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Sheet 1 of 26 Oscillator Basics Tutorial J P Silver E-mail: [email protected] ABSTRACT This paper discusses the basics of oscillator design including the parameters effecting oscillator per- formance, with special emphasis on the causes of phase noise. Theory is given for the two types of os- cillator topography namely feedback and reflection oscillators. INTRODUCTION This tutorial shows how device parameters can effect the performance of oscillators, including output power, os- cillating frequency and probably most important of all – phase noise. An example is given for each of the two types of oscillator the feedback oscillator and the reflec- tion oscillator OSCILLATOR SPECIFICATION OSCILLATOR FREQUENCY The frequency of operation determines the active device to be used as well as the technology. For example, a UHF oscillator would use a device with an f T of a few GHz and would employ a lumped resonator. It would be impracticable to use a dielectric resonator due to size at UHF frequencies. The use of a very high f T device may lead to problems with stability. OSCILLATOR BANDWIDTH Many applications require variable frequency operation for use in synthesisers where a range of frequencies or frequency steps may be needed across a particular band of frequencies. In order to achieve such variations the resonator must to have a variable element, which is often a varactor diode (a device whose depletion layer width and hence its value of capacitance, is directly controlled by the amount of reverse bias applied). The use of varactors can cause problems in that they usually deter- mines the Q of the resonator, which is an important fac- tor in setting the phase noise floor, as we shall see later. OUTPUT POWER The output power requirement is determined by the ap- plication, but the use of a high power device will mean a phase noise performance. After start-up the oscillator will reach a point close to the saturated output power of the device, which is not far from the 1dB compression point. A diagram of a typical compression characteris- tic, for the device used as an amplifier, is shown below in Figure 1. 1dB 1dB compression point Saturated output power Input power Output power Figure 1 Typical compression characteristic of an amplifier. The diagram shows the compres- sion point occurs when the gain has dropped by 1 dB from the linear region. The saturated output power is the maximum power that the amplifier can deliver and occurs several dB’s beyond the 1 dB compression point. The following expressions [1] give empirical formulae for a common-source amplifier output power based on the small signal gain of the device (ie modulus of S21). The objective is to maximise (P out – P in ) of the amplifier, which is the net useful power to the load:- 2 sat sat in sat out S21 ie gain transducer signal small tuned = G power ouput saturated P P P exp 1 P P = ⎛− = G
Transcript
Page 1: Oscillator Basics

Sheet

1 of 26

Oscillator Basics Tutorial J P Silver

E-mail: [email protected]

ABSTRACT This paper discusses the basics of oscillator design including the parameters effecting oscillator per-formance, with special emphasis on the causes of phase noise. Theory is given for the two types of os-cillator topography namely feedback and reflection oscillators.

INTRODUCTION This tutorial shows how device parameters can effect the performance of oscillators, including output power, os-cillating frequency and probably most important of all – phase noise. An example is given for each of the two types of oscillator the feedback oscillator and the reflec-tion oscillator

OSCILLATOR SPECIFICATION OSCILLATOR FREQUENCY

The frequency of operation determines the active device to be used as well as the technology. For example, a UHF oscillator would use a device with an fT of a few GHz and would employ a lumped resonator. It would be impracticable to use a dielectric resonator due to size at UHF frequencies. The use of a very high fT device may lead to problems with stability.

OSCILLATOR BANDWIDTH Many applications require variable frequency operation for use in synthesisers where a range of frequencies or frequency steps may be needed across a particular band of frequencies. In order to achieve such variations the resonator must to have a variable element, which is often a varactor diode (a device whose depletion layer width and hence its value of capacitance, is directly controlled by the amount of reverse bias applied). The use of varactors can cause problems in that they usually deter-mines the Q of the resonator, which is an important fac-tor in setting the phase noise floor, as we shall see later.

OUTPUT POWER The output power requirement is determined by the ap-plication, but the use of a high power device will mean a phase noise performance. After start-up the oscillator will reach a point close to the saturated output power of

the device, which is not far from the 1dB compression point. A diagram of a typical compression characteris-tic, for the device used as an amplifier, is shown below in Figure 1.

1dB

1dB compression

point

Saturated outputpower

Input power

Output power

Figure 1 Typical compression characteristic of an amplifier. The diagram shows the compres-sion point occurs when the gain has dropped by 1 dB from the linear region. The saturated output power is the maximum power that the amplifier can deliver and occurs several dB’s beyond the 1 dB compression point.

