $$, often neglected).

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

• Features: Adjustable 1.25V–37V, current limiting, thermal shutdown.

• Applications: Power supplies, battery chargers.

Switching Regulators (Brief): High efficiency (>80%), use switching element (transistor) and inductor; types: buck, boost, buck-boost.

Dual Power Supply Configuration: Use two regulators (e.g., 7812 & 7912) or virtual ground circuits (e.g., TLE2426).

8. Additional Topics and IC Fundamentals

Integrated Circuits (ICs):

• Definition: Miniaturized circuit with transistors, resistors, capacitors fabricated on semiconductor chip.

• Characteristics: Small size, low cost, high reliability, matched components.

• Advantages over Discrete:

  • Miniaturization, reduced parasitic, lower power, mass production.

• Disadvantages: Limited power handling, design flexibility, high initial cost.

• Basic Building Components:

  • Transistors (BJT, MOSFET) – active.

  • Resistors – diffusion or thin-film.

  • Capacitors – MOS or junction capacitors.

  • Inductors – rarely integrated (use gyrators).

Data Sheets for ICs:

• Importance: Provides complete specifications for design and troubleshooting.

• Typical Information:

  • Absolute Maximum Ratings (voltage, current, temperature)

  • Electrical Characteristics (typical values at specific conditions)

  • Pin Configuration/Diagram

  • Application Circuits

  • Thermal Characteristics

  • Package Information

Special Circuits:

• Clipper: Removes portion of input signal above/below reference.

• Clamper: Shifts entire signal up/down by DC level.

• Voltage-Controlled Oscillator (VCO): Output frequency proportional to input voltage; used in PLLs, FM modulation.

• Effect of Power Supply Variation: Changes op-amp parameters (gain, offset, bandwidth); PSRR quantifies rejection.

[!EXAM TIP]

Past papers frequently ask for comparisons (fixed vs adjustable regulators, feedback topologies) and calculations (oscillator frequency, regulator output, filter design). Always write formulas with units and box final answers. For derivations (e.g., CMRR, oscillator frequency), state assumptions (e.g., matched components, ideal op-amp).