วงจรออสซิลเลเตอร์
ออสซิลเลเตอร์แปลงอินพุต DC (แรงดันไฟฟ้า) เป็นเอาต์พุต AC (รูปคลื่น) ซึ่งสามารถมีรูปร่างและความถี่คลื่นที่แตกต่างกันได้หลากหลายซึ่งอาจมีความซับซ้อนในลักษณะหรือคลื่นไซน์อย่างง่ายขึ้นอยู่กับการใช้งาน นอกจากนี้ยังใช้ออสซิลเลเตอร์ในอุปกรณ์ทดสอบหลายชิ้นที่ผลิตคลื่นไซน์ไซน์รูปสี่เหลี่ยมจัตุรัสฟันเลื่อยหรือรูปสามเหลี่ยมหรือเพียงแค่รถไฟของพัลส์ที่มีความกว้างตัวแปรหรือคงที่ LC Oscillators มักใช้ในวงจรความถี่วิทยุเนื่องจากมีลักษณะสัญญาณรบกวนเฟสที่ดีและใช้งานง่าย
โดยพื้นฐานแล้วออสซิลเลเตอร์เป็นแอมพลิฟายเออร์ที่มี“ ผลตอบรับเชิงบวก” หรือข้อเสนอแนะเชิงปฏิรูป (ในเฟส) และหนึ่งในปัญหามากมายในการออกแบบวงจรอิเล็กทรอนิกส์คือการหยุดแอมพลิฟายเออร์ไม่ให้สั่นในขณะที่พยายามทำให้ออสซิลเลเตอร์สั่น ออสซิลเลเตอร์ทำงานเนื่องจากเอาชนะการสูญเสียของวงจรเรโซเนเตอร์ป้อนกลับไม่ว่าจะในรูปแบบของตัวเก็บประจุตัวเหนี่ยวนำหรือทั้งสองอย่างในวงจรเดียวกันโดยใช้พลังงาน DC ที่ความถี่ที่ต้องการในวงจรเรโซเนเตอร์นี้ กล่าวอีกนัยหนึ่งออสซิลเลเตอร์เป็นเครื่องขยายเสียงที่ใช้ผลตอบรับเชิงบวกที่สร้างความถี่เอาต์พุตโดยไม่ต้องใช้สัญญาณอินพุต
ดังนั้นออสซิลเลเตอร์จึงเป็นวงจรที่สร้างรูปคลื่นเอาต์พุตเป็นระยะ ๆ ที่ความถี่ที่แม่นยำและสำหรับวงจรอิเล็กทรอนิกส์ใด ๆ ที่จะทำงานเป็นออสซิลเลเตอร์จะต้องมีลักษณะสามประการดังต่อไปนี้
การขยายบางรูปแบบ
ข้อเสนอแนะเชิงบวก (การสร้างใหม่)
ความถี่กำหนดเครือข่ายข้อเสนอแนะ
Some form of Amplification
Positive Feedback (regeneration)
A Frequency determine feedback network
ออสซิลเลเตอร์มีแอมพลิฟายเออร์ป้อนกลับสัญญาณขนาดเล็กที่มีอัตราขยายวงเปิดเท่ากับหรือมากกว่าหนึ่งเล็กน้อยเพื่อให้การสั่นเริ่มต้น แต่เพื่อให้การสั่นดำเนินต่อไปการขยายวงเฉลี่ยจะต้องกลับมาเป็นเอกภาพ นอกเหนือจากส่วนประกอบที่ทำปฏิกิริยาเหล่านี้แล้วจำเป็นต้องมีอุปกรณ์ขยายสัญญาณเช่น Operational Amplifier หรือ Bipolar Transistor ต่างจากแอมพลิฟายเออร์ตรงที่ไม่มีอินพุต AC ภายนอกที่จำเป็นในการทำให้ออสซิลเลเตอร์ทำงานเนื่องจากพลังงานของแหล่งจ่ายกระแสตรงจะถูกแปลงโดยออสซิลเลเตอร์เป็นพลังงาน AC ที่ความถี่ที่ต้องการ
Where: β is a feedback fraction.
Oscillators are circuits that generate a continuous voltage output waveform at a required frequency with the values of the inductors, capacitors or resistors forming a frequency selective LC resonant tank circuit and feedback network. This feedback network is an attenuation network which has a gain of less than one ( β <1 ) and starts oscillations when Aβ >1 which returns to unity ( Aβ =1 ) once oscillations commence.
The LC oscillators frequency is controlled using a tuned or resonant inductive/capacitive (LC) circuit with the resulting output frequency being known as the Oscillation Frequency. By making the oscillators feedback a reactive network the phase angle of the feedback will vary as a function of frequency and this is called Phase-shift.
There are basically types of Oscillators
1. Sinusoidal Oscillators – these are known as Harmonic Oscillators and are generally a “LC Tuned-feedback” or “RC tuned-feedback” type Oscillator that generates a purely sinusoidal waveform which is of constant amplitude and frequency.
2. Non-Sinusoidal Oscillators – these are known as Relaxation Oscillators and generate complex non-sinusoidal waveforms that changes very quickly from one condition of stability to another such as “Square-wave”, “Triangular-wave” or “Sawtoothed-wave” type waveforms.
เมื่อใช้แรงดันไฟฟ้าคงที่ แต่มีความถี่แตกต่างกันกับวงจรที่ประกอบด้วยตัวเหนี่ยวนำตัวเก็บประจุและตัวต้านทานปฏิกิริยาของทั้งวงจรตัวเก็บประจุ / ตัวต้านทานและตัวเหนี่ยวนำ / ตัวต้านทานจะเปลี่ยนทั้งแอมพลิจูดและเฟสของสัญญาณเอาต์พุตเมื่อเทียบกับ สัญญาณอินพุตเนื่องจากปฏิกิริยาของส่วนประกอบที่ใช้ ที่ความถี่สูงรีแอคแตนซ์ของตัวเก็บประจุจะต่ำมากซึ่งทำหน้าที่เป็นไฟฟ้าลัดวงจรในขณะที่รีแอคแตนซ์ของตัวเหนี่ยวนำสูงทำหน้าที่เป็นวงจรเปิด ที่ความถี่ต่ำการย้อนกลับเป็นจริงรีแอคแตนซ์ของตัวเก็บประจุทำหน้าที่เป็นวงจรเปิดและรีแอคแตนซ์ของตัวเหนี่ยวนำจะทำหน้าที่ลัดวงจร ระหว่างสองขั้วนี้การรวมกันของตัวเหนี่ยวนำและตัวเก็บประจุจะทำให้เกิดวงจร "Tuned" หรือ "Resonant" ที่มีความถี่เรโซแนนซ์ (ƒr) ซึ่งค่าปฏิกริยาคาปาซิทีฟและอุปนัยมีค่าเท่ากันและตัดกันเหลือเพียงความต้านทานของ วงจรต่อต้านการไหลของกระแส ซึ่งหมายความว่าไม่มีการเปลี่ยนเฟสเนื่องจากกระแสไฟฟ้าอยู่ในเฟสที่มีแรงดันไฟฟ้า พิจารณาวงจรด้านล่าง
วงจรประกอบด้วยขดลวดอุปนัย L และตัวเก็บประจุ C. ตัวเก็บประจุเก็บพลังงานในรูปของสนามไฟฟ้าสถิตและก่อให้เกิดศักย์ (แรงดันไฟฟ้าสถิตย์) บนจานในขณะที่ขดลวดอุปนัยเก็บพลังงานไว้ในรูปของ สนามแม่เหล็กไฟฟ้า ตัวเก็บประจุจะถูกชาร์จถึงแรงดันไฟฟ้ากระแสตรง, V โดยวางสวิตช์ในตำแหน่ง A เมื่อตัวเก็บประจุถูกชาร์จจนเต็มสวิตช์จะเปลี่ยนเป็นตำแหน่ง B
ตอนนี้ตัวเก็บประจุที่มีประจุถูกเชื่อมต่อแบบขนานบนขดลวดอุปนัยดังนั้นตัวเก็บประจุจะเริ่มปล่อยตัวเองผ่านขดลวด แรงดันไฟฟ้าคร่อม C เริ่มลดลงเมื่อกระแสผ่านขดลวดเริ่มสูงขึ้น
กระแสไฟฟ้าที่เพิ่มขึ้นนี้จะสร้างสนามแม่เหล็กไฟฟ้ารอบ ๆ ขดลวดซึ่งต้านทานการไหลของกระแสนี้ เมื่อตัวเก็บประจุ C จะปล่อยพลังงานที่เก็บไว้ในตัวเก็บประจุออกจนหมด C ในฐานะสนามไฟฟ้าสถิตจะถูกเก็บไว้ในขดลวดอุปนัย L เป็นสนามแม่เหล็กไฟฟ้ารอบขดลวดขดลวด
เนื่องจากตอนนี้ไม่มีแรงดันไฟฟ้าภายนอกในวงจรเพื่อรักษากระแสภายในขดลวดจึงเริ่มตกลงเมื่อสนามแม่เหล็กไฟฟ้าเริ่มยุบลง แรงเคลื่อนไฟฟ้าย้อนกลับถูกเหนี่ยวนำในขดลวด (e = -Ldi / dt) ทำให้กระแสไหลไปในทิศทางเดิม
กระแสนี้จะชาร์จตัวเก็บประจุ C ที่มีขั้วตรงข้ามกับประจุเดิม C จะชาร์จต่อไปเรื่อย ๆ จนกว่ากระแสจะลดลงเป็นศูนย์และสนามแม่เหล็กไฟฟ้าของขดลวดจะยุบตัวลงอย่างสมบูรณ์
พลังงานเดิมที่นำเข้าสู่วงจรผ่านสวิตช์จะถูกส่งกลับไปที่ตัวเก็บประจุซึ่งมีศักย์ไฟฟ้าสถิตอีกครั้งแม้ว่าตอนนี้จะอยู่ในขั้วตรงกันข้ามก็ตาม ตอนนี้ตัวเก็บประจุเริ่มปล่อยกลับมาอีกครั้งผ่านขดลวดและทำซ้ำกระบวนการทั้งหมด ขั้วของแรงดันไฟฟ้าจะเปลี่ยนไปเมื่อพลังงานถูกส่งผ่านไปมาระหว่างตัวเก็บประจุและตัวเหนี่ยวนำที่สร้างแรงดันไฟฟ้าไซน์ชนิด AC และรูปคลื่นกระแสไฟฟ้า
จากนั้นกระบวนการนี้จะเป็นพื้นฐานของวงจรรถถัง LC oscillators และในทางทฤษฎีการหมุนวนไปมาจะดำเนินต่อไปอย่างไม่มีกำหนด อย่างไรก็ตามสิ่งต่าง ๆ ไม่สมบูรณ์แบบและทุกครั้งที่มีการถ่ายโอนพลังงานจากตัวเก็บประจุ C ไปยังตัวเหนี่ยวนำ L และกลับจาก L ถึง C การสูญเสียพลังงานบางอย่างเกิดขึ้นซึ่งจะสลายการสั่นเป็นศูนย์เมื่อเวลาผ่านไป