The following expressions [1] give empirical formulae for a common-source amplifier output power based on the small signal gain of the device (ie modulus of S21). The objective is to maximise (Pout – Pin) of the amplifier, which is the net useful power to the load:-

2

sat

sat

insatout

S21 ie gain transducer signal small tuned = G

power ouput saturatedP

P

Pexp1PP

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

G

Page 2: Oscillator Basics

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2 of 26

( )

inoutin

out

inout

inout

P w.r.tP atedifferenti 1 = PP

0 = P-P

-: varied is power input the as that require we, P - P maximise to is objective the Since

∂∂

d

nG11P P

2 & 1 equations Combining PP P

is power output oscillator maxiumum the and

(2) ............. 11P P

is output amplifier the, P-P of value maximum At

(1) ........... nG = PP

G = P

Pexp

1 = P

PexpG =

PP

SATOSC

inout OSC

satout

inout

sat

in

SAT

in

sat

in

in

out

⎟⎠⎞

⎜⎝⎛ −−=

−=

⎟⎠⎞

⎜⎝⎛ −=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

GG

G

G

G

G

l

l

∂∂

Thus, the maximum oscillator output power can be pre-dicted from the common-source amplifier saturated out-put power and the small-signal common source trans-ducer gain G.

OUTPUT POWER EXAMPLE The following example uses the Fujitsu FHX35LG, which is a HEMT GaAs FET device. The data sheets give the value of the 1dB compression point of ~ 15dBm when biased at 3V (Vds) with a drain current 20mA (Ids). The data sheets also give the magnitudes of S21 for various frequencies. We shall assume we want to estimate the output power from a feedback oscillator using one active device. The schematic of the oscillator is shown below in Figure 2.

Amplifier Resonator

Output

Figure 2 Schematic of a simple feedback oscil-lator. The amplifier needs to have enough gain to overcome the loss of the resonator. The output is usually lightly coupled to an at-tenuator to overcome load-pull problems.

From the data sheet the estimated saturated output power is ~ 16.5dBm; S21 Magnitude @ 8GHz 2.659

( )( )( )

9.6dBm P

659.2659.2Ln

659.21116.5 P

LnG11P P

OSC

2

2

2OSC

SATOSC

=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

⎟⎠⎞

⎜⎝⎛ −−=

GG

POWER CONSUMPTION Many applications like portable telephones are reliant on batteries for power so that is essential to minimise the power required by an oscillator and to ensure maximum efficiency. This again can conflict with the phase noise performance, as a good output match is required for the oscillator to minimise the phase noise.

SPURIOUS & HARMONIC OUTPUT Any non-linear device will create harmonics at multiples of the fundamental frequency. These harmonics may interact with out-of-band signals in the system mixer causing spurious responses in the receiver, thereby de-sensitising it. In transmitters the efficiency may be re-duced if strong signals are generated that are not re-quired, as the total power will be shared amongst the different signals generated.

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PHASE NOISE. For a discussion on phase noise read the Phase Noise Tutorial. But in summary Leeson’s equation is given below:-

1Q2f

+12PFkT = )(L

2

Lmavs ⎥⎥⎦

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎞⎜⎜⎝

⎛+

fmfcf

fmfcf o

m

Usually the phase noise is specified in dBc/Hz ie :-

L = 10Log FkT2P

1 +2f Q

dBc / Hz10avs m L

( )f fcfm

f fcfmm

o+⎛

⎝⎜⎜

⎠⎟⎟ +

⎛⎝⎜

⎞⎠⎟

⎢⎢

⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

2

1

The Leeson equation identifies the most significant causes of phase noise in oscillators. Therefore it is possible to highlight the main causes in order to be able to minimise them. In order to minimise the phase noise of an oscillator we therefore need to ensure the following:- (1) Maximise the Q. (2) Maximise the power. This will require a high RF voltage across the resonator and will be limited by the breakdown voltages of the active devices in the circuit. (3) Limit compression. If the active device is driven well into compression, then almost certainly the noise Figure of the device will be degraded. It is normal to employ some form of AGC circuitry on the active de-vice front end to clip and hence limit the RF power in-put. (4) Use an active device with a low noise figure. (5) Phase perturbation can be minimised by using high impedance devices such as GaAs Fet’s and HEMT’s, where the signal-to-noise ratio or the signal voltage rela-tive to the equivalent noise voltage can be very high.

