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ECEN 5014, Spring 2013 – Special Topics: Active Microwave Circuits and MMICs Zoya Popovic, University of Colorado, Boulder

LECTURE 12 – MICROWAVE OSCILLATORS L12.1. INTRODUCTION A microwave oscillator is a circuit that converts DC power to microwave power. Microwave oscillators can be one-ports, two-ports or three-ports, depending on what active device is used and how it is connected to the rest of the circuit. A one-port oscillator is schematically shown in Fig.L12.1a, where s is the reflection coefficient of the device and sL is the reflection coefficient of the load. The oscillation condition is:

s sL 1, or in terms of impedances and applying Kirchoff's voltage law,

Z Z R R X XL L L 0 0.

Since the load is a passive impedance, RL 0 indicates that R is negative. This makes sense, since an oscillator gives RF power, so that the dissipated power in the oscillator impedance is negative, P R I R Iosc | | . Since any generator can be described as a negative resistor, devices used in oscillators are often called negative resistance devices, although they might or might not have a true negative resistance behavior. Transistors alone do not have true negative resistance, but devices such as Gunn diodes do.

(a) (b)

Fig.L12.1. (a) A one-port and (b) a two-port microwave oscillator block diagram. The same analysis as above is true for a two-port oscillator. Let us start from Fig.L12.1b, showing a general amplifier circuit. If there is an RF output with no generator present at port 1, this implies that

'

1

11ss ,

and since |s| is less than unity, this implies that | ' |s11 has to be greater than unity.

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Consider now a two-port oscillator circuit shown in Fig.L1b, where M1 is a lossless resonator circuit, and M2 is a lossless matching circuit that enables all of the external RF power to be delivered to the load. Let us assume that the oscillation condition is satisfied at port 1:

1

11ss

' ,

where s is the reflection coefficient of the resonator load. We can also write:

s ss s s

s s

s s

s sL

L

L

L11 11

12 21

22

11

221 1'

,

so that 1 1

11

22

11s

s s

s sL

L'

,

where is the determinant of the transistor scattering matrix. Further, we get

s s s s s s

ss s

s s

L L

L

11 22

11

22

1

1

,

.

At port 2, we have

s ss s s

s s

s s

s s22 2212 21

11

22

111 1'

.

so that 1 1

22

11

22s

s s

s s'

.

Comparing the expressions for 1 22/ 's and sL , we obtain 1 22/ 's sL ,which tells us that the output port 2 satisfies the oscillation condition as well. A load can be placed in both ports. This result can be generalized to any number of ports, showing that the oscillator is simultaneously oscillating at each of the ports.

L12.2. MICROWAVE OSCILLATOR ANALYSIS There are several methods that can be used to determine the oscillation frequency in an oscillator:

- examining the open-loop gain of the oscillator circuit - examining the closed-loop gain of the oscillator circuit - computing the circular function of the oscillator circuit - examining the input reflection coefficient at the input/output port of the oscillator.

L12.2.1. Open-loop gain analysis Consider the block diagram of an oscillator consisting of a two-port active device (with Sa ) and a two port imbedding (passive) network (with a scattering matrix S), Fig.L12.2. The oscillation condition for a multiport network can be easily generalized from our previous discussion to be:

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det( )S S - I 0a where I is the identity matrix. It is easy to see how for a one-port oscillator this reduces to our previous condition. For the case of a two-port oscillator, the oscillation condition reduces to:

s s s s s s s sa a a a a11 11 12 21 21 12 22 22 1 | || |S S ,

where | |Sa a a a as s s s 11 22 12 21 and | |S s s s s11 22 12 21 .

The open-loop gain method of analyzing an oscillator circuit requires that the loop be broken at some point in the circuit, and the loop gain computed over the frequency range of interest. The circuit will oscillate at the frequency where the loop gain has a magnitude greater than unity and a phase of 0 degrees. This condition really means that in order for an oscillation to build up, the total loop gain must be greater than one and the round-trip phase a multiple of 2. In a circuit simulator, the open loop gain is determined by inserting a probe with the following s-parameters:

000

001

100

.