การดำเนินการสั่นของการส่งผ่านพลังงานไปมาระหว่างตัวเก็บประจุ, C ไปยังตัวเหนี่ยวนำ, L จะดำเนินต่อไปเรื่อย ๆ ถ้าไม่ใช่เพื่อการสูญเสียพลังงานภายในวงจร พลังงานไฟฟ้าจะสูญเสียไปใน DC หรือความต้านทานจริงของขดลวดตัวเหนี่ยวนำในอิเล็กทริกของตัวเก็บประจุและในการแผ่รังสีจากวงจรดังนั้นการสั่นจะลดลงเรื่อย ๆ จนกว่าพวกมันจะตายไปอย่างสมบูรณ์และกระบวนการจะหยุดลง
จากนั้นในวงจร LC ที่ใช้งานได้จริงแอมพลิจูดของแรงดันไฟฟ้าของการสั่นจะลดลงในแต่ละครึ่งรอบของการสั่นและในที่สุดก็จะตายไปเป็นศูนย์ จากนั้นการสั่นจะถูกกล่าวว่า "ทำให้หมาด ๆ " โดยปริมาณการทำให้หมาด ๆ จะถูกกำหนดโดยคุณภาพหรือปัจจัย Q ของวงจร
The frequency of the oscillatory voltage depends upon the value of the inductance and capacitance in the LC tank circuit. We now know that for resonance to occur in the tank circuit, there must be a frequency point were the value of XC, the capacitive reactance is the same as the value of XL, the inductive reactance ( XL = XC ) and which will therefore cancel out each other out leaving only the DC resistance in the circuit to oppose the flow of current.
If we now place the curve for inductive reactance of the inductor on top of the curve for capacitive reactance of the capacitor so that both curves are on the same frequency axes, the point of intersection will give us the resonance frequency point, ( ƒr or ωr ) as shown below.
Where: ƒr is in Hertz, L is in Henries and C is in Farads.
Then the frequency at which this will happen is given as:
Then by simplifying the above equation we get the final equation for Resonant Frequency, ƒr in a tuned LC circuit as:
Where:
L is the Inductance in Henries
C is the Capacitance in Farads
ƒr is the Output Frequency in Hertz
This equation shows that if either L or C are decreased, the frequency increases. This output frequency is commonly given the abbreviation of ( ƒr ) to identify it as the “resonant frequency”.
To keep the oscillations going in an LC tank circuit, we have to replace all the energy lost in each oscillation and also maintain the amplitude of these oscillations at a constant level. The amount of energy replaced must therefore be equal to the energy lost during each cycle.
If the energy replaced is too large the amplitude would increase until clipping of the supply rails occurs. Alternatively, if the amount of energy replaced is too small the amplitude would eventually decrease to zero over time and the oscillations would stop.
The simplest way of replacing this lost energy is to take part of the output from the LC tank circuit, amplify it and then feed it back into the LC circuit again. This process can be achieved using a voltage amplifier using an op-amp, FET or bipolar transistor as its active device. However, if the loop gain of the feedback amplifier is too small, the desired oscillation decays to zero and if it is too large, the waveform becomes distorted.
To produce a constant oscillation, the level of the energy fed back to the LC network must be accurately controlled. Then there must be some form of automatic amplitude or gain control when the amplitude tries to vary from a reference voltage either up or down.
To maintain a stable oscillation the overall gain of the circuit must be equal to one or unity. Any less and the oscillations will not start or die away to zero, any more the oscillations will occur but the amplitude will become clipped by the supply rails causing distortion. Consider the circuit below.
A Bipolar Transistor is used as the LC oscillators amplifier with the tuned LC tank circuit acts as the collector load. Another coil L2 is connected between the base and the emitter of the transistor whose electromagnetic field is “mutually” coupled with that of coil L.
“Mutual inductance” exists between the two circuits and the changing current flowing in one coil circuit induces, by electromagnetic induction, a potential voltage in the other (transformer effect) so as the oscillations occur in the tuned circuit, electromagnetic energy is transferred from coil L to coil L2 and a voltage of the same frequency as that in the tuned circuit is applied between the base and emitter of the transistor. In this way the necessary automatic feedback voltage is applied to the amplifying transistor.
The amount of feedback can be increased or decreased by altering the coupling between the two coils L and L2. When the circuit is oscillating its impedance is resistive and the collector and base voltages are 180o out of phase. In order to maintain oscillations (called frequency stability) the voltage applied to the tuned circuit must be “in-phase” with the oscillations occurring in the tuned circuit.
Therefore, we must introduce an additional 180o phase shift into the feedback path between the collector and the base. This is achieved by winding the coil of L2 in the correct direction relative to coil L giving us the correct amplitude and phase relationships for the Oscillators circuit or by connecting a phase shift network between the output and input of the amplifier.
The LC Oscillator is therefore a “Sinusoidal Oscillator” or a “Harmonic Oscillator” as it is more commonly called. LC oscillators can generate high frequency sine waves for use in radio frequency (RF) type applications with the transistor amplifier being of a Bipolar Transistor or FET.
Harmonic Oscillators come in many different forms because there are many different ways to construct an LC filter network and amplifier with the most common being the Hartley LC Oscillator, Colpitts LC Oscillator, Armstrong Oscillator and Clapp Oscillator to name a few.
An inductance of 200mH and a capacitor of 10pF are connected together in parallel to create an LC oscillator tank circuit. Calculate the frequency of oscillation.