(6) Reduce flicker noise. The intrinsic noise sources in a GaAs FET are the thermally generated channel noise and the induced noise at the gate. There is no shot noise in a GaAs FET, however the flicker noise (1/f noise) is significant below 10 to 50MHz. Therefore it is prefer-able to use bipolar devices for low-noise oscillators due to their much lower flicker noise, for example a 2N5829 Si Bipolar transistor, has a flicker corner frequency of approximately 5KHz with a typical value of 6MHz for a GaAs FET device. The effect of flicker noise can be reduced by RF feedback, eg an un-bypassed emitter re-sistor of 10 to 30 ohms in a bipolar circuit can improve flicker noise by as much as 40dB. (7) The energy should be coupled from the resonator rather than another point of the active device. This will limit the band-width as the resonator will also act as a band pass filter.

Flicker effect

Resonator Q The relationship between loaded Q, noise factor and centre frequency can be used to derive the single-sideband phase noise performance, for a given fre-quency offset in the form of the nomograph shown in Figure 6.

Phase perturbation

Generally then we require to maximise the loaded Q and this can be done by using a coaxial or dielectric resona-tors. However this is all very well for a fixed frequency oscillator where we are able to maximise the Q, we gen-erally require a variable frequency oscillator, (VCO) for use in a phase locked loop, to cover a band of frequen-cies. Such VCO’s require a method of converting the PLL control voltage to frequency and this is normally done with a varactor diode (Vari-capacitance diode). Unfortunately any noise on the PLL control voltage and any internally generated noise will modulate the carrier, increasing the overall phase noise performance. The equivalent noise voltage modulating the varactor is given by Nyquist’s equation [4]:-

Hz volts/root TR4V enrn k=

The peak phase deviation in a 1 Hz bandwidth which results from the varactor noise resistance is :-

( ) ( ) 2

220L ie

2d20L

-: is dBc/Hz in noise phase resulting The

Hz/volt. in constant gain VCOthe is K where2

= d v

m

nvmm

m

nv

fVK

LogfLogf

fVK

==ϑ

ϑ

Therefore, the total single-sideband phase noise will be the power sum of the oscillator phase noise given by the

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Leeson equation added to the varactor phase noise just given. A quick lookup chart for the additive single sideband phase noise is given in Figure 3 and Figure 4 where the phase noise is given for a VCO tuning constant range of 1000MHz/V to 0.001MHz/V for a given effective noise resistance of 3.3KΩ and 1KΩ respectively

Phase Noise Contribution from Varactor

-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

100 1000 10000 100000 1000000 10000000

fm (Hz)

Phas

e N

oise

(dB

c/H

z) 100

1000

10000

100000

1000000

10000000

Figure 3 Varactor phase modulation contribu-tion to single sideband phase noise perform-ance, assumimg an equivalent varactor noise resistance of 3300ohms.

Phase Noise Contribution from Varactor

-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

100 1000 10000 100000 1000000 10000000

fm (Hz)

Phas

e N

oise

(dB

c/H

z) 100

1000

10000

100000

1000000

10000000

Figure 4 Varactor phase modulation contribu-tion to single side-band phase noise perform-ance, assuming an equivalent varactor noise resistance of 1000ohms.

As can be seen from the proceeding graphs the contribu-tion of phase noise from the varactor can be very sig-nificant and mask the performance of a good low noise oscillator employing a high Q resonator. For this reason

it is not preferable to use a large gain control constant, but to use a narrower range varactor and physically switch capacitance in and out using PIN switching di-odes to cover the required range.

OSCILLATOR FUNDAMENTALS

FEEDBACK OSCILLATOR An amplifier provides an output that is a replica of the input. An oscillator provides an output at a specific fre-quency with no input signal required. Figure 5 shows the three fundamental parts of a feedback oscillator ie the amplifier (capable of amplifying at the frequency of interest) a resonator (the frequency selective component) and an output load. The resonator may contain trans-formers or other impedance transforming components such as coupling capacitors. Amplifier

resonator

Output load ~ 50 ohms

X

Z = Zin Rload

+Zin

Vout

a

b

Figure 5 Closed loop (a) and open loop (b) oscillator mod-els. Figure a shows the closed loop model with the three main parts of the oscillator the resonator, active device and output load. To aid analysis the loop is often broken at point X to form the open loop model shown in figure b. The open loop model can be analysed, for insertion magni-tude and phase difference. At the required oscillator fre-quency the phase difference through the loop must be 0 or 360 degrees (or multiples of 0 and 360 degrees) and that the corresponding loop gain magnitude is maximum and greater than unity. There will be no output when power is initially applied, but even if the amplifier were noise free, noise would still be generated in the resonator at the resonant fre-quency. This noise will be applied to the input of the

