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<b>Designing Low and High Frequency Oscillator Circuits</b> === Objective What You Will Learn ===
[[File:TUT12-1.png|thumb|400px| The Wien Bridge Oscillator.]]
In the first part of this tutorial lesson, you will build a Wien Bridge Oscillator using an LM741 Op-Amp and will analyze its oscillatory behavior. In the second part, you will build and test a high frequency Colpitts oscillator using a parallel resonant LC circuit. The primary objective of this tutorial lesson is to underline the challenges of analyzing oscillator circuits.
=== Designing and Testing a Wien Bridge Oscillator===
The following is a list of parts needed for this part of the tutorial lesson:
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Voltage Source VCC: 15V
Voltage Source VEE: -15V
LM741 Op-Amp: Accessible form "Parts > Integrated Circuits" menu
Four Resistors: R1, R2, R3 and R4
Two Capacitors: C1 and C2
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The Wien Bridge Oscillator comprises an Op-Amp, four resistors and two capacitors. The oscillator can also be viewed as a positive gain amplifier combined with a bandpass filter that provides positive feedback. At the resonant frequency, the reactance of the series R2–C2 arm will be an exact multiple of the shunt R1–C1 arm. If the two R3 and R4 arms are adjusted to the same ratio, then the bridge is balanced. These conditions can be written as:
<math>\omega^2 = \frac {1}{R_1 R_2 C_1 C_2}</math>
and
<math> \frac {C_1}{C_2} = \frac {R_4}{R_3} - \frac {R_2}{R_1}</math>
This can be further simplified by setting R1 = R2 = R and C1 = C2 = C. In that case, setting R4 = 2 R3 will satisfy the above conditions and will lead to the following simple equation for the oscillation frequency:
<math>f_o = \frac {1}{2 \pi RC}</math>
For this project, you will initially choose R1 = R2 = 10k, R3 = 10k, R3 = 2 R3 = 20k, and C1 = C3 = 100pF. Place and connect all the parts as shown in the above figure. Place a voltage probe marker at the output of the Op-Amp. The oscillation frequency of this circuit must be f<sub>o</sub> = 1 / (2π . 1e+4 . 1e-9) = 15.9kHz.
Run a Transient Test of this circuit with start and stop times set to 0 and 25ms and a Step Ceiling of 1μs. You will get a graph like the one shown below.
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[[File:TUT12-2.png|thumb|800px|The output voltage of the Wien Bridge Oscillator when R4 = 20k.]]
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The circuit is certainly not oscillating in a sustainable way as the output voltage fluctuates around 3.3mV. Next, change the value of R4 to 21k and run a new Transient Test. This time you will get a fairly large oscillation between -5V and +5V as show below.
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[[File:TUT12-3.png|thumb|800px|The output voltage of the Wien Bridge Oscillator when R4 = 21k.]]
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In order to calculate the oscillation frequency, first zoom in the graph window by clicking the "Zoom In" button of the [[Graph toolbar|Graph Toolbar]]. Then use the "Delta Tool" of the [[Graph toolbar|Graph Toolbar]] to measure the distance between two successive peaks of the sinusoidal signal. The period is about 78.5μs, which gives an oscillation frequency of 12.75kHz.
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[[File:TUT12-4.png|thumb|800px|Measuring the signal period on a close-up of the output voltage of the Wien Bridge Oscillator when R4 = 21k.]]
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Next, reduce the value of R4 further to 20.25k and run a new Transient Test again, but this time with a stop time of 50ms. You will notice that the sustainable oscillations start at much later time around 30ms with a reduced peak-to-peak level. If you zoom in the graph and measure the signal period, you will note that it has reduced to 71.5μs, which gives an oscillation frequency of about 14kHz, fairly close to the design frequency.
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[[File:TUT12-5.png|thumb|800px|The output voltage of the Wien Bridge Oscillator when R4 = 20.25k.]]
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=== Analyzing the Wien Bridge Oscillator and Explaining Its Behavior ===
The above behavior of the circuit can be explain using the feedback theory. The Wien Bridge Oscillator is indeed an non-inverting Op-Amp amplifier with positive feedback provided using two series and shunt RC impedances:
<math>Z_1(s) = \frac {R . \frac{1}{sC}}{R + \frac{1}{sC}} </math>
<math>Z_2(s) = R + \frac{1}{sC} </math>
The gain of the non-inverting amplifier is given by:
<math>A = 1 + \frac{R_4}{R_3} </math>
The feedback gain can be found from the voltage divider:
<math> \beta (s) = \frac{Z_1(s)}{Z_1(s) + Z_2(s)} = \frac {sRC} {s^2 R^2 C^2 + 3sRC + 1} </math>
The Barkhausen criterion for oscillation requires that T(s) = Aβ(s) ≥ 1. Substituting s = jω, one can write:
<math> T(j\omega) = A \beta(j\omega) = \frac {j\omega RC} {(1-\omega ^2 R^2 C^2)+ 3j\omega RC} </math>
A zero phase shift can be achieved only if
<math> 1-\omega ^2 R^2 C^2 = 0 </math>
which lead to the oscillation frequency ω<sub>o</sub> = 1/(RC). This further simplifies the Barkhausen criterion to
<math> T(j\omega) = \frac {A}{3} \ge 1</math>
which means that A = 1 + R<sub>4</sub>/R<sub>3</sub> ≥ 3 or R<sub>4</sub> ≥ 2R<sub>3</sub>. This shows that setting R<sub>4</sub> = 2R<sub>3</sub> provides the threshold of oscillation. For R<sub>4</sub> > 2R<sub>3</sub>, positive feedback causes the oscillations to grow indefinitely.
=== Designing a BJT Colpitts Oscillator ===
[[File:TUT12-6.png|thumb|400px| The Colpitts Oscillator.]]
<math>A \beta(s) = \left( \frac{C_3}{C_3 + C_4} \right). \left( \frac{I_C}{V_T} \right). R_3 \ge 1</math>
=== Building and & Testing the Colpitts Oscillator ===
For this project, you will choose L1 = 27μH, C1 = 100nF, C2 = 1nF, C3 = 1nF, C4 = 15nF, R1 = 22k, R2 = 10k, R3 = 2k and R4 = 1k. Place and connect all the parts as shown in the above figure. Place a voltage probe marker at the output of the oscillator circuit at the collector of Q1. For the given inductance and capacitance value, the oscillation frequency of this Colpitts circuit is found to be f<sub>o</sub> = 1MHz.
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=== Changing the Tuning of the Colpitts Oscillator ===
In this part, let's change the tuning of the parallel LC circuit to get an oscillation frequency of 4MHz. Due to the presence of the square root in the denominator of the formula for f<sub>o</sub>, you need to make the product of L1 and (C3 || C4) 16 times smaller. To do so, set C3 to 0.25nF and L1 to 6.75μH. Run a new transient test of your modified oscillator with the stop time set to 50μs. As you can see from the figure below, the output voltage this time oscillates between 3V and 11V with a period of about 260ns, which corresponds to an oscillation frequency of 3.85MHz.
