Unit 3 - Notes

ECE221

Unit 3: Oscillators and Introduction to Operational amplifiers

Part A: Sinusoidal Oscillators

1. Introduction to Oscillators

An electronic oscillator is a circuit that generates a repetitive, periodic waveform (AC signal) without the need for an external input signal. It converts DC energy from the power supply into AC energy.

Positive Feedback (Regenerative Feedback):
Oscillators rely on positive feedback, where a portion of the output signal is fed back to the input in phase with the original signal.

The Barkhausen Criterion:
For sustained oscillations to occur, two conditions must be met:

Loop Gain: The magnitude of the loop gain ( $A\beta$ ) must be equal to or slightly greater than unity.
$|A\beta| \geq 1$
(Where $A$ is amplifier gain and $\beta$ is feedback factor)
Phase Shift: The total phase shift around the closed loop must be $0^\circ$ or $360^\circ$ (integer multiple of $2\pi$ ).

2. General Form of Oscillator Circuit

Most oscillators consist of an amplifier and a feedback network comprising reactive components (Inductors $L$ and Capacitors $C$ ).

Generalized Circuit Structure:

Amplifier: Usually a transistor (BJT/FET) or Op-Amp.
Feedback Network: Three reactive elements ( $Z_1, Z_2, Z_3$ ) arranged in a Pi-network.

General Conditions for Oscillation:
For a standard transistor configuration where the ground is connected to the junction of $Z_1$ and $Z_2$ :

$Z_1$ and $Z_2$ must have the same reactance type (both capacitors or both inductors).
$Z_3$ must have the opposite reactance type to $Z_1$ and $Z_2$ .

3. RC Oscillators (Audio Frequency)

RC oscillators are generally used for low frequencies (Audio Frequency range: 20 Hz – 20 kHz) because Inductors become bulky at low frequencies.

A. RC Phase Shift Oscillator

This circuit uses a standard Inverting Amplifier and a feedback network consisting of three RC sections.

Principle: The inverting amplifier produces a $180^\circ$ phase shift. To satisfy Barkhausen's criterion ( $360^\circ$ total), the feedback network must provide an additional $180^\circ$ phase shift.
Network: Three cascaded RC sections. Each section contributes theoretically $60^\circ$ phase shift ( $60^\circ \times 3 = 180^\circ$ ).
Frequency of Oscillation:
$f_o = \frac{1}{2\pi RC\sqrt{6}}$
Condition for Sustained Oscillation:
The gain of the amplifier ( $A_v$ ) must be at least 29 to overcome the attenuation of the feedback network ( $\beta = 1/29$ ).
$|A_v| \geq 29$

B. Wien Bridge Oscillator

This is one of the most popular audio frequency oscillators due to its stability and ease of tuning.

Structure: It uses a non-inverting amplifier (0 phase shift) and a Wien Bridge feedback network. The feedback network consists of a series RC arm and a parallel RC arm.
Phase Shift: At the resonant frequency, the feedback network produces a $0^\circ$ phase shift. Since the amplifier is non-inverting ( $0^\circ$ ), the total loop phase shift is $0^\circ$ .
Frequency of Oscillation:
$f_o = \frac{1}{2\pi RC}$
Condition for Sustained Oscillation:
The feedback factor $\beta$ at resonance is $1/3$ . Therefore, the closed-loop gain of the amplifier must be:
$A_v = 1 + \frac{R_f}{R_1} \geq 3$

4. LC Oscillators (Radio Frequency)

LC oscillators are used for high frequencies (Radio Frequency range: > 100 kHz). They use a tank circuit (LC parallel circuit) for resonance.

A. Hartley Oscillator

Identification: The tank circuit consists of two inductors in series (often a tapped coil) and one capacitor in parallel.
Feedback: The center tap of the inductors is grounded (or connected to the source/emitter). Feedback is taken across one inductor ( $L_2$ ) and input is applied across the other ( $L_1$ ).
Frequency of Oscillation:
$f_o = \frac{1}{2\pi \sqrt{L_{eq}C}}$
Where $L_{eq} = L_1 + L_2 + 2M$ ( $M$ is mutual inductance).
Advantages: easy to tune by varying $C$ .

B. Colpitts Oscillator

Identification: The tank circuit consists of two capacitors in series and one inductor in parallel.
Feedback: The voltage divider is formed by the two capacitors ( $C_1$ and $C_2$ ).
Frequency of Oscillation:
$f_o = \frac{1}{2\pi \sqrt{LC_{eq}}}$
Where $C_{eq} = \frac{C_1 C_2}{C_1 + C_2}$ (Series combination).
Advantages: Better frequency stability than Hartley because capacitors are less prone to stray magnetic field interference than tapped inductors.

5. Crystal Oscillator

Crystal oscillators provide the highest frequency stability. They utilize the Piezoelectric Effect of quartz crystals.

Piezoelectric Effect: When mechanical pressure is applied to a crystal, a voltage appears across its faces. Conversely, applying an AC voltage causes mechanical vibrations.

Equivalent Circuit:
A crystal behaves like an RLC circuit:

Series Arm: $L$ (mass), $C_s$ (compliance), $R$ (friction).
Parallel Arm: $C_m$ (mounting capacitance of electrodes).