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factorfeedback gain loopopen A

feedbackafter gain Af 1

= Af

==

=−

β

βAA

amplifier where it will be amplified and fed back in phase at the resonant frequency and further amplified, building up each time. Eventually the signal will cause the amplifier to limit, ensuring that the oscillator output power eventually peaks, usually at the saturated output power of the amplifier.

The gain A will be infinite when the loop gain µβ is unity and the phase shift is 360°. This is known as the Barkhausen criterion for oscillation [5].

The oscillator loop gain is given by:

-50

-70 0.0

1000

100

10

1

0.1

1000

100

10

1

0.1

-30

-20 -10 0 +10

-167

- -90

-157 -110

-177 -130 -187 Fo/2 Fm

-197

Power-NF Floor

-150 SSB Ø-NOISE

Figure 6 Nomograph for calculating the phase noise of an oscillator. The nomograph is valid for offset frequencies 1/fc to fo/(2QL), where fc = flicker corner frequency of the active device and QL = loaded Q of the resonator.

Page 6: Oscillator Basics

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To evaluate the circuit it is easier to split the loop at ‘X’ in Figure 5a. Figure 5b shows the open loop equivalent circuit for the oscillator, where the circuit is analysed for the voltage gain & phase appearing across the terminat-ing impedance. If the phase shift is 0 degrees and the voltage magnitude ratio at the same point greater than unity then oscillation will occur. Figure 7 shows a typi-cal voltage and phase plot for a open-loop oscillator.

1

2

Voltage ratio

magnitude frequency

0

180

-180

fo

Phase degrees

Voltage response through the loop

Phase response

through the

Figure 7 Typical magnitude & phase response for an open-loop oscillator, showing that at the maxi-mum magnitude (greater than unity) the phase dif-ference through the loop is zero.

The resonator can take a variety of different forms, which will be described later. The idea is to design the resonator with a light enough coupling to give the Q required to meet a particular phase noise performance.

NEGATIVE RESISTANCE OSCILLATOR.

These types of oscillator also employ feedback, but as a way of providing a negative impedance at an input com-plex load. At microwave frequencies, devices are usu-ally characterised by using S-parameters and these can be used to calculate stability, and for our purposes insta-bility. This is can be achieved by using a conditionally stable device, or by using a non-conditionally stable device in a different configuration eg common-source or by using positive feedback eg source feedback. The general conditions for oscillation are [7] :- K < 1 ΓG.S11’ = 1 where ΓG = input load reflection coefficient and S11 = modified input reflection coefficient.

ΓL.S22’ = 1 where ΓL = output load reflection coefficient and S22 = modified output reflection coefficient The stability factor K must be below one for any chance of oscillation. K may be optimised to be below one by the methods of configuration and feedback. Passive terminations need to be selected to resonate the input and output ports at the frequency of oscillation ie ΓG.S11’ = 1 or ΓL.S22’ = 1 It can be shown that if one port meets the above criteria for oscillation, then the other expression must be satis-fied ie if one port is oscillating then so is the other. Since the loads are passive then this will imply that | S11’ | > 1 and | S22’ | > 1 From transmission line theory we can generate expres-sions for the modified input and output return losses due to the addition of a load

MODIFIED S PARAMETERS - MISMATCHED SOURCE & LOAD [8]

The diagram below (Figure 8) is a diagram showing the S-parameters for a two-port device and how they are modified at the input port when a mismatched output load is added. By definition S-parameters are measured in a 50 ohm system so if the output load is mismatched it will alter S11 to S11’. This is a useful result for the oscil-lator designer as it means that it should be possible to modify the input return loss to be greater than 0dB by simply mismatching the output. The following theory describes how the input return loss can be modified.

a1

b2 b1

a2

S12

S21

S11’ modified

S11

S21

S12

S22

ΓL ZL ≠ Zo mismatched load

Figure 8 Modified input return loss by applying a mis-matched output load. This diagram is used in the fol-lowing analysis.