The idea is that the probe is perfectly matched, lossless and operates as a part of a circulator. This kind of device cannot be implemented in reality. The probe injects a known signal a1' which travels around the loop in the clockwise direction, returning to the probe as b1 ' . The reflection coefficient seen at port 1’ of the probe is the open-loop gain, which can be found to be

Gs s

s so

a

a

21 12

22 221.

Fig.L12.2. Generalized 2-port oscillator circuit. There are a few problems with the open-loop gain analysis. First, setting the open-loop gain to be Go 1 0 , gives

s s s sa a21 12 22 22 1 ,

which is not the oscillation condition we found above. Another problem of this method is that the frequency of oscillation depends on where the probe is placed in the circuit, what impedance it is normalized to, how it is oriented in the circuit, etc. All of these problems are consequences of the fact that the probe breaks and therefore changes the loop.

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L12.2.2. Closed-loop gain analysis The closed-loop gain approach uses a probe with a scattering matrix equal to:

010

101

100

.

This probe is not physical (since power is not conserved), but it is not invasive, since it maintains closure of the loop. When the closed-loop gain is calculated using this probe, the denominator is exactly the expression from the oscillation condition, which means that when the oscillation condition is satisfied, the closed-loop gain blows up, Gc . The frequency dependence determines whether the circuit can oscillate or not. Briefly, if | |Gc 1, the circuit is passive. If | |Gc 1 and in frequency travels counter-clockwise around the Smith chart, the circuit is capable of oscillation. If the direction is clockwise, it is an amplifier. Since the function tends to infinity, it is not a precise indicator of the oscillation frequency, but rather an indicator of stability. The closed-loop gain is invariant to probe orientation and position in the circuit. L12.2.3. Circular function analysis If a standard ideal circulator scattering matrix is used for the probe, one gets a circular function for the loop gain as:

Cs s s s

s s s s

a a a

a a

11 11 21 12

12 21 22 221

| || |S S,

using a probe matrix

010

001

100

.

Under steady-state conditions, when C 1 0 , the expression for C reduces to the oscillation condition for a two-port. The circular function is non-invasive, the result does not depend on its position and orientation and in addition, when looking at a polar plot of C as a function of frequency, it is easy to determine what the oscillation condition is by looking at the intersection of the function with the real axis. L12.2.4. Input/Output s-parameter analysis In circuit simulators such as ADS/MWO, oscillators in Harmonic Balance simulations are analyzed by looking at the open loop gain in the feedback loop. The magnitude of the open loop gain needs to be greater than unity, and the phase zero at the oscillation frequency. If you consider an oscillator from a matched external port, the reasoning is a bit different. You will examine this in your homework and we will discuss it afterwards in class.

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A standard oscillator design procedure is: 1. Pick a transistor with enough gain at the frequency you want. 2. Select a circuit that gives K 1 at the operating frequency. Add feedback if this is not

satisfied. 3. Design an output port matching circuit that gives | ' |s11 1 in the desired frequency range. 4. Place a resonator at the input port so that s s11 1' . The value of s22 ' should be greater than

one if the input resonator is chosen properly. L12.3. RESONATORS Resonators are used in a variety of filters and in oscillator circuits for frequency control. Frequency control refers to both the frequency of oscillation and the purity of the output sinusoid. A resonator which has the ability to precisely control the frequency of oscillation has a high Q factor. Note that in general Q 1/tuning-range and Q resonator size. There are a few types of resonators commonly used at microwave frequencies:

- Lumped element resonators are high Q capacitors and inductors with associated parasitics (packaged inductors have some parasitic capacitance and resistance and packaged capacitors have parasitic inductance and resistance). Also, devices with a capacitance value controlled by an applied DC voltage are available for a tunable lumped element resonator. Oscillators using such a component (e.g. a varactor diode) for frequency control are called VCOs.

- Microstrip resonators can be open or shorted sections of line that provide the right

impedance for instability, rectangular /2 resonating line sections, circular disks, circular rings, triangular microstrip etc.

- Metallic cavity resonators are often used for high-Q and convenient mechanical

tunability. Waveguide cavity resonators are usually /4 shorted stubs. The output can be coupled out either with a short loop (magnetic coupling) or a short monopole (electric coupling). Often there is a mechanical tuning screw near the open circuit end of the cavity. The lowest order rectangular cavity resonator is a TE101 mode, where the width and the length of the cavity are g/2 at the resonant frequency.

