Unit 3 - Notes

ECE221 7 min read

Unit 3: Oscillators and Introduction to Operational amplifiers

Part 1: Oscillators

1. Sinusoidal Oscillators

An oscillator is an electronic circuit that generates a periodic, oscillating electronic signal, typically a sine wave or a square wave, without requiring an external input signal. Sinusoidal oscillators are specifically designed to produce continuous sine wave outputs.

Principle of Operation:
Oscillators employ positive feedback. A portion of the output signal is fed back to the input in phase with the original signal.

The Barkhausen Criterion:
For a circuit to sustain continuous, undamped oscillations, it must satisfy two conditions (Barkhausen Criterion):

Loop Gain Magnitude: The magnitude of the loop gain ( $|A \beta|$ ) must be equal to or slightly greater than 1 (where $A$ is amplifier gain and $\beta$ is feedback fraction).
Phase Shift: The total phase shift around the closed loop must be $0^\circ$ or $360^\circ$ (or integer multiples of $360^\circ$ ).

2. General Form of Oscillator Circuit

A general oscillator consists of two main blocks:

Active Device (Amplifier): Provides voltage or current gain (e.g., BJT, FET, or Op-Amp).
Feedback Network: A passive network (composed of R, C, L, or crystals) that determines the frequency of oscillation and provides the required phase shift.

The basic equation of the closed-loop gain $A_f$ is:
$A_f = \frac{A}{1 - A\beta}$
When $A\beta = 1$ , the closed-loop gain approaches infinity, meaning the circuit produces an output even with zero input.

3. RC Phase Shift Oscillator

Used for low-frequency (audio frequency) oscillations.

Construction: Consists of an inverting amplifier (which provides a $180^\circ$ phase shift) and an RC feedback network.
Feedback Network: To achieve the required $360^\circ$ total phase shift, the feedback network must provide an additional $180^\circ$ shift. This is done using three identical RC sections cascaded together, each providing a $60^\circ$ phase shift at the frequency of oscillation.
Frequency of Oscillation ( $f_o$ ):
$f_o = \frac{1}{2\pi RC\sqrt{6}}$
Gain Requirement: The amplifier gain must be at least 29 ( $A \ge 29$ ) to overcome the attenuation ( $1/29$ ) of the 3-stage RC network.

4. Wien Bridge Oscillator

A highly stable, low-distortion audio frequency oscillator.

Construction: Uses a non-inverting amplifier and a Wien Bridge feedback network. The bridge consists of a series RC circuit and a parallel RC circuit.
Phase Shift: The Wien bridge produces a $0^\circ$ phase shift at the resonant frequency, so the amplifier must also be non-inverting ( $0^\circ$ phase shift).
Frequency of Oscillation ( $f_o$ ):
$f_o = \frac{1}{2\pi RC}$
(assuming equal resistors and capacitors in the frequency-determining arms).
Gain Requirement: The attenuation of the feedback network at $f_o$ is $1/3$ . Therefore, the amplifier must have a gain of exactly 3 to maintain $|A\beta| = 1$ .

5. Hartley LC Oscillator

Used for high-frequency (Radio Frequency - RF) generation.

Construction: The feedback network is a parallel LC circuit (tank circuit) consisting of two inductors ( $L_1$ , $L_2$ ) and one capacitor ( $C$ ). The connection between the inductors is tapped and grounded.
Phase Shift: The tapped inductor provides a $180^\circ$ phase shift, paired with an inverting amplifier (another $180^\circ$ ).
Frequency of Oscillation ( $f_o$ ):
$f_o = \frac{1}{2\pi \sqrt{L_{eq}C}}$
Where $L_{eq} = L_1 + L_2 + 2M$ ( $M$ is mutual inductance if coils are magnetically coupled).

6. Colpitts LC Oscillator

Similar to the Hartley oscillator but utilizes a tapped capacitance instead of a tapped inductance.

Construction: The tank circuit consists of two capacitors ( $C_1$ , $C_2$ ) and one inductor ( $L$ ). The connection between the capacitors is grounded.
Advantage: Offers better frequency stability and a purer sine wave than the Hartley oscillator because the capacitors provide a low-reactance path for high-frequency harmonics, effectively filtering them.
Frequency of Oscillation ( $f_o$ ):
$f_o = \frac{1}{2\pi \sqrt{L C_{eq}}}$
Where $C_{eq} = \frac{C_1 C_2}{C_1 + C_2}$ (capacitors in series).

7. Crystal Oscillator

Provides extreme frequency stability (up to parts per million, ppm). Used in watches, microcontrollers, and communication systems.

Piezoelectric Effect: Quartz crystals exhibit the piezoelectric effect; applying mechanical stress generates a voltage, and applying a voltage causes mechanical vibration.
Equivalent Circuit: A crystal behaves like an RLC circuit. It has two resonant frequencies:
- Series Resonance ( $f_s$ ): Formed by equivalent $R, L, C$ in series.
- Parallel Resonance ( $f_p$ ): Formed by the series RLC branch in parallel with the mounting capacitance ( $C_m$ ).
Operation: The crystal is usually operated in the inductive region between $f_s$ and $f_p$ to replace the inductor in a Colpitts oscillator (known as a Pierce oscillator).