We can define the load reflection coefficient as:

Page 7: Oscillator Basics

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7 of 26

L22

212121L2221

1

121L222

1

2L222

1

121

1

221

1

221

1

111

L2221212221212

L2121112121111

L2

2

L

LL

.1S = 'S ).-(1'S

.1

..ab gives rearrange

1...

ab = 'S

ab 'S ;

ab 'S

..........

b2. = a2 ZZ

Γ−∴=Γ

−Γ

+=

==

Γ+=+=

Γ+=+=

Γ∴=Γ+−

SSS

aaS

abS

abS

aaS

bSaSaSaSbbSaSaSaSb

ba

ZZ

Lo

o

( ) ( )

( )

b (divide by on both sides)

S1 - S

Multiply both sides by 1 - S

S 1 - S + = S 1 - S

S - S S + = S 1 - S

1 LL

11L

22 L22 L

11 22 L L 11 22 L

11 11 22 L L 11 22 L

= +−

⎛⎝⎜

⎞⎠⎟

= +

S a SSS

a a

SS S

S S

S S

11 1 1221

221 1

1121 12

21 12

21 12

1. . .

..

'. .

..

. . . ' .

. . . ' .

ΓΓ

ΓΓ

Γ

Γ Γ Γ

Γ Γ Γ

( )

( )

L22

L1111

L11L2211

21122211

L2211L1221L221111

.S-1D.S ='S

D.S = .S-1'S

SSSS = D

.S-1'S = .. + .SS-S

ΓΓ−

Γ−Γ

ΓΓΓ SS

This is the final result to define the modified input re-flection coefficient.

MISMATCHED SOURCE Similarly we can perform the same analysis by mis-matching the source load to modify the output return loss. The S-parameter diagram for the analysis is shown below in Figure 9.

S12

S22’modifiedΓs

Figure 9 Modified output return loss by applying a mismatched source load. This diagram is used in the following analysis.

S11

S21122222

S11

1212

.1.. '

.S-1S = 'S

Γ−Γ

+=

Γ

SSSSS

s

s DΓΓ−

⇒11

2222 S-1

.S 'S

It can be proved that simultaneous oscillation will occur if one port is oscillating as follows:

12212211

22

11

22

21121111

L22

s11

SSSS = D where

S1

S = S1SSS = 'S above from

begin to noscillatio for = '

1 = '

1

Γ−Γ−

Γ−Γ

+

ΓΓ

L

L

L

L D

SS

Page 8: Oscillator Basics

Sheet

8 of 26

( )

LΓ = 'S

1 22

S

S

S

S

S

S

S

SL

SSL

LSLS

L

L

D

D

D

D

D

sD

Γ−Γ

ΓΓ−

ΓΓ

Γ−Γ

Γ

ΓΓ−Γ

ΓΓΓ−Γ

ΓΓ−Γ−

=

SS - 1 =

'S1

S - 1S =

S - 1SS+S22 = 'S

SS - 1 =

S - 1 = S

S - 1 = S

-: get weexpandingBy

= S

S1 'S

1

22

11

22

11

22

11

211222

22

11

1122

2211

11

22

11

Therefore this proves that if conditions exist for one port to oscillate then they must exist at the other port. Note ΓL is normally designated the output termination. Two-port oscillator design may be summarised as fol-lows: (1) Select a transistor/FET with sufficient gain and out-put power capability for the frequency of operation. This may be based on oscillator data sheets, amplifier performance, or S-Parameter calculation. (2) Select a topography that gives K < 1 at the operating frequency. Add feedback if K < 1 has not been achieved. (3) Select an output load matching circuit that gives | S11 ‘| > 1 over the desired frequency range. In the simplest case this could be a 50 ohm load. (4) Resonate the input port with a lossless termination so that ΓGS’11 = 1. The value of S’22 will be greater than unity with the input properly resonated.

RESONATORS [3] The resonator is the core component of the oscillator, in that it is the frequency selective component and its Q is the dominating factor for the phase noise performance of the oscillator. This section discusses the range of resonators, that can be used for an oscillator covering, dielectric, cavity, transmission line, lumped element and coaxial resona-tors.