Then we can see from the above example that by decreasing the value of either the capacitance, C or the inductance, L will have the effect of increasing the frequency of oscillation of the LC tank circuit.
The basic conditions required for an LC oscillator resonant tank circuit are given as follows.
For oscillations to exist an oscillator circuit MUST contain a reactive (frequency-dependant) component either an “Inductor”, (L) or a “Capacitor”, (C) as well as a DC power source.
In a simple inductor-capacitor, LC circuit, oscillations become damped over time due to component and circuit losses.
Voltage amplification is required to overcome these circuit losses and provide positive gain.
The overall gain of the amplifier must be greater than one, unity.
Oscillations can be maintained by feeding back some of the output voltage to the tuned circuit that is of the correct amplitude and in-phase, (0o).
Oscillations can only occur when the feedback is “Positive” (self-regeneration).
The overall phase shift of the circuit must be zero or 360o so that the output signal from the feedback network will be “in-phase” with the input signal.
In the next tutorial about Oscillators, we will examine the operation of one of the most common LC oscillator circuits that uses two inductance coils to form a centre tapped inductance within its resonant tank circuit. This type of LC oscillator circuit is known commonly as a Hartley Oscillator.
The Hartley Oscillator design uses two inductive coils in series with a parallel capacitor to form its resonance tank circuit producing sinusoidal oscillations
One of the main disadvantages of the basic LC Oscillator circuit we looked at in the previous tutorial is that they have no means of controlling the amplitude of the oscillations and also, it is difficult to tune the oscillator to the required frequency. If the cumulative electromagnetic coupling between L1 and L2 is too small there would be insufficient feedback and the oscillations would eventually die away to zero.
Likewise if the feedback was too strong the oscillations would continue to increase in amplitude until they were limited by the circuit conditions producing signal distortion. So it becomes very difficult to “tune” the oscillator.
However, it is possible to feed back exactly the right amount of voltage for constant amplitude oscillations. If we feed back more than is necessary the amplitude of the oscillations can be controlled by biasing the amplifier in such a way that if the oscillations increase in amplitude, the bias is increased and the gain of the amplifier is reduced.
If the amplitude of the oscillations decreases the bias decreases and the gain of the amplifier increases, thus increasing the feedback. In this way the amplitude of the oscillations are kept constant using a process known as Automatic Base Bias.
One big advantage of automatic base bias in a voltage controlled oscillator, is that the oscillator can be made more efficient by providing a Class-B bias or even a Class-C bias condition of the transistor. This has the advantage that the collector current only flows during part of the oscillation cycle so the quiescent collector current is very small. Then this “self-tuning” base oscillator circuit forms one of the most common types of LC parallel resonant feedback oscillator configurations called the Hartley Oscillator circuit.
In the Hartley Oscillator the tuned LC circuit is connected between the collector and the base of a transistor amplifier. As far as the oscillatory voltage is concerned, the emitter is connected to a tapping point on the tuned circuit coil.
The feedback part of the tuned LC tank circuit is taken from the centre tap of the inductor coil or even two separate coils in series which are in parallel with a variable capacitor, C as shown.
The Hartley circuit is often referred to as a split-inductance oscillator because coil L is centre-tapped. In effect, inductance L acts like two separate coils in very close proximity with the current flowing through coil section XY induces a signal into coil section YZ below.
An Hartley Oscillator circuit can be made from any configuration that uses either a single tapped coil (similar to an autotransformer) or a pair of series connected coils in parallel with a single capacitor as shown below.
Hartley Oscillator Tank Circuit
When the circuit is oscillating, the voltage at point X (collector), relative to point Y (emitter), is 180o out-of-phase with the voltage at point Z (base) relative to point Y. At the frequency of oscillation, the impedance of the Collector load is resistive and an increase in Base voltage causes a decrease in the Collector voltage.
Thus there is a 180o phase change in the voltage between the Base and Collector and this along with the original 180o phase shift in the feedback loop provides the correct phase relationship of positive feedback for oscillations to be maintained.
The amount of feedback depends upon the position of the “tapping point” of the inductor. If this is moved nearer to the collector the amount of feedback is increased, but the output taken between the Collector and earth is reduced and vice versa. Resistors, R1 and R2 provide the usual stabilizing DC bias for the transistor in the normal manner while the capacitors act as DC-blocking capacitors.
In this Hartley Oscillator circuit, the DC Collector current flows through part of the coil and for this reason the circuit is said to be “Series-fed” with the frequency of oscillation of the Hartley Oscillator being given as.
Note: LT is the total cumulatively coupled inductance if two separate coils are used including their mutual inductance, M.
The frequency of oscillations can be adjusted by varying the “tuning” capacitor, C or by varying the position of the iron-dust core inside the coil (inductive tuning) giving an output over a wide range of frequencies making it very easy to tune. Also the Hartley Oscillator produces an output amplitude which is constant over the entire frequency range.
As well as the Series-fed Hartley Oscillator above, it is also possible to connect the tuned tank circuit across the amplifier as a shunt-fed oscillator as shown below.
In the shunt-fed Hartley oscillator circuit, both the AC and DC components of the Collector current have separate paths around the circuit. Since the DC component is blocked by the capacitor, C2 no DC flows through the inductive coil, L and less power is wasted in the tuned circuit.
The Radio Frequency Coil (RFC), L2 is an RF choke which has a high reactance at the frequency of oscillations so that most of the RF current is applied to the LC tuning tank circuit via capacitor, C2 as the DC component passes through L2 to the power supply. A resistor could be used in place of the RFC coil, L2 but the efficiency would be less.
A Hartley Oscillator circuit having two individual inductors of 0.5mH each, are designed to resonate in parallel with a variable capacitor that can be adjusted between 100pF and 500pF. Determine the upper and lower frequencies of oscillation and also the Hartley oscillators bandwidth.
From above we can calculate the frequency of oscillations for a Hartley Oscillator as:
The circuit consists of two inductive coils in series, so the total inductance is given as:
As well as using a bipolar junction transistor (BJT) as the amplifiers active stage of the Hartley oscillator, we can also use either a field effect transistor, (FET) or an operational amplifier, (op-amp). The operation of an Op-amp Hartley Oscillator is exactly the same as for the transistorised version with the frequency of operation calculated in the same manner. Consider the circuit below.
The advantage of constructing a Hartley Oscillator using an operational amplifier as its active stage is that the gain of the op-amp can be very easily adjusted using the feedback resistors R1 and R2. As with the transistorised oscillator above, the gain of the circuit must be equal too or slightly greater than the ratio of L1/L2. If the two inductive coils are wound onto a common core and mutual inductance M exists then the ratio becomes (L1+M)/(L2+M).
Then to summarise, the Hartley Oscillator consists of a parallel LC resonator tank circuit whose feedback is achieved by way of an inductive divider. Like most oscillator circuits, the Hartley oscillator exists in several forms, with the most common form being the transistor circuit above.
This Hartley Oscillator configuration has a tuned tank circuit with its resonant coil tapped to feed a fraction of the output signal back to the emitter of the transistor. Since the output of the transistors emitter is always “in-phase” with the output at the collector, this feedback signal is positive. The oscillating frequency which is a sine-wave voltage is determined by the resonance frequency of the tank circuit.
In the next tutorial about Oscillators, we will look at another type of LC oscillator circuit that is the opposite to the Hartley oscillator called the Colpitts Oscillator. The Colpitts oscillator uses two capacitors in series to form a centre tapped capacitance in parallel with a single inductance within its resonant tank circuit.
In many ways, the Colpitts oscillator is the exact opposite of the Hartley Oscillator we looked at in the previous tutorial. Just like the Hartley oscillator, the tuned tank circuit consists of an LC resonance sub-circuit connected between the collector and the base of a single stage transistor amplifier producing a sinusoidal output waveform.
The basic configuration of the Colpitts Oscillator resembles that of the Hartley Oscillator but the difference this time is that the centre tapping of the tank sub-circuit is now made at the junction of a “capacitive voltage divider” network instead of a tapped autotransformer type inductor as in the Hartley oscillator.