8. Reading a Datasheet of a 1 MHz Crystal

When evaluating a 1 MHz crystal datasheet, look for the following parameters:

Nominal Frequency: 1.000000 MHz.
Frequency Tolerance: Maximum deviation from nominal at a specific temperature (e.g., $\pm 30$ ppm at $25^\circ\text{C}$ ).
Load Capacitance ( $C_L$ ): The external capacitance required by the oscillator circuit to oscillate at exactly 1 MHz (typically $18\text{pF}$ to $32\text{pF}$ ).
Equivalent Series Resistance (ESR): Maximum resistance at resonance (e.g., $500\ \Omega$ ). Lower ESR means easier oscillation startup.
Drive Level: Maximum power the crystal can dissipate without damage (e.g., $100\ \mu\text{W}$ ).
Operating Temperature Range: Standard ranges (e.g., $-20^\circ\text{C}$ to $+70^\circ\text{C}$ ).

Part 2: Introduction to Operational Amplifiers

1. The Operational Amplifier (Op-Amp)

An Operational Amplifier is a highly stable, high-gain, directly coupled voltage amplifier. Originally designed to perform mathematical operations (addition, subtraction, integration, differentiation) in analog computers, it is now the fundamental building block of analog electronics.

2. Schematic Symbol

The op-amp is represented by a triangle pointing in the direction of signal flow.

Inverting Input (-): Signal applied here is amplified and inverted ( $180^\circ$ out of phase).
Non-Inverting Input (+): Signal applied here is amplified but not inverted ( $0^\circ$ phase shift).
Output: The amplified difference between the two inputs.
Power Supply Pins: Typically $+V_{CC}$ (positive supply) and $-V_{EE}$ (negative supply).

3. Block Diagram of a Typical Op-Amp

Internally, a standard op-amp consists of several cascaded stages:

Input Stage (Dual-Input, Balanced-Output Differential Amplifier): Provides high input impedance, high Common-Mode Rejection Ratio (CMRR), and most of the voltage gain.
Intermediate Stage (Dual-Input, Unbalanced-Output Differential Amplifier): Provides additional voltage gain.
Level Shifting Stage: Op-amps are directly coupled, meaning DC voltages build up. The level shifter (usually an emitter follower with a constant current source) shifts the DC level back to zero volts relative to ground.
Output Stage (Push-Pull Amplifier): Provides low output impedance, high current sourcing/sinking capability, and voltage swing near the supply rails.

4. Characteristics of Operational Amplifiers

Ideal Op-Amp Characteristics:

Voltage Gain ( $A_v$ ): Infinite ( $\infty$ )
Input Impedance ( $Z_{in}$ ): Infinite ( $\infty$ ) - draws zero current.
Output Impedance ( $Z_{out}$ ): Zero ( $0\ \Omega$ ) - can drive any load.
Bandwidth: Infinite ( $\infty$ ) - amplifies all frequencies equally.
Common-Mode Rejection Ratio (CMRR): Infinite ( $\infty$ ) - perfectly rejects noise common to both inputs.
Slew Rate: Infinite ( $\infty\ \text{V}/\mu\text{s}$ ) - output changes instantaneously.
Offset Voltage: Zero ( $0\text{V}$ ) - zero output when inputs are zero.

Practical Op-Amp Characteristics (e.g., General Purpose):

Voltage Gain: High (typically $10^5$ to $10^6$ ).
Input Impedance: High (typically $1\ \text{M}\Omega$ to $10^{12}\ \Omega$ for FET inputs).
Output Impedance: Low (typically $10\ \Omega$ to $100\ \Omega$ ).
Bandwidth: Limited (defined by Gain-Bandwidth Product, e.g., $1\text{ MHz}$ ).
CMRR: High (typically $80\text{ dB}$ to $120\text{ dB}$ ).
Slew Rate: Finite (e.g., $0.5\ \text{V}/\mu\text{s}$ ).

5. Reading Datasheet of the 741 Op-Amp

The LM741 is an industry-standard, general-purpose operational amplifier. Key datasheet parameters include:

Supply Voltage ( $V_{CC}/V_{EE}$ ): Typically $\pm 15\text{V}$ (Maximum $\pm 22\text{V}$ ).
Input Offset Voltage ( $V_{OS}$ ): The DC voltage that must be applied between the input terminals to force the output to zero. (Typical: $1\text{mV}$ to $5\text{mV}$ for 741).
Input Bias Current ( $I_B$ ): The average of the currents flowing into both inputs. (Typical: $80\text{nA}$ ).
Input Offset Current ( $I_{OS}$ ): The difference between the two input bias currents. (Typical: $20\text{nA}$ ).
Large Signal Voltage Gain ( $A_{VD}$ ): Open-loop DC gain. (Typical: $200,000\text{ V/V}$ or $106\text{ dB}$ ).
Slew Rate (SR): The maximum rate of change of the output voltage. For the 741, it is a relatively slow $0.5\text{ V}/\mu\text{s}$ , making it unsuitable for high-frequency applications.
Gain Bandwidth Product (GBW): The frequency at which the open-loop gain drops to 1 (unity). (Typical: $1\text{ MHz}$ for 741).
CMRR: Ability to reject signals common to both inputs. (Typical: $90\text{ dB}$ ).
Pin Configuration (8-pin DIP):
- Pin 2: Inverting Input
- Pin 3: Non-Inverting Input
- Pin 6: Output
- Pin 7: $+V_{CC}$
- Pin 4: $-V_{EE}$ (or Ground)
- Pins 1 & 5: Offset Null (used to zero out $V_{OS}$ via external potentiometer).

Unit 2

Unit 4