- Dielectric resonators look like an aspirin pill (often called a dielectric puck). These are

low cost resonators used externally in microstrip oscillators. They are made out of barium titanate compounds, with relative dielectric constants between 30 and 90. The increased dielectric constant gives high energy concentration, but also higher losses. The mode used in commercially available resonators is a hybrid mode referred to as TE01 mode, where the (l,m,n) subscripts usually used in waveguide resonators are modified to (l,m,), where the indicates that the rod is a bit larger than half of a period of the field variation. The Q factors of these resonators are often specified at several thousand. The term DRO is used often and refers to a dielectric resonator oscillator.

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- YIG resonators are high Q ferrite spheres made of yttrium iron garnet, 2 2 4( )Y Fe FeO .

They can be tuned over a wide range by varying a DC magnetic field, and making use of a magnetic resonance, which ranges between 500MHz and 50GHz depending on the material and field used. YIG resonators typically have unloaded Q factors of 1000 or greater. Their high tunability and Q-factor as well as small size (excluding the magnetic field control) make them a special case of the tunability/Q/size laws mentioned earlier. The drawback of a YIG resonator is the slower tuning speed. A varactor based resonator is a faster solution if tuning speed is important.

Fig. L12.3. Comparison of microwave resonators from J. Dick, Proc. IEEE FCS, 1992.

- Whispering Gallery Mode resonators have been used more recently in high precision oscillators. The term whispering gallery refers to the field concentration along the dielectric air interface (think of light in a ring of fiber-optic cable). Sapphire is the main dielectric material used for such a resonator because it has the lowest microwave losses for any known solid. In fact Q-factors are now quoted at 3105 at room temperature, 3107 at 77K and 11010 around 10K.

- It is also important to acknowledge the use of surface acoustic wave (SAE) resonators

at sub-microwave frequencies. These types of resonators make use of the mechanical, or acoustic, resonance of a material to achieve high-Q piezoelectric response. Quartz crystals, used from the kHz range up to about 100 MHz, are the workhorse of low-frequency signal generators. There has also been a lot of work recently in mechanical

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resonators that can reach GHz frequencies, but it is a field in development and not many products exist at this point.

- optical fiber resonators are also used for high quality factors (very long fibers), but

require electrooptic transducers to be used in microwave circuits. FigureL12.3. shows how most of the above resonators compare in terms of realizable Q-factors and rough size (note that this graph is over 20 years old, but I could not find a new one, if you find one, please let me know). L10.3. ANALYSIS OF RESONATORS In some cases it may be enough for us to resort to our knowledge of circuit theory and simply model the resonator as a parallel or series RLC circuit coupled to the oscillator using a series or parallel capacitor. Lumped element and often microstrip resonators can be simulated using circuit simulators such as Serenade or ADS. Microstrip resonators are better analyzed using a field simulator such as Momentum, however. Resonators such as dielectric pucks or metallic cavities can not be designed in a circuit simulator but an equivalent circuit can sometimes be used. A closer analysis of resonators may require an analysis of the field configurations and modes in the 3-dimensional geometry. In still other cases we may be strictly interested in the bandpass properties of the resonator. In this case we not care about the equivalent circuit or mode distribution, just the Q, input impedance, and resonant frequency. Assuming we have coupled the resonator to a 50-Ohm line, the resonant frequency and input impedance are easily found using a network analyzer. Finding the Q is a more difficult task, however. One method for calculating resonator Q is summarized below. Calculating Resonator Q Factors An open or shorted half-wavelength section of line is the most often used microstrip resonator. One or both ends of the line can be coupled to the rest of the circuit. The resonator can be fed in series or from the side. Usually, it is fed through a small gap capacitor. For an open circuited resonator, the voltage is maximum and the current zero at the edges, and vice versa is true in the middle. This means that the impedance is high at the edges and zero in the middle, and it is 50 ohms somewhere in between. If you fed the resonator in the middle, you would probably see nothing interesting on the network analyzer, since very little power would be coupled into the resonator. The parameters that describe a resonator are the resonant frequency, 0f , the coupling coefficient