The Colpitts oscillator uses a capacitive voltage divider network as
its feedback source. The two capacitors, C1 and C2 are placed across a single common inductor, L as shown. Then C1, C2 and L form the tuned tank circuit with the condition for oscillations being: XC1 + XC2 = XL, the same as for the Hartley oscillator circuit.
The advantage of this type of capacitive circuit configuration is that with less self and mutual inductance within the tank circuit, frequency stability of the oscillator is improved along with a more simple design.
As with the Hartley oscillator, the Colpitts oscillator uses a single stage bipolar transistor amplifier as the gain element which produces a sinusoidal output. Consider the circuit below.
Colpitts Oscillator
Tank Circuit
The emitter terminal of the transistor is effectively connected to the junction of the two capacitors, C1 and C2 which are connected in series and act as a simple voltage divider. When the power supply is firstly applied, capacitors C1 and C2 charge up and then discharge through the coil L. The oscillations across the capacitors are applied to the base-emitter junction and appear in the amplified at the collector output.
Resistors, R1 and R2 provide the usual stabilizing DC bias for the transistor in the normal manner while the additional capacitors act as a DC-blocking bypass capacitors. A radio-frequency choke (RFC) is used in the collector circuit to provide a high reactance (ideally open circuit) at the frequency of oscillation, ( ƒr ) and a low resistance at DC to help start the oscillations.
The required external phase shift is obtained in a similar manner to that in the Hartley oscillator circuit with the required positive feedback obtained for sustained undamped oscillations. The amount of feedback is determined by the ratio of C1 and C2. These two capacitances are generally “ganged” together to provide a constant amount of feedback so that as one is adjusted the other automatically follows.
The frequency of oscillations for a Colpitts oscillator is determined by the resonant frequency of the LC tank circuit and is given as:
where CT is the capacitance of C1 and C2 connected in series and is given as:
The configuration of the transistor amplifier is of a Common Emitter Amplifier with the output signal 180o out of phase with regards to the input signal. The additional 180o phase shift require for oscillation is achieved by the fact that the two capacitors are connected together in series but in parallel with the inductive coil resulting in overall phase shift of the circuit being zero or 360o.
The amount of feedback depends on the values of C1 and C2. We can see that the voltage across C1 is the the same as the oscillators output voltage, Vout and that the voltage across C2 is the oscillators feedback voltage. Then the voltage across C1 will be much greater than that across C2.
Therefore, by changing the values of capacitors, C1 and C2 we can adjust the amount of feedback voltage returned to the tank circuit. However, large amounts of feedback may cause the output sine wave to become distorted, while small amounts of feedback may not allow the circuit to oscillate.
Then the amount of feedback developed by the Colpitts oscillator is based on the capacitance ratio of C1 and C2 and is what governs the the excitation of the oscillator. This ratio is called the “feedback fraction” and is given simply as:
A Colpitts Oscillator circuit having two capacitors of 24nF and 240nF respectively are connected in parallel with an inductor of 10mH. Determine the frequency of oscillations of the circuit, the feedback fraction and draw the circuit.
The oscillation frequency for a Colpitts Oscillator is given as:
As the colpitts circuit consists of two capacitors in series, the total capacitance is therefore:
The inductance of the inductor is given as 10mH, then the frequency of oscillation is:
The frequency of oscillations for the Colpitts Oscillator is therefore 10.8kHz with the feedback fraction given as:
Just like the previous Hartley Oscillator, as well as using a bipolar junction transistor (BJT) as the oscillators active stage, we can also an operational amplifier, (op-amp). The operation of an Op-amp Colpitts Oscillator is exactly the same as for the transistorised version with the frequency of operation calculated in the same manner. Consider the circuit below.
Note that being an inverting amplifier configuration, the ratio of R2/R1 sets the amplifiers gain. A minimum gain of 2.9 is required to start oscillations. Resistor R3 provides the required feedback to the LC tank circuit.
The advantages of the Colpitts Oscillator over the Hartley oscillators are that the Colpitts oscillator produces a more purer sinusoidal waveform due to the low impedance paths of the capacitors at high frequencies. Also due to these capacitive reactance properties the FET based Colpitts oscillator can operate at very high frequencies. Of course any op-amp or FET used as the amplifying device must be able to operate at the required high frequencies.
Then to summarise, the Colpitts Oscillator consists of a parallel LC resonator tank circuit whose feedback is achieved by way of a capacitive divider. Like most oscillator circuits, the Colpitts oscillator exists in several forms, with the most common form being similar to the transistor circuit above.
The centre tapping of the tank sub-circuit is made at the junction of a “capacitive voltage divider” network to feed a fraction of the output signal back to the emitter of the transistor. The two capacitors in series produce a 180o phase shift which is inverted by another 180o to produce the required positive feedback. The oscillating frequency which is a purer sine-wave voltage is determined by the resonance frequency of the tank circuit.
In the next tutorial about Oscillators, we will look at RC Oscillators which uses resistors and capacitors as its tank circuit to produce a sinusoidal waveform.
RC Oscillators use a combination of an amplifier and an RC feedback network to produce output oscillations due to the phase shift between the stages
In the amplifier tutorials we saw that a single stage transistor amplifier can produce 180o of phase shift between its output and input signals when connected as a common-emitter type amplifier and that its output signal across the collector load depends entirely on the input signal injected into the transistors base terminal.
But we can configure transistor stages to operate as oscillators by placing resistor-capacitor (RC) networks around the transistor to provide the required regenerative feedback without the need for a tank circuit. Frequency selective RC coupled amplifier circuits are easy to build and can be made to oscillate at any desired frequency by selecting the appropriate values of resistance and capacitance.
For an RC oscillator to sustain its oscillations indefinitely, sufficient feedback of the correct phase, that is positive (in-phase) Feedback must be provided along with the voltage gain of the single transistor amplifier being used to inject adequate loop gain into the closed-loop circuit in order to maintain oscillations allowing it to oscillates continuously at the selected frequency.
In an RC Oscillator circuit the input is shifted 180o through the feedback circuit returning the signal out-of-phase and 180o again through an inverting amplifier stage to produces the required positive feedback. This then gives us “180o + 180o = 360o” of phase shift which is effectively the same as 0o, thereby giving us the required positive feedback. In other words, the total phase shift of the feedback loop should be “0” or any multiple of 360o to obtain the same effect.
In a Resistance-Capacitance Oscillator or simply known as an RC Oscillator, we can make use of the fact that a phase shift occurs between the input to a RC network and the output from the same network by using interconnected RC elements in the feedback branch, for example.
The circuit on the left shows a single resistor-capacitor network whose output voltage “leads” the input voltage by some angle less than 90o. In a pure or ideal single-pole RC network. it would produce a maximum phase shift of exactly 90o, and because 180o of phase shift is required for oscillation, at least two single-poles networks must be used within an RC oscillator design.
However in reality it is difficult to obtain exactly 90o of phase shift for each RC stage so we must therefore use more RC stages cascaded together to obtain the required value at the oscillation frequency. The amount of actual phase shift in the circuit depends upon the values of the resistor (R) and the capacitor (C), at the chosen frequency of oscillations with the phase angle ( φ ) being given as:
Where: XC is the Capacitive Reactance of the capacitor, R is the Resistance of the resistor, and ƒ is the Frequency.
In our simple example above, the values of R and C have been chosen so that at the required frequency the output voltage leads the input voltage by an angle of about 60o. Then the phase angle between each successive RC section increases by another 60o giving a phase difference between the input and output of 180o (3 x 60o) as shown by the following vector diagram.
So by cascading together three such RC networks in series we can produce a total phase shift in the circuit of 180o at the chosen frequency and this forms the bases of a “RC Oscillator” otherwise known as a Phase Shift Oscillator as the phase angle is shifted by an amount through each stage of the circuit. Then the phase shift occurs in the phase difference between the individual RC stages. Conveniently op-amp circuits are available in quad IC packages. For example, the LM124, or the LM324, etc. so four RC stages could also be used to produce the required 180o of phase shift at the required oscillation frequency.
We know that in an amplifier circuit either using a Bipolar Transistor or an Inverting Operational Amplifier configuration, it will produce a phase-shift of 180o between its input and output. If a three-stage RC phase-shift network is connected as a feedback network between the output and input of an amplifier circuit, then the total phase shift created to produce the required regenerative feedback is: 3 x 60o + 180o = 360o = 0o as shown.