LUMPED ELEMENT As discussed in the design example of section Error! Reference source not found. lumped element resona-tors are configured to form either a low, high or band pass filter, and the given number of elements is directly related to the Q and loss of the resonator. The simplest resonators can consist of just two elements an inductor and a capacitor ie:-

TWO ELEMENT RESONATOR CIRCUITS Figure 10 shows a schematic diagram of a two-element resonator. This circuit is seldom used in oscillators as the loaded Q will be very low as the source and load impedances will directly load the tuned cicuit.

Q = LR

ω..2

Q = 2.R.Lω

Figure 10 Schematic of a two element, lumped resona-tor, together with loaded Q equations.

At resonance the transmission phase is zero and the net-work is loss less (except for the resistance of the induc-tor). The series resonator impedes signal transmission while the parallel network allows signal transmission. The main problem with such a simple resonator is achieving a required Q, for example if we want a Q of 30 we would need the following series inductor & ca-pacitor at 1GHz:-

Page 9: Oscillator Basics

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0.05pF = 9477

91*21

= f21

= C

477nH = 1E9*2

30*50*2 = 2.R.Q = L

22

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

EE

Lππ

πω

Although the inductor is a realised value the capacitor could not be realised except in perhaps inter-digital form. This could be used if the oscillator is designed for fixed frequency but the value is impracticable as a varactor in a voltage controlled oscillator. The situation can be improved by using more than two elements eg 3 or 4 as described in the next section.

THREE ELEMENT RESONATOR CIRCUITS The diagram below shows a range of three element lumped resonators - Figure 11.

L

2L

2

C

L

X2XRX

RX

= Q

+=

Q RX

X X

LL

C L

=

= 2.

C

2C

2

L

C

X2XR

X

RX

= Q

+=

Q RX

X X

CC

L C

=

= 2.

X L & XCL C= =2 1

π. .

. .f

f

Figure 11 Schematic diagram of a range of three ele-ment resonators together with equations to calculate the reactive components and loaded Q.

FOUR ELEMENT RESONATOR CIRCUITS Four element resonators are used most commonly in oscillators as the loaded Q of the resonator can be set independently of the resonant circuit so that sensible component values can be calculated. Figure 12 shows a four element lumped resonator and Figure 13 shows an alternative configuration.

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C shunt

C series L

Figure 12 Schematic diagram of a four element lumped resonator

( )( )

Q unloaded L the is Q where

Q1

Q1

1 = Q where

12

R = X

-:elyapproximat is Q loaded given a for reactance The.C of function a is Q Loaded

1 = L

-:by given is f at resonate to inductance Required

resistance load tinput/oupu = R

1RR21

1C

-:is L inductor series the withresonates whichecapacitanc Effective

u

uL

e

2/1

ocshunt

shunt

2series

o

o

2o

2o

e

series

⎟⎟⎠

⎞⎜⎜⎝

⎛−

++

=

L

eo

eo

shunto

oshunt

series

XQR

C

CC

C

ω

ωω

C shunt

C series

L

Figure 13 Schematic diagram of the alternative four element lumped resonator

Page 11: Oscillator Basics

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11 of 26

( )

resistance load tinput/oupu = R

1R

2C

-:is L inductor shunt the withresonates whichecapacitanc Effective

L.f2

1

= Ce

-: inductor shunt resonate to eCapacitanc

admittance inductor shunt given a isB & Q unloaded L the is Q where

Q1

Q1

1 = Q where

X..21C 1

2R = X

o

2o

shunt

series

2

L

u

uL

e

cseriesseries

2/1

ocseries

+−=

⎟⎠⎞

⎜⎝⎛

=∴⎟⎟⎠

⎞⎜⎜⎝

⎛−

serieso

series

L

eo

CC

Ce

fBQR

ω

π

π

COAXIAL CABLE RESONATOR [10] A quarter-wave coaxial resonator is formed by shorting the centre conductor of a coaxial line to its shield at one end, leaving the other end open-circuited. The physical length of the resonator is equal to one quarter the wave-length (90 degrees electrical length) in the medium fill-ing the resonator. A diagram of a coaxial resonator is shown below in Figure 14.