, and the quality factor Q. The unloaded quality factor Q describes a resonator alone in the universe, so it cannot be measured directly. The Q is the ratio of the stored reactive energy and

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the lost power. The loaded quality factor LQ includes the rest of the circuit, so it describes the

coupling, and is related to the unloaded Q as

(1 )L eQ Q Q

and eQ is the external quality factor due to resistive loading. How do we measure these

parameters? The network analyzer measures the reflection coefficient of a loaded resonator. Let us look at an equivalent parallel resonant LCR circuit. The input impedance in this case is

1 1 1

1 2in

Rj C

Z R j L j Q

where 0 0( ) / is the frequency detuning parameter. The points on the Smith chart of

the impedance as the frequency varies are shown in Fig.L12.4a for several coupling cases. When

0, 1R Z , which is called the case of critical coupling and is the circle A in the figure. For

0, 1R Z the resonator is overcoupled, corresponding to circle labeled C, and for

0, 1R Z the resonator is undercoupled, circle B. The coupling coefficient can be

measured by measuring the reflection coefficient 110s of the coupled resonator at resonance:

(a) (b)

Fig.L12.4. (a) Resonator impedance circles on the Smith chart for different coupling conditions. (b) Finding the different Q factors from the Smith chart and a network analyzer measurement of the resonator

110

110

1 1

1undercoupledovercoupled

s

s

9

When we find the number for from the measurement, we have the intersection of the impedance circle with the real axis. To identify the different quality factors, we write the normalized impedance as

1 2 1 2 (1 ) 1 2inL e

zj Q j Q j Q

where the corresponding normalized frequency deviations can be expressed as

1 1 1, ,

2 2 2L eL eQ Q Q

The intersection of the arc B=G (all points for which the conductance equals the susceptance) with the impedance circle gives two frequencies which determine the unloaded Q as:

0

1 2

fQ

f f

Similarly, the intersection of the arcs B=G+1 and B=1 with the impedance circle determine the loaded and external quality factors, respectively, as shown in Fig.L12.4b. Resonators in the time domain: It may have occurred to some of you that there might be a relationship between resonator Q-factor and its time domain behavior: in fact there is an interesting relationship which can be derived if we consider the assumption that the rate of energy decay from a resonator (if we turn the original energy source off) is proportional to the amount of energy stored in the resonator. This statement leads us to the differential equation

Wdt

dW 2

which has solutions teWtW 2

0)(

where the solution with negative exponent refers to the decay rate of energy leaving the resonator and the solution with positive exponent refers to the rate of energy build-up when we turn the driving source on. It can then be shown that the decay constant is related to the resonator Q such that the solution actually is

QteWtW /0)(

where higher Q means slower rates of energy decay or energy build up. This gives some insight into why higher Q helps stabilize the frequency when coupled to an oscillator.

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L12.4. VARIOUS MICROWAVE OSCILLATOR EXAMPLES Next, we look at the active device portion of the oscillator. Transistors can be configured with feedback, similarly to low-frequency circuits, but there are also a few devices such as Gunn diodes, IMPATT diodes and tunnel diodes which are two-terminal devices that exhibit negative resistance. We will first look at a Gunn diode, which is still used extensively at millimeter-wave frequencies, and then we will analyze two simple transistor oscillators. L12.4.1. Gunn diode device characteristics and operation The Gunn diode is a two-terminal device that has a true RF negative resistance. Also, if you measure its dc I-V curve, you can observe ac negative resistance behavior. The Gunn diode is not a diode in the real sense of the word, but it is called a diode because it has two terminals. It was named after J.B. Gunn who discovered the bulk negative resistance effect in GaAs in 1963. Gunn diodes are sometimes called transferred electron devices because electrons are transferred from a lower to a higher-energy conduction band when oscillations are produced. Gunn diodes are bulk devices, i.e. they do not have a semiconductor junction.

