Voltage Controlled Oscillators High Speed Electronic Circuits Voltage Controlled Oscillators By: Abanoub Mamdouh Andrew Fahim Ramez Mamdouh Michael Samir Mounir Basel Mohamed Ahmed April 27, 2016

Outline Introduction to Voltage Controlled Oscillators Varactors Negative resistance realization using oscillators Voltage controlled colpitts oscillators Varactor tuned differential oscillator Applications of VCOs

Voltage Controlled Oscillators: Introduction Generally, oscillators are electronic circuits designed to generate repetitive electronic signals. Many types of oscillators exist nowadays and every type has his own limitations. Quartz crystal oscillators are used for frequencies less than 100 MHz. When the frequency surpasses 300 MHz, physical limitations take place. Voltage controlled LC oscillators can operate from tens of gigahertz to hundreds of gigahertz Ring oscillators can operate above hundreds of gigahertz and they’re generally used in supercomputers and microprocessors.

Voltage Controlled Oscillators: Introduction We can intuitively deduct from the name how these oscillators work: the frequency of oscillation of this type of oscillators depends on the input voltage. The output signal can be sinusoidal or sawtooth like shown below.

Varactors Varactors Before proceeding into the details of the VCO, a new element must be introduced. This element is called varactor. The name is divided into 2 parts: 1st part “var” stands for variable, the 2nd part “actor” stands for capacitor. So varactor is a variable capacitor. The basic implementation of a varactor is simply a reversed biased diode. When you reverse bias a diode, you vary the depletion region width which create a capacitance between the n-junction and the p-junction. When the voltage changes, the depletion region width changes with it and thus you create a variable capacitor. Some illustrations regarding the varactor will be shown in the next slide

Varactors Varactors Increasing the voltage and observing the change in depletion region width the voltage

Varactors Varactors How does the voltage affect the value of the capacitance? The capacitance is inversely proportional to the depletion region thickness (T), C α 1 𝑇 The depletion region thickness (T) is proportional to the square root of the applied voltage (V), T α 𝑉 1 2 Hence the capacitance C α 1 𝑉 1 2 .

Hyperabrupt varactors Varactor Diodes types Varactor diode Abrupt varactors Hyperabrupt varactors Different varactor diodes, have different values and parameters of PN junction . Abrupt varactors and hyper-abrupt varactors have different properties as detailed below. Different doping profiles could be applied to the PN junction of the varactor diode , to achieve certain C-V relations.

Abrupt Diodes Varactors For an abrupt varactor diode the doping concentration is held constant, i.e constant doping level as far as reasonably possible. Disadvantage : In applications where a linear dependence is required, a linearizer is needed. This takes additional circuitry that may be an additional burden for some applications, not only in terms of circuitry, but also the slower response speed caused by the linearizer.

Hyper-abrupt varactor diodes Varactors Hyper-abrupt varactor diodes This provides a narrow band linear frequency variation. much greater capacitance change for the given voltage change Disadvantage : Low Q factor, only used for microwave applications . Up to a few GHZ at most .

MOS Varactors Varactors The MOS capacitor can be used as a varactor. The capacitance varies dramatically from accumulation Vgb < Vfb to depletion. In accumulation: majority carriers (holes) form the bottom plate of a parallel plate capacitor. In depletion, the presence of a depletion region with dopant atoms creates a non-linear capacitor that can be modeled as two series capacitors (Cdep and Cox), effectively increasing the plate thickness to tox + tdep

MOS Varactor (Inversion) Varactors MOS Varactor (Inversion) For a quasi-static excitation, thermal generation leads to minority carrier generation. Thus the channel will invert for VGB > VT and the capacitance will return to Cox. The transition around threshold is very rapid. If a MOSFET MOS-C structure is used (with source/drain junctions), then minority carriers are injected from the junctions and the high-frequency capacitance includes the inversion transition.

Negative resistance realization using oscillators In a parallel LC Oscillator setup (Often referred to as an LC tank) as shown in the figure below , there’re internal losses associated with the Capacitor modelled as a parallel resistor Rpc and Inductor modelled as a parallel resistor Rpl. You can also note that there’s another resistance Rp to model other losses in the circuit (interconnects, heat, etc..). The presence of these resistors will destroy the oscillation for this circuit because the signal will decay over time due to the existence of a lossy element.

Negative resistance realization using oscillators To compensate the previously mentioned losses sources, we need to put – if existent – a negative resistance. This could be achieved using transistors according to the setup depicted below: 𝑖 𝑥 ≈ 𝑖 𝑐 = 𝑔 𝑚 𝑣 𝑖𝑛 = − 𝑔 𝑚 𝑣 𝑥 𝑛 𝐺 𝑥 = 𝑖 𝑥 𝑣 𝑥 =− 𝑔 𝑚 𝑛 In this setup, the collector of the transistor is connected to a block called “Impedance Transformer” (Do not conflict with normal transformer!)