λ /4

b

a

Figure 14 Schematic diagram of a coaxial cable resonator showing the critical dimensions.

coaxcoax 41 length Resonator = ;

f

2.99E8 =

λελ

λ

λ

==r

air

air

The unloaded Q of the resonator is a function of the conductor losses, the dielectric losses and the physical dimensions of the coaxial cable ie:

1-12-r

rD

C

DCU

Fm8.854x10=y;permitivit relative

;1 ie dielectric ofty conductivi =

..f.2 = factor) ssipationTangent/Di (Loss tan. = Q

by given is conductors the separates that dielectric the from oncontributi Q The

conductors the ofty conductivi = and ty permeabili = where

b1

a1

abLn....

2. = Q

by given is and conductors the inflow current to due lost energy to due is conductor from oncontributi Q The

Dielectric = D & Conductor = C e wherQ1

Q1

Q1

o

o

f

εε

ρσ

εεπσδ

σµ

σµπ

=

+

+=

DESIGN EXAMPLE OF A COAXIAL CABLE RESONATOR

The following example is for the design of a coaxial resonator to operate in an oscillator at 1GHz. The reso-nator is made from semi-rigid coaxial cable that contains a dielectric of PTFE, which has a relative permittivity of ~ 2.2 and a tanδ of 0.0004.

5.04cm = 36090.

2.21E9

2.99E8

= length Resonator

CALCULATION OF RESONATOR Q FACTOR The Q factor of the resonator determines the phase noise performance of the oscillator. Loss in the coaxial cable from the conductivity of the sheath and the loss tangent of the dielectric will set the Q of the resonator. Most coaxial cables especially semi-rigid cables use copper as the conductor, therefore the equation for the Q contribu-tion for the conductor ie Qcc is given by:

Page 12: Oscillator Basics

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12 of 26

The dielectric of the cable also effects the Q of the reso-nator and is given by:

92.95 Q (0.000358) 3.58mm = b example above For3.58mm or 0.141" is cable rigid-semi

typical of diameter Overall

f8.398.b. =

Q unloaded to oncontributi Conductor = Q

cc

cc

=∴

The dielectric of the cable also effects the Q of the reso-nator and is given by:

6.98 2500

192.95

1 =

Q1+

Q1 =

Q1 unloaded Total

2500 0.0004

1 Q

10GHz @ 0.0004 ~ PTFE for tan

material dielectric of tangent loss tan.

1 = Q

unloaded to oncontributi loss Dielectric = Q

dcc

d

d

=+

==∴

δ

δ

Note the Qcc term dominates the overall Q factor of the resonator at this frequency. The table below shows (Table 1) design data for a range of common materials used in the construction of coaxial cables:-

Material εr ρ tanδ Copper - 1.56E-8Ω.m - Gold - 2.04E-8Ω.m - Silver - 1.63 E-8Ω.m - Nylon 3.0 109-1011Ω.m 0.012@3GHz PTFE 2-2.1 1E-16 0.0004@10GHz

Polythene HD

2.25 >1014Ω.m 0.0004@10GHz

PVC flexi 4.5 109-1012Ω.m

Table 1 Design data for a range of materials com-monly used in the construction of coaxial cables. The parameters shown are relative permittivity (εr), resistivity ρ (1/ρ = conductivity) and tan delta (tanδ).

COAXIAL RESONATOR [11] A quarter-wave coaxial resonator is formed, by plating a piece of dielectric material with a high relative permit-tivity using a highly conductive metal. A cylindrical hole is formed along the axis of a cylinder of high relative permittivity dielectric material. All sur-faces, apart from the end surface, are coated with a good conductor to form the coaxial resonator. The physical length of the resonator is equal to one quarter the wave-length (90 degrees electrical length) in the medium fill-ing the resonator. The diagram (Figure 15) below shows the key dimensions of a coaxial resonator.

λ/4

W d

End of resonator platedOuter surface plated

Inner surface plated

RC

L ≡

Figure 15 Schematic diagram of a coaxial resonator showing the key dimensions. Note the resonator is plated with silver except for one end to allow it to be grounded.

The expression for the unloaded Q of such a resonator is

( ) ⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛

dW.079.1.60 =Z Impedance Input

88.5 of withdielectric sivered a for 200 = 38.6 of withdielectric silvered a for 240 = k

mm in diameter inside = d mm, in diameter outside = where

d1

W14.25

dW.079.1Ln

.ok. =

rin

L

Ln

W

f

r

r

ε

εε

Page 13: Oscillator Basics

Sheet

13 of 26

π

ε

πε

4.Zo.Q = Resistance

.103*2*4.25.