Fig.L12.5. Sketch of the energy band diagram of bulk GaAs. The negative differential resistance in GaAs is present because of its specific energy band structure, shown in Fig.L12.5 In an n-type slab of GaAs, the energy band gap is direct in momentum space, which means that carriers can move directly from the valence into the conduction band when they acquire an energy equal to the band gap energy. Further, there are two conduction bands – one direct band at 1.4eV above the valence band, and a satellite band 0.31eV higher in energy that the main conduction band. The effective mass of electrons in the upper band is larger ( m m2 00 55* . ) than the effective mass in the lower band ( m m1 00 067* . ). The electron mobility in the lower band (1 8500 cm2/(Vs)) is about 50 times greater than that of the higher band ( 2 150 cm2/(Vs)). From the expression that relates the effective mass and mobility,

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q

m*

the carrier collision times must be shorter in the upper band, since the mobility ratio is larger than the effective mass ratio.

Fig.L12.6. (a) The electron velocity in an n-type bulk GaAs versus applied DC electric field, and electron distribution in the different bands for different applied field values (b), (c), (d). What happens when two ohmic contacts are made on a slab of n-type GaAs, and a dc electric field is applied across them? Fig.L12.6 shows the electron velocity versus applied electric field, which is equivalent to the dependence of the current through the slab versus the applied voltage (why?). For certain values of the field, up to point 1 in Fig.12.6a, corresponding to about several kV/cm, the slab behaves like a normal resistor. The electron distribution corresponding to this case is shown in Fig.L12.6b, most of the electrons that gained enough energy from the field are now sitting in the lower conduction band, they are fast and have a mobility 1 . As the magnitude of the applied dc field increases, some of the electrons will gain enough energy to get transferred up into the higher satellite band, but then they loose in mobility and slow down. This corresponds to the region between points 1 and 2 in Fig.L12.6a, and the electron distribution is

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shown in Fig.L12.6c. If the electrons that form the current in the slab are slowing down with applied field. This in turn means that the current is decreasing with applied voltage, which gives a true negative resistance. As the field is increased further, there are no more electrons to be slowed down, since the whole lower band will be depleted, so the slab is again a resistor, and all the electrons have a mobility 2 . The two common materials used for transferred electron devices are GaAs and InP. The critical electric field value in GaAs is 3.2kV/cm, and in InP it is 10.5kV/cm. The respective peak carrier velocities 2 2 107. cm/s and 2 5 107. cm/s. Is it possible to calculate the negative differential resistance (or negative differential mobility), or equivalently to calculate the function v(E)? We can start from the expression for the current density as a function of electric field, assuming the material is linear:

J = E = + )E q n n( 1 1 2 2 ,

where the densities n1 and n2 are different in the two valleys of the conduction band, and in GaAs the ratio of the upper-to-lower band concentration is 94. (This number is obtained by taking into account that the density is proportional to the effective mass to the power 3/2, and also remembering the total number of valleys and the number of Brillouin zones. This is beyond the level of this course, however, so we will just assume that we know that the upper band has a higher concentration.) For very high and very low fields, all of the electrons, with density n n n0 1 2 , have respective mobilities of 1 and 2 , as shown in Fig.L12.6a. For the region in between,

q n n qn( )1 1 2 2 0 ,

where the average mobility is introduced as

n n

n1 1 2 2

0

.

If the transition from 1 to 2 occurs quickly, there will be a portion of the curve with negative slope. Is there a static (dc) negative resistance in Gunn diodes with a uniform electric field distribution? Starting from Poisson’s equation,

1)(

0

00

N

nqNNnq

x

E

where the electric field is assumed to be in the negative x direction (as in Fig.L12.7) and N0 is the doping (i.e. the equilibrium concentration). For a static solution, the current density is constant with x (the coordinate along the device length):

J qn v E ( ) , where J does not depend on E and is directed in the negative x-direction (not intuitive). Therefore,

13

1)(

_

0

0

EvqN

JqN

x

E

.

We assume that initially a voltage V is applied across the sample of GaAs of length L such that V/L is greater than the critical field. The only quantity in the equation above that depends on the electric field is the velocity. For the contact region (x<0), the doping is very large and the electric field must be small. Because of continuity, E must start out low and increase fast with x, which in turn means that the average velocity must increase. This in turn means that dE/dx must decrease from the high initial value. The resulting conclusion is that E is not uniform and at some coordinate x reaches the critical field. In order for the integral of E to be consistent with the applied voltage, it has to continue to increase.