Negative resistance realization using oscillators Let’s do some simple circuits analysis to the previous circuit: Consider a test voltage source is connected to the collector terminal as previously shown, we can conclude that: 𝑖 𝑥 ≈ 𝑖 𝑐 = 𝑔 𝑚 𝑣 𝑖𝑛 = − 𝑔 𝑚 𝑣 𝑥 𝑛 The denominator n represents the transformer’s ratio that will act on the impedance and thus will act on the voltage (because ic is approximately equals ix) Thus we can conclude from the previous equation, the conductance seen by the test voltage source is 𝐺 𝑥 = 𝑖 𝑥 𝑣 𝑥 =− 𝑔 𝑚 𝑛

Negative resistance realization using oscillators Now let’s see how we can realize this impedance transformer. Let’s recall the Colpitts oscillator but in a slightly different way. Analyzing this circuit we can see that: Where 𝐶 2 ′ = 𝐶 2 + 𝐶 π , simplifying we can see that: Consequently:

Negative resistance realization using oscillators The MOS or BJT cross coupled pair generate a negative resistance.

Voltage Controlled Colpitts Oscillators Voltage Controlled Colpitts Oscillators (For reading only) The Colpitts topology employs only one transistor and finds wide application in discrete design. This is because transistors are least expensive and they occupy a small area. They can be used for high frequency discrete oscillators.

Voltage Controlled Colpitts Oscillator (For reading) In the following figure, a common-gate topology whose output (the collector voltage) is fed back to the input (the emitter node) is presented. The current source is used as the bias current of Q1 , and Vb ensures Q1 is in the forward active region. R1 models the loss of the inductor

Voltage Controlled Colpitts Oscillator (For reading) In order to analyze the colpitts oscillator, we have to break the feedback loop. We note that Q1 operates as an ideal voltage –dependent current source, injecting its small-signal current into the node. We therefore break the loop at the collector as shown in the figure , where an independent current source Itest is drawn from the node , and the current returned by the transistor , Iret , is measured as the quantity of interest. The transfer function Iret/Itest must exhibit a phase of 360 and a magnitude of at least unity at the frequency of oscillation.

Voltage Controlled Colpitts Oscillator (For reading) We observe that Itest is divided between and Z1, which is given by The current flowing through C1 is then:

Voltage Controlled Colpitts Oscillator (For reading) This current is now multiplied by the parallel combination of 1/(C2 s) and 1/gm to yield Vx. Since Iret = gm Vpi = -gm Vx , we have: We now equate this transfer function to unity ( which is equivalent to setting its phase to 360 and its magnitude to 1) and cross-multiply, obtaining : At the oscillation frequency, s = jw , both real and imaginary parts of the left-hand side must be equal to zero:

Voltage Controlled Colpitts Oscillator (For reading) From the equation , we obtain the oscillation frequency: The second term on the right is typically negligible, yielding That is the oscillation occurs at the resonance of L1 and the series combination of C1 and C2, using the result from the previous equation gives the startup condition: The transistor must then provide sufficient transconductance to satisfy or exceed this requirement. Since the right-hand side is minimum if C1 = C2, we conclude that gmRp must be at least = 4

Voltage Controlled Colpitts Oscillator (For reading) As for the capacitance, a varactor can be used to tune the desired frequency. Which is the capacitance of the reversed biased PN-junction which is a function of the reverse bias voltage Vr according to the following equation: Where Cj0 denotes the capacitance corresponding to zero bias and V0 is the built in potential. Substituting back in the frequency equation we get : Using Macluarin series , we get: Which gives a direct relation between the frequency and the applied voltage to use this oscillator as a VCO.

Varactor tuned differential oscillator Combining what we saw in the cross coupled differential resistance, the varactors and the LC tank, we can create a varactor tuned differential oscillator. Why Vc<Vdd? The frequency of oscillation of this oscillator is: ω 2 = 1 2𝐿∗( 𝐶 1 ∗ 𝐶 2 𝐶 1 + 𝐶 2 ) We can control C1 and C2 by varying Vc, the capacitance will then change according to the following formula

Varactor tuned differential oscillator

Varactor tuned differential oscillator

Applications Tone generators Frequency shift keying FM modulation Clock, signal and function generators Frequency synthesizers, navigation systems, instrumentation systems and telecommunication devices. High-frequency VCOs are usually used in phase-locked loops for radio receivers Voltage-to-frequency converters, with a highly linear relation between voltage and frequency They are used to convert a slow analog signal into a digital signal over a long distance

VCO requirements VCO tuning range: It is obvious that the voltage controlled oscillator must be able to tune over the range that the loop is expected to operate over. This requirement is not always easy to meet and may require the VCO or resonant circuit to be switched in some extreme circumstances. VCO tuning gain: The gain of the voltage controlled oscillator is important. It is measured in terms of volts per Hz (or V/MHz, etc). As implied by the units it is the tuning shift for a given change in voltage. The voltage controlled oscillator gain affects some of the overall loop design considerations and calculations. Phase noise performance: The phase noise performance of the voltage controlled oscillator is of particular importance in some PLL applications - particularly where they are used in frequency synthesizers. Here the phase noise performance of the VCO determines many of the overall phase noise performance characteristics of the overall loop and the overall synthesizer if used in one

Razavi, fundamentals of Microelectronics – chapter 13 RF and Microwave Transistor Oscillator Design – A. Grebennikov University of California, Berkeley- EECS 142  US 1624537, Colpitts, Edwin H., "Oscillation generator", published 1 February 1918, issued 12 April 1927  Razavi, B. Design of Analog CMOS Integrated Circuits. McGraw-Hill. 2001.