6.4.2.2.1 Core Components and Tank Circuit

The central part of a Colpitts oscillator's configuration is its LC tank circuit. Unlike the Hartley oscillator which uses a tapped inductor, the Colpitts oscillator employs a capacitive voltage divider in its resonant tank.

• Inductor (L): A single inductor, typically connected in parallel with the series combination of the two capacitors.
• Capacitors ($C_1$ and $C_2$): Two capacitors connected in series. These two capacitors effectively form a capacitive voltage divider.

The parallel combination of the inductor L and the series combination of $C_1$ and $C_2$ constitutes the frequency-determining resonant tank circuit. The total capacitance of this series combination is $C_{eq} = \frac{C_1 C_2}{C_1 + C_2}$.

6.4.2.2.2 Amplifier and Feedback Connection

The LC tank circuit is coupled with an active amplifying device, which can be a Bipolar Junction Transistor (BJT), a Field-Effect Transistor (FET), or an Operational Amplifier (Op-Amp). The configuration depends on the chosen amplifier type.

• Transistor-based Colpitts (Common-Emitter Configuration Example):
  • The collector of the transistor is typically connected to one end of the inductor (L) or through a Radio Frequency Choke (RFC) to the supply.
  • The base of the transistor (input to the amplifier) receives the feedback signal from the junction of $C_1$ and $C_2$.
  • The emitter of the transistor is often connected to the common point of the tank circuit (the junction between $C_1$ and $C_2$), which is also usually grounded for AC signals via a bypass capacitor ($C_E$). This creates a $180^\circ$ phase shift between the output (collector) and input (base) of the common-emitter amplifier.
  • The voltage across one capacitor (e.g., $C_1$) is usually connected to the input of the amplifier (e.g., base), while the output of the amplifier (e.g., collector) is coupled to the other end of the tank circuit. The voltage feedback is derived across $C_2$.
• Op-Amp based Colpitts (Non-inverting Configuration Example):
  • The LC tank circuit is placed in the feedback loop of the op-amp.
  • The non-inverting input of the op-amp provides $0^\circ$ phase shift. Therefore, the feedback network must also provide $0^\circ$ phase shift. This is achieved by the specific phase relationship between the voltages across $C_1$ and $C_2$ in the tank circuit at resonance.
  • The output of the op-amp is connected to one side of the tank circuit (e.g., the top of $L$).
  • The feedback signal to the non-inverting input is taken from the junction of $C_1$ and $C_2$.
  • The inverting input of the op-amp is typically connected to a voltage divider that sets the gain.

6.4.2.2.3 Phase Shift Requirements

For sustained oscillations, the Barkhausen Criteria must be met:
1. Loop Gain = 1: The total gain around the loop must be unity or slightly greater (to start oscillations).
2. Total Phase Shift = $0^\circ$ (or $360^\circ$): The total phase shift around the feedback loop must be zero degrees.

• Inverting Amplifier (e.g., Common Emitter BJT): The amplifier provides a $180^\circ$ phase shift. To achieve a total $360^\circ$ phase shift, the feedback network (the LC tank) must provide an additional $180^\circ$ phase shift. This is inherent in the design of the Colpitts feedback from the capacitive divider. The voltage across $C_1$ is out of phase with the voltage across $C_2$ relative to the common ground.
• Non-inverting Amplifier (e.g., Non-inverting Op-Amp): The amplifier provides $0^\circ$ phase shift. In this case, the LC tank circuit itself must act as a filter that provides $0^\circ$ phase shift at the resonant frequency, ensuring positive feedback.

6.4.2.2.4 Frequency of Oscillation

The oscillation frequency ($f_r$) of the Colpitts oscillator is determined by the resonance of its LC tank circuit:

$f_r = \frac{1}{2\pi \sqrt{L C_{eq}}}$

Where $C_{eq}$ is the equivalent series capacitance of $C_1$ and $C_2$:

$C_{eq} = \frac{C_1 C_2}{C_1 + C_2}$

Substituting $C_{eq}$ into the frequency formula:

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

6.4.2.2.5 Advantages of Colpitts Configuration

• Good Frequency Stability: Compared to Hartley oscillators, Colpitts often offers better frequency stability, especially at higher frequencies. This is because the capacitors in the tank circuit tend to swamp out (dominate over) the effects of parasitic capacitances of the active device (transistor or op-amp), making the frequency less sensitive to device variations.
• Easier Tuning: Tuning can be achieved by varying the inductor L or by using a ganged variable capacitor for $C_1$ and $C_2$ (though varying L is often more practical for high frequencies).
• Pure Sinusoidal Output: Capable of generating relatively pure sine waves.

6.4.2.2.6 Disadvantages

• Limited Frequency Tuning Range: If using fixed L and variable C, the range of frequency tuning is limited because $C_1$ and $C_2$ are in series and must maintain a specific ratio for proper feedback.
• Starting Issues: Can sometimes be difficult to start oscillating, especially if the loop gain is not sufficient.

Exercise Solutions

Easy:

1. The primary function of the Wien Bridge network in a Wien Bridge Oscillator is to act as the frequency-selective feedback network that determines the oscillation frequency. Its frequency-determining arms are composed of Resistors (R) and Capacitors (C).
2. The resonant frequency (fr) of the Wien Bridge network is the specific frequency at which the network provides exactly zero degrees (0°) of phase shift between its input and output. This zero phase shift condition is crucial for meeting one of the Barkhausen Criteria for sustained oscillations when combined with a non-inverting amplifier.