Fig.L12.7. A charge layer forms in a Gunn diode (a) and gets pushed towards the anode by the electric field (b). We were so far talking only about dc electric fields. How does this Gunn diode produce rf energy? When a group of electrons slow down at a defect (imperfection) in a piece of GaAs under dc bias, these electrons make a “traffic jam”, and more electrons pile up to form a charge layer, Fig.L12.7a. This layer now produces an electric field that decreases the original field to the left of them, and increases the field to the right, Fig.12.7b, so that there is an electric force on the charge bunch which is pushed towards the electrode on the right, forming a current pulse. When this pulse gets to the electrode, the electric field goes back to the original value, another traffic jams happens, another pulse forms, and so oscillations occur. This is the main mode of oscillation in a piece of GaAs called the Gunn mode and it was experimentally measured by Gunn using capacitive probes along a sample. A figure from Gunn’s paper is reproduced in

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Fig.L12.8 (“Instabilities of current and potential distribution in GaAs and INP,” J.B. Gunn, 7th Conf. Phys. Semicond., Academic Press, p199, 1964). The frequency will depend on the length of the piece of GaAs between electrodes, as well as the concentration of electrons. The space-charge bunch, usually called a domain, must grow to its stable condition before it reaches the ohmic contact at the anode of the device. This means that the dielectric relaxation time d must be less than the transit time through the sample length L:

q n

L

v| | , or nL

v

q

| |

,

. where 13 0 is the dielectric constant of GaAs, 100 cm2/(Vs) is the mobility of the

electrons, and v 107 cm/s is the stable electron velocity. This implies that the nL product must be greater than 1012 cm-2 . The frequency of operation is determined by the transit time through the device:

fT

v

Lt

1

or fL 107 cm / s .

At 10GHz, a typical Gunn diode will have an active region 10m long and a doping level of 1016 m-3 . They are usually biased at around 8V and 100mA, and have low efficiencies of only a few percent. This means that the diode dissipates a lot of heat and needs to be packaged for good heat-sinking. A M/A-Com Gunn diode package and basic specifications are shown in Fig.L12.9. Sometimes the chip diode is packaged on a small diamond heat sink mounted on the metal (compare diamond thermal conductivity to that of copper). The largest achieved power levels from commercial Gunn diodes are a few kW with less than 10% efficiency at 10GHz in pulsed mode, and 1W with less than 5% efficiency in continuous wave (CW) mode. Indium phosphate has the same energy band diagram shape as GaAs, and InP Gunn diodes have achieved about 0.1W CW at 100GHz with about 3% efficiency. Fig.L12.8. Direct measurement of motion of a high-field domain in a bulk GaAs device, by J.B. Gunn. The recording shows the derivative of the potential with respect to time taken at successive time intervals, versus position between cathode and anode.

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Fig.L12.9. A commercial MaCom Gunn diode package (a) and specifications (b). L12.4.2. A Gunn diode microstrip oscillator How does one design a two-terminal device oscillator in practice? There are basically two kinds of embedding circuits that can be used: waveguide cavities and tuners (such as shown in Kurokawa’s paper) and microstrip (or other planar) circuits with lower-Q resonators. In the first kind, the device is mounted in a post in the waveguide, and the embedding impedance is a function of all the evanescent modes that this post excites. We will study later a classical paper by Eisenhart and Kahn that presents an excellent method for calculating this impedance partly analytically and with great insight. An example of the second type, a microstrip Gunn-diode oscillator, is shown in Fig.L12.10. In designing this simple-looking circuit, the device needs to be provided with: proper mounting, biasing, a resonator and a coupling mechanism to a load.

Fig.L12.10. Layout of a Gunn-diode microstrip oscillator.