The three RC stages are cascaded together to obtain the required slope for a stable oscillation frequency. The feedback loop phase shift is -180o when the phase shift of each stage is -60o. This occurs when ω = 2pƒ = 1.732/RC as (tan 60o = 1.732). Then to achieve the required phase shift in an RC oscillator circuit is to use multiple RC phase-shifting networks such as the circuit below.
The basic RC Oscillator which is also known as a Phase-shift Oscillator, produces a sine wave output signal using regenerative feedback obtained from the resistor-capacitor (RC) ladder network. This regenerative feedback from the RC network is due to the ability of the capacitor to store an electric charge, (similar to the LC tank circuit).
This resistor-capacitor feedback network can be connected as shown above to produce a leading phase shift (phase advance network) or interchanged to produce a lagging phase shift (phase retard network) the outcome is still the same as the sine wave oscillations only occur at the frequency at which the overall phase-shift is 360o.
By varying one or more of the resistors or capacitors in the phase-shift network, the frequency can be varied and generally this is done by keeping the resistors the same and using a 3-ganged variable capacitor because capacitive reactance (XC) changes with a change in frequency as capacitors are frequency-sensitive components. However, it may be required to re-adjust the voltage gain of the amplifier for the new frequency.
If the three resistors, R are equal in value, that is R1 = R2 = R3, and the capacitors, C in the phase shift network are also equal in value, C1 = C2 = C3, then the frequency of oscillations produced by the RC oscillator is simply given as:
Where:
ƒr is the oscillators output frequency in Hertz
R is the feedback resistance in Ohms
C is the feddback capacitance in Farads
N is the number of RC feedback stages.
This is the frequency at which the phase shift circuit oscillates. In our simple example above, the number of stages is given as three, so N = 3 (√2*3 = √6). For a four stage RC network, N = 4 (√2*4 = √8), etc.
Since the resistor-capacitor combination in the RC Oscillator ladder network also acts as an attenuator, that is the signal reduces by some amount as it passes through each passive stage. It could be assumed that the three phase shift sections are independent of each other but this is not the case as the total accumulative feedback attenuation becomes -1/29th ( Vo/Vi = β = -1/29 ) across all three stages. Thus the voltage gain of the amplifier must be sufficiently high enough to overcome these passive RC losses. Clearly then in order to produce a total loop gain of -1, in our three stage RC network above, the amplifier gain must be equal too, or greater than, 29 to compensate for the attenuation of the RC network.
The loading effect of the amplifier on the feedback network has an effect on the frequency of oscillations and can cause the oscillator frequency to be up to 25% higher than calculated. Then the feedback network should be driven from a high impedance output source and fed into a low impedance load such as a common emitter transistor amplifier but better still is to use an Operational Amplifier as it satisfies these conditions perfectly.
When used as RC oscillators, Operational Amplifier RC Oscillators are more common than their bipolar transistors counterparts. The oscillator circuit consists of a negative-gain operational amplifier and a three section RC network that produces the 180o phase shift. The phase shift network is connected from the op-amps output back to its “inverting” input as shown below.
As the feedback is connected to the inverting input, the operational amplifier is therefore connected in its “inverting amplifier” configuration which produces the required 180o phase shift while the RC network produces the other 180o phase shift at the required frequency (180o + 180o). This type of feedback connection with the capacitors in series and the resistors connected to ground (0V) potential is known as a phase-lead configuration. In other words, the output voltage leads the input voltage producing a positive phase angle.
But we can also create a phase-lag configuration by simply changing the positions of the RC components so that the resistors are connected in series and the capacitors are connected to ground (0V) potential as shown. This means that the output voltage lags the input voltage producing a negative phase angle.
However, due to the reversal of the feedback components, the orginal equation for the frequency output of the phase-lead RC oscillator is modified to:
Although it is possible to cascade together only two single-pole RC stages to provide the required 180o of phase shift (90o + 90o), the stability of the oscillator at low frequencies is generally poor.
One of the most important features of an RC Oscillator is its frequency stability which is its ability to provide a constant frequency sine wave output under varying load conditions. By cascading three or even four RC stages together (4 x 45o), the stability of the oscillator can be greatly improved.
RC Oscillators with four stages are generally used because commonly available operational amplifiers come in quad IC packages so designing a 4-stage oscillator with 45o of phase shift relative to each other is relatively easy.
RC Oscillators are stable and provide a well-shaped sine wave output with the frequency being proportional to 1/RC and therefore, a wider frequency range is possible when using a variable capacitor. However, RC Oscillators are restricted to frequency applications because of their bandwidth limitations to produce the desired phase shift at high frequencies.
An operational amplifier based 3-stage RC Phase Shift Oscillator is required to produce a sinusoidal output frequency of 4kHz. If 2.4nF capacitors are used in the feedback circuit, calculate the value of the frequency determining resistors and the value of the feedback resistor required to sustain oscillations. Also draw the circuit.
The standard equation given for the phase shift RC Oscillator is:
The circuit is to be a 3-stage RC oscillator which will therefore consist of equal resistors and three equal 2.4nF capacitors. As the frequency of oscillation is given as 4.0kHz, the value of the resistors are calculated as:
The operational amplifiers gain must be equal to 29 in order to sustain oscillations. The resistive value of the oscillation resistors are 6.8kΩ, therefore the value of the op-amps feedback resistor Rƒ is calculated as:
The Wien Bridge Oscillator uses uses two RC networks connected together to produce a sinusoidal oscillator
In the RC Oscillator tutorial we saw that a number of resistors and capacitors can be connected together with an inverting amplifier to produce an oscillating circuit.
One of the simplest sine wave oscillators which uses a RC network in place of the conventional LC tuned tank circuit to produce a sinusoidal output waveform, is called a Wien Bridge Oscillator.
The Wien Bridge Oscillator is so called because the circuit is based on a frequency-selective form of the Wheatstone bridge circuit. The Wien Bridge oscillator is a two-stage RC coupled amplifier circuit that has good stability at its resonant frequency, low distortion and is very easy to tune making it a popular circuit as an audio frequency oscillator but the phase shift of the output signal is considerably different from the previous phase shift RC Oscillator.
The Wien Bridge Oscillator uses a feedback circuit consisting of a series RC circuit connected with a parallel RC of the same component values producing a phase delay or phase advance circuit depending upon the frequency. At the resonant frequency ƒr the phase shift is 0o. Consider the circuit below.
The above RC network consists of a series RC circuit connected to a parallel RC forming basically a High Pass Filter connected to a Low Pass Filter producing a very selective second-order frequency dependant Band Pass Filter with a high Q factor at the selected frequency, ƒr.
At low frequencies the reactance of the series capacitor (C1) is very high so acts a bit like an open circuit, blocking any input signal at Vin resulting in virtually no output signal, Vout. Likewise, at high frequencies, the reactance of the parallel capacitor, (C2) becomes very low, so this parallel connected capacitor acts a bit like a short circuit across the output, so again there is no output signal.
So there must be a frequency point between these two extremes of C1 being open-circuited and C2 being short-circuited where the output voltage, VOUT reaches its maximum value. The frequency value of the input waveform at which this happens is called the oscillators Resonant Frequency, (ƒr).
At this resonant frequency, the circuits reactance equals its resistance, that is: Xc = R, and the phase difference between the input and output equals zero degrees. The magnitude of the output voltage is therefore at its maximum and is equal to one third (1/3) of the input voltage as shown.
It can be seen that at very low frequencies the phase angle between the input and output signals is “Positive” (Phase Advanced), while at very high frequencies the phase angle becomes “Negative” (Phase Delay). In the middle of these two points the circuit is at its resonant frequency, (ƒr) with the two signals being “in-phase” or 0o. We can therefore define this resonant frequency point with the following expression.
Where:
ƒr is the Resonant Frequency in Hertz
R is the Resistance in Ohms
C is the Capacitance in Farads
We said previously that the magnitude of the output voltage, Vout from the RC network is at its maximum value and equal to one third (1/3) of the input voltage, Vin to allow for oscillations to occur. But why one third and not some other value. In order to understand why the output from the RC circuit above needs to be one-third, that is 0.333xVin, we have to consider the complex impedance (Z = R ± jX) of the two connected RC circuits.