Resonance Frequencies:

Series Resonance ( $f_s$ ): Impedance is minimum (Resistive).
Parallel Resonance ( $f_p$ ): Impedance is maximum.
- The crystal operates in the Inductive region between $f_s$ and $f_p$ .

Reading a Datasheet: 1 MHz Crystal

When selecting a 1 MHz crystal, the datasheet includes key parameters:

Nominal Frequency: 1.000000 MHz.
Frequency Tolerance: Expressed in ppm (parts per million), e.g., $\pm 30$ ppm. Indicates deviation at $25^\circ$ C.
Load Capacitance ( $C_L$ ): The external capacitance required across the crystal pins for it to oscillate exactly at the nominal frequency (typically 18pF or 20pF).
Equivalent Series Resistance (ESR): The resistive loss in the crystal (typically $20\Omega$ to $100\Omega$ ). Lower is better for easy startup.
Drive Level: The maximum power dissipated by the crystal (e.g., $100 \mu W$ ). Exceeding this damages the crystal.

Part B: The Operational Amplifier (Op-Amp)

1. Introduction and Schematic Symbol

An Operational Amplifier is a high-gain, direct-coupled, differential voltage amplifier. Originally designed to perform mathematical operations (addition, integration), it is now the fundamental building block of analog electronics.

Schematic Symbol:
A triangle pointing to the right.

Inverting Input (-): Signal applied here appears inverted at the output ( $180^\circ$ shift).
Non-Inverting Input (+): Signal applied here appears in-phase at the output.
Power Supply: Dual supply, $+V_{CC}$ and $-V_{EE}$ .

2. Block Diagram of a Typical Op-Amp

An Op-Amp consists of four main cascaded stages:

Input Stage (Dual Input, Balanced Output Differential Amplifier):
- Provides high input impedance and high Common Mode Rejection Ratio (CMRR).
- Suppress noise common to both inputs.
- Determines most of the Op-Amp's characteristics.
Intermediate Stage (Dual Input, Unbalanced Output Differential Amplifier):
- Driven by the output of the first stage.
- Provides high voltage gain.
- Converts the differential signal to a single-ended signal.
Level Shifting Stage (Emitter Follower with Current Source):
- The direct coupling of the first two stages raises the DC level of the signal above ground.
- This stage shifts the DC level back to zero volts with respect to ground so that output is 0V when input is 0V.
Output Stage (Push-Pull Complementary Symmetry Amplifier):
- Provides low output impedance.
- Increases the current supplying capability (to drive loads).
- Provides large output voltage swing.

3. Characteristics of Operational Amplifier

Parameter	Ideal Op-Amp Characteristics	Practical Op-Amp (IC 741)
Open Loop Gain ( $A_{OL}$ )	Infinite ( $\infty$ )	Very High (~ $2 \times 10^5$ )
Input Impedance ( $Z_{in}$ )	Infinite ( $\infty$ )	High (~ $2 M\Omega$ )
Output Impedance ( $Z_{out}$ )	Zero ($0$)	Low (~ $75 \Omega$ )
Bandwidth (BW)	Infinite (flat gain for all freq)	~1 MHz (unity gain BW)
Offset Voltage ( $V_{os}$ )	Zero ($0$)	Few mV (adjustable)
CMRR	Infinite	High (~90 dB)
Slew Rate	Infinite	Slow (~0.5 V/ $\mu$ s)

4. Reading the Datasheet of IC 741

The uA741 is the industry-standard general-purpose Op-Amp.

Pin Configuration (8-Pin DIP)

Pin 2: Inverting Input (-)
Pin 3: Non-Inverting Input (+)
Pin 6: Output
Pin 7: $+V_{CC}$ (Positive Supply)
Pin 4: $-V_{EE}$ (Negative Supply)
Pin 1 & 5: Offset Null (Used to zero the output voltage)
Pin 8: Not Connected (NC)

Key Datasheet Parameters Explained:

Absolute Maximum Ratings:
- Supply Voltage: Typically $\pm 18V$ . Exceeding this destroys the IC.
- Power Dissipation: Max power the package can handle (e.g., 500mW).
Input Offset Voltage ( $V_{IO}$ ):
- The small voltage required at the input to force the output to zero. For 741, typical is 2mV to 6mV.
Input Bias Current ( $I_{IB}$ ):
- The average of the currents flowing into the two input terminals. Practical op-amps require a small current to bias the internal transistors. Typical: 80nA.
Slew Rate (SR):
- The maximum rate of change of output voltage per unit of time.
- $SR = \frac{dV_o}{dt}|_{max}$
- For IC 741, $SR = 0.5 V/\mu s$ . This limits the op-amp's use in high-frequency or fast-switching applications.
Common Mode Rejection Ratio (CMRR):
- The ability of the op-amp to reject signals present on both inputs simultaneously (like noise).
- $CMRR = 20 \log (\frac{A_d}{A_{cm}})$
- For 741, typical is 90dB.
Gain Bandwidth Product (GBWP):
- The frequency at which the open-loop gain drops to unity (1). For 741, approx 1 MHz.