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IMPATT diodes – Another two-terminal device used for microwave generation is the impact-ionization avalanche transit-time diode, first proposed by Read in 1958 for a n pip diode structure. Although three-terminal devices are now more commonly used at microwaves, the IMPATT diode is still very useful at millimeter-wave frequencies for high power. It is just a pn diode under reverse bias. The applied reverse bias voltage is large enough, 70-100V, so that an avalanche process takes place. This means that some of the electrons get violently accelerated and as they hit the atoms, they form electron-hole pairs that are now new carriers, which in turn get accelerated and an avalanche process happens. The fact that there is a saturation velocity of the charges in every semiconductor accounts for oscillations in an IMPATT diode. The mechanism of oscillation of an IMPATT diode does not depend on the carrier mobility, so silicon IMPATTs are made as well. They dissipate a lot of heat, are usually used in pulsed mode, have higher efficiencies than Gunn diodes (about 20%), and are very noisy because of the avalanche mechanism. Commercial IMPATTs are packaged in the same way as Gunn diodes, and the maximum available powers are 50W pulsed at 10GHz and 0.2W CW at 100GHz. L12.4.3. A Negative Resistance Transistor VCO Example Figure L12.11 shows a circuit diagram of a Clapp oscillator topology and a simple first-order equivalent circuit, where it is assumed that the values of C1 and C2 either dominate or take into account the intrinsic transistor capacitances.

(a) (b) Figure L12.11. (a) Clapp transistor oscillator with varactor diode for tuning. (b) Equivalent circuit for transistor with feedback. By analyzing the circuit in Fig.L12.11b, we obtain the following approximate expression for the input impedance:

21

21

212

)(

CC

CCj

CC

gZ m

in

which shows that the real part of the input impedance is negative, indicating a reflection coefficient larger than unity, while the imaginary part is capacitive, and in fact is the series combination of the two external feedback capacitors.

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If the series resistance of the external inductor or combination of inductor is RLS, then for the maximal value of C1=C2=Cm, the following condition of oscillation is obtained:

m

LS

m g

R

C

1

This shows that for oscillations to be maintained, the minimal value of the external capacitance is a function of the resistance of the resonator (which could just be an inductor) and the transconductance. If a varactor diode is added in series to the inductor, the total capacitance is reduced, and the value of the inductor will change for a given resonant frequency. As an example, assume that the large-signal gm is 20mS and that you chose a varactor that has a capacitance ratio of 4:1 and a parasitic inductance of 0.45nH. The inductor is implemented with a microstrip transmission line with Q=200 and the capacitances are C1=2pF and C2=0.5pF. The dynamic negative resistance is

921

2 CC

gr m

This means that the loss in the resonator needs to be smaller than 9 ohms, this is the combined resistance of the varactor diode and the inductor. A typical medium value for the tuning capacitance is 1pF. For an assumed oscillation frequency of e.g. 7.5GHz, this gives an inductance value of about 1.57nH. From this and the Q factor, we find that the resistance of the inductor is about a quarter of an ohm. Now we can find the tuning bandwidth assuming e.g. 0.4-1.6pF capacitance values of the diode in reverse bias. Note that it is also possible to use a high-Q resonator, such as a dielectric resonator coupled to the microstrip line, instead of the broadband varactor-tuned version. In that case, the phase noise would be optimized and the oscillator would not be tunable. A dielectric resonator can be modeled as a resonant circuit which is transformer-coupled to the microstrip line. A full-wave simulation is suggested for precise modeling, and varying three parameters can control the Q: distance from end of open or short-circuited microstrip line (L), distance from line that enables coupling (d) and height above substrate (h). In your next project, you will analyze a three-port transistor oscillator implented in microstrip technology. A description of such an oscillator is given in Project 5.

12.5. INJECTION LOCKING OF MICROWAVE OSCILLATORS

Injection locking of an oscillator means that a signal is injected into the oscillator to control some of its parameters. The injected signal is usually about 20dB below the free-running oscillator power and it is usually a very clean signal (low noise). Its frequency is close to the free-running frequency (typically within a few percent). If the frequency and power of the injected signal are adequate, the oscillator will lock to the injected signal. This means that its frequency will change

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and become equal to the injected frequency. It turns out that its noise characteristics will also improve if the injected signal has lower noise. Injection-locking is commonly used for FM noise improvement and also for modulation by FM modulating the injected signal instead of the oscillator directly.