We know from our AC Theory tutorials that the real part of the complex impedance is the resistance, R while the imaginary part is the reactance, X. As we are dealing with capacitors here, the reactance part will be capacitive reactance, Xc.
If we redraw the above RC network as shown, we can clearly see that it consists of two RC circuits connected together with the output taken from their junction. Resistor R1 and capacitor C1 form the top series network, while resistor R2 and capacitor C2 form the bottom parallel network.
Therefore the total DC impedance of the series combination (R1C1) we can call, ZS and the total impedance of the parallel combination (R2C2) we can call, ZP. As ZS and ZP are effectively connected together in series across the input, VIN, they form a voltage divider network with the output taken from across ZP as shown.
Lets assume then that the component values of R1 and R2 are the same at: 12kΩ, capacitors C1 and C2 are the same at: 3.9nF and the supply frequency, ƒ is 3.4kHz.
The total impedance of the series combination with resistor, R1 and capacitor, C1 is simply:
We now know that with a supply frequency of 3.4kHz, the reactance of the capacitor is the same as the resistance of the resistor at 12kΩ. This then gives us an upper series impedance ZS of 17kΩ.
For the lower parallel impedance ZP, as the two components are in parallel, we have to treat this differently because the impedance of the parallel circuit is influenced by this parallel combination.
The total impedance of the lower parallel combination with resistor, R2 and capacitor, C2 is given as:
At the supply frequency of 3400Hz, or 3.4kHz, the combined DC impedance of the RC parallel circuit becomes 6kΩ (R||Xc) with the vector sum of this parallel impedance being calculated as:
So we now have the value for the vector sum of the series impedance: 17kΩ, ( ZS = 17kΩ ) and for the parallel impedance: 8.5kΩ, ( ZP = 8.5kΩ ). Therefore the total output impedance, Zout of the voltage divider network at the given frequency is:
Then at the oscillation frequency, the magnitude of the output voltage, Vout will be equal to Zout x Vin which as shown is equal to one third (1/3) of the input voltage, Vin and it is this frequency selective RC network which forms the basis of the Wien Bridge Oscillator circuit.
If we now place this RC network across a non-inverting amplifier which has a gain of 1+R1/R2 the following basic Wien bridge oscillator circuit is produced.
The output of the operational amplifier is fed back to both the inputs of the amplifier. One part of the feedback signal is connected to the inverting input terminal (negative or degenerative feedback) via the resistor divider network of R1 and R2 which allows the amplifiers voltage gain to be adjusted within narrow limits.
The other part, which forms the series and parallel combinations of R and C forms the feedback network and are fed back to the non-inverting input terminal (positive or regenerative feedback) via the RC Wien Bridge network and it is this positive feedback combination that gives rise to the oscillation.
The RC network is connected in the positive feedback path of the amplifier and has zero phase shift a just one frequency. Then at the selected resonant frequency, ( ƒr ) the voltages applied to the inverting and non-inverting inputs will be equal and “in-phase” so the positive feedback will cancel out the negative feedback signal causing the circuit to oscillate.
The voltage gain of the amplifier circuit MUST be equal too or greater than three “Gain = 3” for oscillations to start because as we have seen above, the input is 1/3 of the output. This value, ( Av ≥ 3 ) is set by the feedback resistor network, R1 and R2 and for a non-inverting amplifier this is given as the ratio 1+(R1/R2).
Also, due to the open-loop gain limitations of operational amplifiers, frequencies above 1MHz are unachievable without the use of special high frequency op-amps.
Determine the maximum and minimum frequency of oscillations of a Wien Bridge Oscillator circuit having a resistor of 10kΩ and a variable capacitor of 1nF to 1000nF.
The frequency of oscillations for a Wien Bridge Oscillator is given as:
A Wien Bridge Oscillator circuit is required to generate a sinusoidal waveform of 5,200 Hertz (5.2kHz). Calculate the values of the frequency determining resistors R1 and R2 and the two capacitors C1 and C2 to produce the required frequency.
Also, if the oscillator circuit is based around a non-inverting operational amplifier configuration, determine the minimum values for the gain resistors to produce the required oscillations. Finally draw the resulting oscillator circuit.
The frequency of oscillations for the Wien Bridge Oscillator was given as 5200 Hertz. If resistors R1 = R2 and capacitors C1 = C2 and we assume a value for the feedback capacitors of 3.0nF, then the corresponding value of the feedback resistors is calculated as:
For sinusoidal oscillations to begin, the voltage gain of the Wien Bridge circuit must be equal too or greater than 3, ( Av ≥ 3 ). For a non-inverting op-amp configuration, this value is set by the feedback resistor network of R3 and R4 and is given as:
If we choose a value for resistor R3 of say, 100kΩ’s, then the value of resistor R4 is calculated as:
While a gain of 3 is the minimum value required to ensure oscillations, in reality a value a little higher than that is generally required. If we assume a gain value of 3.1 then resistor R4 is recalculated to give a value of 47kΩ. This gives the final Wien Bridge Oscillator circuit as:
Then for oscillations to occur in a Wien Bridge Oscillator circuit the following conditions must apply.
With no input signal a Wien Bridge Oscillator produces continuous output oscillations.
The Wien Bridge Oscillator can produce a large range of frequencies.
The Voltage gain of the amplifier must be greater than 3.
The RC network can be used with a non-inverting amplifier.
The input resistance of the amplifier must be high compared to R so that the RC network is not overloaded and alter the required conditions.
The output resistance of the amplifier must be low so that the effect of external loading is minimised.
Some method of stabilizing the amplitude of the oscillations must be provided. If the voltage gain of the amplifier is too small the desired oscillation will decay and stop. If it is too large the output will saturate to the value of the supply rails and distort.
With amplitude stabilisation in the form of feedback diodes, oscillations from the Wien Bridge oscillator can continue indefinitely.
In our final look at Oscillators, we will examine the Crystal Oscillator which uses a quartz crystal as its tank circuit to produce a high frequency and very stable sinusoidal
One of the most important features of any oscillator is its frequency stability, or in other words its ability to provide a constant frequency output under varying load conditions.
Some of the factors that affect the frequency stability of an oscillator generally include: variations in temperature, variations in the load, as well as changes to its DC power supply voltage to name a few.
Frequency stability of the output signal can be greatly improved by the proper selection of the components used for the resonant feedback circuit, including the amplifier. But there is a limit to the stability that can be obtained from normal LC and RC tank circuits.
To obtain a very high level of oscillator stability a Quartz Crystal is generally used as the frequency determining device to produce another types of oscillator circuit known generally as a Quartz Crystal Oscillator, (XO).
When a voltage source is applied to a small thin piece of quartz crystal, it begins to change shape producing a characteristic known as the Piezo-electric effect. This Piezo-electric Effect is the property of a crystal by which an electrical charge produces a mechanical force by changing the shape of the crystal and vice versa, a mechanical force applied to the crystal produces an electrical charge.
Then, piezo-electric devices can be classed as Transducers as they convert energy of one kind into energy of another (electrical to mechanical or mechanical to electrical). This piezo-electric effect produces mechanical vibrations or oscillations which can be used to replace the standard LC tank circuit in the previous oscillators.
There are many different types of crystal substances that can be used as oscillators with the most important of these for electronic circuits being the quartz minerals, due in part to their greater mechanical strength.
The quartz crystal used in a Quartz Crystal Oscillator is a very small, thin piece or wafer of cut quartz with the two parallel surfaces metallised to make the required electrical connections. The physical size and thickness of a piece of quartz crystal is tightly controlled since it affects the final or fundamental frequency of oscillations. The fundamental frequency is generally called the crystals “characteristic frequency”.
Once cut and shaped, the crystal can not be used at any other frequency. In other words, its size and shape determines its fundamental oscillation frequency.
The crystals characteristic or characteristic frequency is inversely proportional to its physical thickness between the two metallised surfaces. A mechanically vibrating crystal can be represented by an equivalent electrical circuit consisting of low resistance R, a large inductance L and small capacitance C as shown below.
Quart Crystal
Oscillator
The equivalent electrical circuit for the quartz crystal shows a series RLC circuit, which represents the mechanical vibrations of the crystal, in parallel with a capacitance, Cp which represents the electrical connections to the crystal. Quartz crystal oscillators tend to operate towards their “series resonance”.