Medium:

1. Key characteristics of the Wien Bridge network at its resonant frequency (fr):
   a) Phase Shift: The phase shift introduced by the RC network is exactly zero degrees (0°). This is the critical condition for its use in the positive feedback path of a non-inverting amplifier.
   b) Voltage Transfer Ratio (Attenuation): At this resonant frequency, the network significantly attenuates the signal. Its voltage transfer ratio (output voltage / input voltage) is typically 1/3 (when R1=R2=R and C1=C2=C). This means the output signal from the bridge is one-third of the input signal to the bridge.
2. Given: R1 = R2 = R = 10 kOhm = 10,000 Ohm, C1 = C2 = C = 10 nF = 10 x 10^-9 F.
   a) Calculate resonant frequency (fr):
   fr = 1 / (2 * pi * R * C)
   fr = 1 / (2 * 3.14159 * 10000 Ohm * 10 x 10^-9 F)
   fr = 1 / (2 * 3.14159 * 100 x 10^-6)
   fr = 1 / (0.000628318)
   fr approx = 15915.5 Hz (or approximately 15.92 kHz) b) Minimum amplifier gain:
   At its resonant frequency, the Wien Bridge network attenuates the signal by a factor of 3 (transfer ratio = 1/3). To satisfy the Barkhausen criterion of a loop gain of 1, the amplifier must compensate for this attenuation.
   Loop Gain = Amplifier Gain (Av) * Feedback Network Transfer Ratio
   1 = Av * (1/3)
   Av = 3
   So, the minimum voltage gain the amplifier must provide is 3. In practice, it's set slightly higher to ensure oscillations start.

Hard:

1. The Barkhausen Criteria for sustained sinusoidal oscillations are:
   1. Loop Gain = 1 (or slightly greater than 1 initially): The magnitude of the loop gain, which is the product of the amplifier's gain (Av) and the feedback network's transfer ratio (Beta), must be equal to or slightly greater than unity (Av * Beta >= 1).
   2. Total Phase Shift = 0° (or 360°): The total phase shift around the entire feedback loop (amplifier phase shift + feedback network phase shift) must be zero degrees or a multiple of 360 degrees.
   How the Wien Bridge Network and Amplifier Satisfy these Criteria:
   • Phase Shift Condition:
     • The non-inverting amplifier configuration inherently provides 0° phase shift between its input and output.
     • The Wien Bridge network is specifically designed such that at its unique resonant frequency (fr), the phase shift it introduces is exactly 0°.
     • Therefore, at fr, the total phase shift around the loop (0° from amplifier + 0° from Wien Bridge) is 0°, satisfying the second Barkhausen criterion.
   • Loop Gain Condition:
     • At its resonant frequency (fr), the Wien Bridge network attenuates the signal, providing a transfer ratio (Beta) of 1/3.
     • To achieve a loop gain of 1 (Av * Beta = 1), the non-inverting amplifier must provide a voltage gain (Av) of at least 3.
     • In practice, the amplifier gain is initially set slightly above 3 to ensure oscillations start, and then an amplitude stabilization mechanism reduces the effective gain to precisely 3 for stable, low-distortion output once oscillations build up. This satisfies the first Barkhausen criterion.
2. Three significant advantages of using the Wien Bridge network in an oscillator design:
   1. No Inductors Required: Unlike LC oscillators, the Wien Bridge Oscillator uses only resistors and capacitors. This is a major advantage because inductors can be bulky, expensive, difficult to fabricate in integrated circuits, and susceptible to electromagnetic interference. Using only R and C makes the circuit simpler, smaller, and often more cost-effective.
   2. Wide Frequency Range and Easy Tunability: The oscillation frequency (fr = 1 / (2 * pi * R * C)) is determined by the values of R and C. By using dual-ganged variable resistors or capacitors, the frequency can be easily tuned over a very wide range (e.g., several decades), making Wien Bridge Oscillators ideal for audio frequency generators and function generators.
   3. Low Distortion Sinusoidal Output: With proper design and especially with the inclusion of an amplitude stabilization mechanism (like a thermistor, lamp, or diodes), Wien Bridge Oscillators can produce very pure, low-distortion sinusoidal waveforms. This makes them suitable for applications requiring high-quality sine waves, such as audio testing equipment.
   (Other advantages include: Good frequency stability, simple design with readily available components.)

Quiz Answers

1. b) Resistors and Capacitors (RC)
2. False.
   • At very high frequencies, both capacitors act like short circuits. The series capacitor acts like a short, and the parallel capacitor shunts the output to ground, leading to significant attenuation, but the phase shift approaches 0 degrees.
3. At its resonant frequency (fr), the Wien Bridge network provides a phase shift of exactly 0 degrees.
4. c) 3
5. c) To ensure the output is a pure, low-distortion sinusoidal waveform.
6. c) Approximately 1.59 kHz
   • fr = 1 / (2 * pi * R * C) = 1 / (2 * 3.14159 * 1000 Ohm * 0.0000001 F)
   • fr = 1 / (0.000628318) approx = 1591.55 Hz = 1.59 kHz
7. c) It can be easily tuned over a wide frequency range.