(a) (b) Fig.L12.12. (a) The equivalent circuit of a one-port injection-locked oscillator. The generator e is the injected signal. (b) The equivalent wave representation of an injection-locked one-port oscillator. The injected wave is c, and the reflection coefficient of the device is a function of both power and frequency. A classical paper on injection locked oscillators is that by Kurokawa, attached to your lecture. This is a difficult paper, but it is the most systematic and the clearest paper on injection-locking. In the paper, a one port oscillator is represented by the impedance equivalent circuit shown in Fig.L12.12a, with the condition of free-running oscillation given by Z f Z A I eD( ) ( ) . In

this equation, Z f( ) is the impedance of the circuit seen from the device terminals (the embedding circuit), and Z AD ( ) is the impedance of the active device, where A is the current (or voltage) amplitude of oscillation. Kurokawa’s paper uses current as the parameter, which is appropriate for IMPATT diodes*. Here you can notice that there is a rather severe assumption: the circuit impedance is only a function of frequency, and the device impedance is only a function of amplitude (power). While this might be true for Gunn and IMPATT diodes, it is usually not true for transistor oscillators. Even if the oscillator circuit is purely reactive, the transistor parameters are a function of both power and frequency even in a small region around the oscillating frequency (the injection-locking range). Another way to look at injection locking is shown in Fig.L12.12b. The oscillator is represented by a reflection coefficient s p fo ( , ) , which is a function of the output power p and frequency f. The matched load has a reflection coefficient of zero. The output signal from the oscillator is represented by a wave amplitude a, where the magnitude of a is the square root of the output power. The injected signal is written as c, where the magnitude of c is the square root of the injected power. From the circuit model we can write the relation

s p fa

co ( , ) .

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Fig.L12.13. (a) Contours of constant p and f in the c a s p fo/ / ( , ) 1 complex plane. (b) Stable oscillations correspond to the bottom half of the c/a circle, and unstable ones to the top half of the circle. The power gradient is in the direction of the arrow. Fig.L12.13a is a plot of 1/ ( , )s p fo in the complex plane. This is a convenient representation, because the origin (c=0) represents the free-running oscillation condition. The contours of constant power and frequency may be drawn as straight lines in a small neighborhood about the origin. As the frequency varies, the tip of the c/a arrow traces out a semicircle, provided the injected power remains the same, and that the deviation of the radiated power is small. For a given |c/a| the frequency deviation varies symmetrically around the free-running frequency from f max to f max , where 2fmax is the injection-locking bandwidth. The power deviation p inside the locking range is not zero and is not symmetrical in general. If injection locking is used for frequency modulation, this corresponds to AM noise. If injection locking is used for

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stabilization, the power at the stabilized frequency will change and it is important to know by how much. From Fig.L12.13a, the frequency and amplitude of an injection-locked oscillator are given by:

f f

p p

p

p

max

max

max

sin

sin( )

arcsin

0

where is the angle between the contours of constant f and p , and changes from / 2 to / 2 inside the locking range. p0 is the change in oscillator power at the free-running frequency after it has been injection-locked. This means that it is sufficient to measure fmax , pmax and p0 in order to predict the injection-locked oscillator behavior. The tip of the c/a arrow traces a whole circle in the complex plane, but only one half of this circle corresponds to stable solutions. The stability of oscillation is discussed in Fig.L12.13b. The contours of growing power are indicated with the direction of the arrow. For oscillations in the top half of the circle (left part of Fig.L12.13b), a perturbation a at a frequency f causes c/a to move from the contour p to a contour of larger power p p . This corresponds to an unstable oscillation. The same perturbation will cause a decrease in power for stable solutions laying on the bottom half of the circle (right hand part of Fig.L12.13b). To decide which half of the c/a circle corresponds to stable solutions, we draw a line from f max to f max , and choose the semicircle which the power-gradient arrow points to. In a measurement scenario, in order to experimentally obtain a Rieke diagram for the injection locking behavior, the following steps can be taken:

- set the free-running oscillation, measure the output power - set the injected signal to a known power level and at the free-running frequency - the length of the c/a vector is given by the square root of the ratio of the free-running to

the injected power - injection lock the oscillator at the free-running frequency. Measure the change in power,

this gives you p0 - sweep the frequency of the injection-locking signal until you reach the two ends of the

locking range – this gives you fmax - during that process, record the peak power of the oscillation – the maximum or minimum

value of the entire power range gives you pmax This procedure gives you , and assuming the constant power and frequency lines are straight, by varying from –90 to 90 degrees you can predict the oscillator behavior for other injected power levels. If you wish to see some examples of this, let me know.