The equivalent impedance of the crystal has a series resonance where Cs resonates with inductance, Ls at the crystals operating frequency. This frequency is called the crystals series frequency, ƒs. As well as this series frequency, there is a second frequency point established as a result of the parallel resonance created when Ls and Cs resonates with the parallel capacitor Cp as shown.
The slope of the crystals impedance above shows that as the frequency increases across its terminals. At a particular frequency, the interaction of between the series capacitor Cs and the inductor Ls creates a series resonance circuit reducing the crystals impedance to a minimum and equal to Rs. This frequency point is called the crystals series resonant frequency ƒs and below ƒs the crystal is capacitive.
As the frequency increases above this series resonance point, the crystal behaves like an inductor until the frequency reaches its parallel resonant frequency ƒp. At this frequency point the interaction between the series inductor, Ls and parallel capacitor, Cp creates a parallel tuned LC tank circuit and as such the impedance across the crystal reaches its maximum value.
Then we can see that a quartz crystal is a combination of a series and parallel tuned resonance circuits, oscillating at two different frequencies with the very small difference between the two depending upon the cut of the crystal. Also, since the crystal can operate at either its series or parallel resonance frequencies, a crystal oscillator circuit needs to be tuned to one or the other frequency as you cannot use both together.
So depending upon the circuit characteristics, a quartz crystal can act as either a capacitor, an inductor, a series resonance circuit or as a parallel resonance circuit and to demonstrate this more clearly, we can also plot the crystals reactance against frequency as shown.
The slope of the reactance against frequency above, shows that the series reactance at frequency ƒs is inversely proportional to Cs because below ƒs and above ƒp the crystal appears capacitive. Between frequencies ƒs and ƒp, the crystal appears inductive as the two parallel capacitances cancel out.
Then the formula for the crystals series resonance frequency, ƒs is given as:
The parallel resonance frequency, ƒp occurs when the reactance of the series LC leg equals the reactance of the parallel capacitor, Cp and is given as:
A quartz crystal has the following values: Rs = 6.4Ω, Cs = 0.09972pF and Ls = 2.546mH. If the capacitance across its terminal, Cp is measured at 28.68pF, Calculate the fundamental oscillating frequency of the crystal and its secondary resonance frequency.
The crystals series resonant frequency, ƒS
The crystal’s parallel resonant frequency, ƒP
We can see that the difference between ƒs, the crystal’s fundamental frequency and ƒp is small at about 18kHz (10.005MHz – 9.987MHz). However during this frequency range, the Q-factor (Quality Factor) of the crystal is extremely high because the inductance of the crystal is much higher than its capacitive or resistive values. The Q-factor of our crystal at the series resonance frequency is given as:
Then the Q-factor of our crystal example, about 25,000, is because of this high XL/R ratio. The Q-factor of most crystals is in the area of 20,000 to 200,000 as compared to a good LC tuned tank circuit we looked at earlier which will be much less than 1,000. This high Q-factor value also contributes to a greater frequency stability of the crystal at its operating frequency making it ideal to construct crystal oscillator circuits.
So we have seen that a quartz crystal has a resonant frequency similar to that of a electrically tuned LC tank circuit but with a much higher Q factor. This is due mainly to its low series resistance, Rs. As a result, quartz crystals make an excellent component choice for use in oscillators especially very high frequency oscillators.
Typical crystal oscillators can range in oscillation frequencies from about 40kHz to well over 100MHz depending upon their circuit configuration and the amplifying device used. The cut of the crystal also determines how it will behave as some crystals will vibrate at more than one frequency, producing additional oscillations called overtones.
Also, if the crystal is not of a parallel or uniform thickness it may have two or more resonant frequencies both with a fundamental frequency producing what are called and harmonics, such as second or third harmonics.
Generally though the fundamental oscillating frequency for a quartz crystal is much more stronger or pronounced than that of and secondary harmonics around it so this would be the one used. We have seen in the graphs above that a crystals equivalent circuit has three reactive components, two capacitors plus an inductor so there are two resonant frequencies, the lowest is a series resonant frequency and the highest is the parallel resonant frequency.
We have seen in the previous tutorials, that an amplifier circuit will oscillate if it has a loop gain greater or equal to one and the feedback is positive. In a Quartz Crystal Oscillator circuit the oscillator will oscillate at the crystals fundamental parallel resonant frequency as the crystal always wants to oscillate when a voltage source is applied to it.
However, it is also possible to “tune” a crystal oscillator to any even harmonic of the fundamental frequency, (2nd, 4th, 8th etc.) and these are known generally as Harmonic Oscillators while Overtone Oscillators vibrate at odd multiples of the fundamental frequency, 3rd, 5th, 11th etc). Generally, crystal oscillators that operate at overtone frequencies do so using their series resonant frequency.
Crystal oscillator circuits are generally constructed using bipolar transistors or FETs. This is because although operational amplifiers can be used in many different low frequency (≤100kHz) oscillator circuits, operational amplifiers just do not have the bandwidth to operate successfully at the higher frequencies suited to crystals above 1MHz.
The design of a Crystal Oscillator is very similar to the design of the Colpitts Oscillator we looked at in the previous tutorial, except that the LC tank circuit that provides the feedback oscillations has been replaced by a quartz crystal as shown below.
This type of Crystal Oscillators are designed around a common collector (emitter-follower) amplifier. The R1 and R2 resistor network sets the DC bias level on the Base while emitter resistor RE sets the output voltage level. Resistor R2 is set as large as possible to prevent loading to the parallel connected crystal.
The transistor, a 2N4265 is a general purpose NPN transistor connected in a common collector configuration and is capable of operating at switching speeds in excess of 100Mhz, well above the crystals fundamental frequency which can be between about 1MHz and 5MHz.
The circuit diagram above of the Colpitts Crystal Oscillator circuit shows that capacitors, C1 and C2 shunt the output of the transistor which reduces the feedback signal. Therefore, the gain of the transistor limits the maximum values of C1 and C2. The output amplitude should be kept low in order to avoid excessive power dissipation in the crystal otherwise could destroy itself by excessive vibration.
Another common design of the quartz crystal oscillator is that of the Pierce Oscillator. The Pierce oscillator is very similar in design to the previous Colpitts oscillator and is well suited for implementing crystal oscillator circuits using a crystal as part of its feedback circuit.
The Pierce oscillator is primarily a series resonant tuned circuit (unlike the parallel resonant circuit of the Colpitts oscillator) which uses a JFET for its main amplifying device as FET’s provide very high input impedances with the crystal connected between the Drain and Gate via capacitor C1 as shown below.
In this simple circuit, the crystal determines the frequency of oscillations and operates at its series resonant frequency, ƒs giving a low impedance path between the output and the input. There is a 180o phase shift at resonance, making the feedback positive. The amplitude of the output sine wave is limited to the maximum voltage range at the Drain terminal.
Resistor, R1 controls the amount of feedback and crystal drive while the voltage across the radio frequency choke, RFC reverses during each cycle. Most digital clocks, watches and timers use a Pierce Oscillator in some form or other as it can be implemented using the minimum of components.
As well as using transistors and FETs, we can also create a simple basic parallel-resonant crystal oscillator similar in operation to the Pierce oscillator by using a CMOS inverter as the gain element. The basic quartz crystal oscillator consists of a single inverting Schmitt trigger logic gate such as the TTL 74HC19 or the CMOS 40106, 4049 types, an inductive crystal and two capacitors. These two capacitors determine the value of the crystals load capacitance. The series resistor helps limit the drive current in the crystal and also isolates the inverters output from the complex impedance formed by capacitor-crystal network.
The crystal oscillates at its series resonance frequency. The CMOS inverter is initially biased into the middle of its operating region by the feedback resistor, R1. This ensures that the Q-point of the inverter is in a region of high gain. Here a 1MΩ value resistor is used, but its value is not critical as long as it is more than 1MΩ. An additional inverter is used to buffer the output from the oscillator to the connected load.
The inverter provides 180o of phase shift and the crystal capacitor network the additional 180o required for oscillation. The advantage of the CMOS crystal oscillator is that it will always automatically readjust itself to maintain this 360o phase shift for oscillation.
Unlike the previous transistor based crystal oscillators which produced a sinusoidal output waveform, as the CMOS Inverter oscillator uses digital logic gates, the output is a square wave oscillating between HIGH and LOW. Naturally, the maximum operating frequency depends upon the switching characteristics of the logic gate used.
We can not finish a Quartz Crystal Oscillators tutorial without mentioning something about Microprocessor crystal clocks. Virtually all microprocessors, micro-controllers, PICs and CPU’s generally operate using a Quartz Crystal Oscillator as its frequency determining device to generate their clock waveform because as we already know, crystal oscillators provide the highest accuracy and frequency stability compared to resistor-capacitor, (RC) or inductor-capacitor, (LC) oscillators.
The CPU clock dictates how fast the processor can run and process the data with a microprocessor, PIC or micro-controller having a clock speed of 1MHz means that it can process data internally one million times per second at every clock cycle. Generally all that’s needed to produce a microprocessor clock waveform is a crystal and two ceramic capacitors of values ranging between 15 to 33pF as shown below.
Most microprocessors, micro-controllers and PIC’s have two oscillator pins labelled OSC1 and OSC2 to connect to an external quartz crystal circuit, standard RC oscillator network or even a ceramic resonator. In this type of microprocessor application the Quartz Crystal Oscillator produces a train of continuous square wave pulses whose fundamental frequency is controlled by the crystal itself. This fundamental frequency regulates the flow of instructions that controls the processor device. For example, the master clock and system timing.
A quartz crystal has the following values after being cut, Rs = 1kΩ, Cs = 0.05pF, Ls = 3H and Cp = 10pF. Calculate the crystals series and parallel oscillating frequencies.
The series oscillating frequency is given as:
The parallel oscillating frequency is given as:
Then the frequency of oscillation for the crystal will be between 411kHz and 412kHz.
The Twin-T Oscillator is another RC oscillator circuit which uses two parallel connected RC networks to produce a sinusoidal output waveform of a single frequency
Twin-T Oscillators are another type of RC oscillator which produces a sinewave output for use in fixed-frequency applications similar to the Wein-bridge oscillator. The twin-T oscillator uses two “Tee” shaped RC networks in its feedback loop (hence the name) between the output and input of an inverting amplifier.
As we have seen, an oscillator is basically an amplifier with positive feedback which has a fixed amount of voltage gain required to maintain oscillations, and the twin-T oscillator is no different. Feedback is provided by the twin-T configured RC network allowing some of the output signal to be fed back to the amplifier’s input terminal. Thus the twin-T RC network provides the 180o phase-shift and the amplifier providing another 180o of phase-shift. These two conditions create 360o in total of phase-shift allowing for sustained oscillations.
Unlike the typical RC Phase-shift Oscillator which configures the feedback resistors and capacitors into a ladder network, or the standard Wien-bridge Oscillator which uses the resistors and capacitors in a bridge configuration, the twin-T oscillator (sometimes known as a parallel-T oscillator) uses a passive resistance-capacitance (RC) network with two interconnected “T” sections (having their R and C elements in opposite formation) connected together in parallel as shown.
Clearly we can see that one of the RC passive networks has a low-pass response, while the other has a high-pass response and we have seen this RC network arrangement before in our tutorial about the Notch Filter. The difference this time is that we are using the combined parallel RC T-configured networks to produce a notch type response which has a center frequency ƒc equal to the desired null frequency of oscillation.
The result is that oscillations cannot occur at frequencies above or below the tuned notch frequency due to the negative feedback path created through the twin-T network. However, at the tuned frequency any negative feedback becomes negligible, thus allowing the positive feedback path created by the amplifying device to dominate creating oscillations at one single frequency (unlike the Wien bridge oscillator which can be adjusted over a large frequency range).
Then the twin-T oscillator’s frequency selective twin-T network produces an output transfer function were the frequency, depth and phase-shift of the notch is determined by the component values used. Thus the individual twin-T networks which make up the RC network are defined by the following equations:
For the low-pass R-C-R network:
For the high-pass C-R-C network:
Combining these two sets of equations together will give us the final equation for the null or centre frequency of the notch resulting in oscillations for a twin-T network.
Where:
ƒC is the frequency of oscillations in Hertz
R is the feedback resistance in Ohms
C is the feddback capacitance in Farads
π (pi) is a constant with a value of about 3.142
Having determined the twin-T network for the oscillator that produces the required 180o of phase shift, which occurs at the null frequency between -90o to +90o (as opposed to the zero to 180o for the Wien-bridge oscillator), we need an amplifier circuit to provide the voltage gain. Twin-T oscillator cicruits are best implimented by combining the RC feedback network with an operational amplifier, as due to their high input impedance caharacteristics, op-amps tend to work better with this type of oscillator compared to transistors.
Standard operational amplifiers can provide high voltage gain, a high input impedance as well as a low output impedance and are therefore excellent amplifiers for twin-T oscillators. At the oscillating frequency, ƒc the feedback gain drops to almost zero so we require an amplifier with a voltage gain much greater than one (unity).
The positive feedback required for oscillation is provided by the feedback resistor R1 while resistor R2 ensures start-up. As a general rule of thumb, to ensure that the circuit oscillates as close to the required frequency as possible, the ratio of these two resistors needs to be greater than one-hundred (>100).
To obtain the required positive gain at the oscillating fequency, we can use a non-inverting amplifier configuration where a small part of the output voltage signal is applied directly to the non-inverting ( + ) input terminal via a suitable voltage divider network. The negative feedback produce by the twin-T osccilator circuit is connected to the inverting ( – ) input terminal. This closed-loop configuration produces a non-inverting oscillator circuit with very good stability, a very high input impedance, and low output impedance as shown.
Then we can see that the twin-T oscillator receives its positive feedback to the non-inverting input through the voltage divider network and its negative feedback through the twin-T RC network. To ensure that the circuit oscillates at the required single frequency, the “Tee-leg” resistor R/2 can be an adustable trimmer potentiometer, but can also be adjusted to compensate for capacitor tolerances so that the circuit oscillates at start-up.
A twin-T oscillator circuit is required to produce a 1kHz sinusoidal output signal for use in an electronic circuit. If an operational amplifier with a gain ratio of 200 is used, calculate the values of the frequency determining components R and C, and the values of the gain resistors.
The frequency of oscillation is to be 1kHz, if we select a reasonable value for the two feedback resistors, R of 10kΩ (remember that these two resistors must have identical values) we can calculate the value of capacitor required using the formula for the frequency of oscillations from above.
Thus R = 10kΩ, and C = 16nF. The center Tee-leg capacitor 2C = 2 x 16nF = 32nF, so the nearest preferred value of 33nF is used.
As the value of the high-pass branch tee-capacitor is 33nF and therefore not exactly equal to 2C (2 x 16nF), we can adjust for this variation and ensure the correct start-up of oscillations by adjusting the low-pass branch tee-resistor by the same amount. Thus the exact value of R(leg) would be 10kΩ/2 = 5kΩ, but the calculated value of this resistor is given as: R(leg) = R/(33nF/16nF) = 4.85kΩ. Clearly then the use of a 5kΩ trim-pot would meet our requirements in this example.
The loop gain of the operation amplifier is required to be 200, so if we choose a value of 1kΩ for R2 then resistor R1 will be 200kΩ as shown.
We have seen in this tutorial that Twin-T Oscillator Circuits can easily be constructed using some passive components and an operational amplifier. The twin-T oscillator circuit uses a tuned RC network for the feedback circuit to produce the required sinusoidal output waveform. Being two T-networks connected together in parallel, they operate in anti-phase to each other creating zero output at the null frequency, but a finite output at all other frequencies.
As a result, the circuit will not oscillate at frequencies above or below the tuned frequency due to the negative feedback through the twin-T RC network. Therefore at the null frequency, the voltage at the non-inverting input of the op-amp is in phase with its output voltage, giving rise to continuous oscillations at the desired frequency.
To ensure that the oscillation frequency is close to the null frequency as possible, a trim-pot can be used in the tee-leg resistor of the low-pass stage to balance the RC network for start up and purity of output waveform as one of the major disadvantage of the “twin-T oscillator” is that the oscillation frequency and quality of the output waveform is much dependent on the interaction of the resistors and capacitors in the twin-T network then clearly the values and selection of these components must be accurate to ensure oscillation at the desired null frequency.