= eCapacitanc

mm in length Physical = 103.4.25

.8.Zo. = Inductance

8r

82r

Zox

x

l

ll

Below resonance, such short-circuited coaxial line ele-ments simulate high-Q, temperature stable ‘ideal’ induc-tors. They will only realise an ‘ideal’ inductor over a narrow range as shown in the diagram Figure 16.

X L

X C

S elf R eson an tF req u en cy

Freq u en cy →

‘Ideal’Inductance

R egion

Frequency

Figure 16 Frequency response of a coaxial resonator. The first region shows an area of inductance followed by a point of resonance followed by a region of capaci-tance. The resonator is usually used below the self-resonant frequency so that in a VCO the varactor can be used to resonate with the coaxial resonator.

In order to use the coaxial resonator as a ‘ideal’ inductor the resonator must be used below the self-resonant fre-quency.

DESIGN EXAMPLE OF A COAXIAL RESONATOR [12,13,14]

The following section describes the design of a coaxial resonator to be used in a varactor controlled oscillator at 900MHz. We need therefore to select a suitable resona-tor that is inductive at 900MHz.

Assume an ‘ideal’ starting inductance of 4nH at 900MHz. The material chosen is a silver-plated ceramic resonator with a relative permittivity of 38.6 from Transtech. It has a tab inductance of 1nH, a W/h ratio of 2.57, a width of 6mm and a characteristic impedance of 9.4Ω.

9.74mm = 9.415.1tan.

26036.0 =

Z

Ztan.

2 = resonator of Length

.900MHz at 15.1 is reactance whose3nH= 1-4 ie inductance

required the from inductance tab the subtract We

60.36mm = 6.38

00E8/3E = c/ = Wavelength

1

o

input1g

68

r

o

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

Ω

π

πλ

εf

long 0.161 = 0.60360.0973 is line coaxial the Therefore

1241MHz = 0973.01.

4800*6036.0 =

MHz 1.4.

=Frequency Resonant Self

415.7 =

0.002461

0.00614.25

0.002460.006.079.1Ln

.6E800240. = Q

=

d1

W14.25

dW.079.1Ln

.ok. = Q

g

g

λ

λl

of

f

⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛

The part resonance could be tested to ensure that it oc-curs at the self-resonant frequency of 1.241GHz.

DIELECTRIC RESONATOR [16] At lower frequencies the length of W/d ratio of a coaxial resonator becomes too big to realise so a dielectric ‘puck’ is used instead. The dielectric resonator is often made from the same material as the coaxial resonators except that they are not plated with a low-loss metal. In addition they are mounted on planer circuits as shown below (figure 35) and are coupled to a transmission line without a direct connection. As with other resonators, standing TE waves will be set up within the resonator,

Page 14: Oscillator Basics

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14 of 26

which will be dependent on the physical dimensions of the cylinder. The diagram of a dielectric resonator is shown below in Figure 17

a

b

Figure 17 Schematic diagram of a dielectric resona-tor showing the key dimensions.

The most common resonant mode in dielectric resona-tors is the TE01δ mode and when the relative dielectric constant is around 40, more than 95% of the stored en-ergy are located within the resonator. For an approxi-mate estimation of the resonant frequency in TE01δ mode of an isolated dielectric resonator, the following simple formula can be used:

⎟⎠⎞

⎜⎝⎛ += 45.3La.

.a34 F

(mm)GHz

The above equation is accurate to about 2% in the range 0.5 < a/L < 2 and 30 < εr < 50 The approximate Q factor of the resonator is directly related to the dielectric loss ie tanδ.

( )ro εεωσδ

δ .. = tan

tan1 Q unloaded =

DESIGN EXAMPLE OF A DIELECTRIC RESONATOR

The following section describes the design of a dielec-tric resonator for a frequency of ~ 7GHz. A manufac-turer of dielectric resonators – Transtech can supply two relative permittivities of 30 and 38. The Trans-Tech D8733-0305-137 puck was selected with the following parameters, εr = 30, Diameter = 7.75mm, Height = 3.48mm, the resonant frequency can be estimated using:

7.313GHz = 45.33.479

3.8735.30.8735.3

34

45.3La.

.a34 F

(mm)GHz

⎟⎠⎞

⎜⎝⎛ +

=⎟⎠⎞

⎜⎝⎛ +=

This calculated figure assumes that the resonator is in free-space. If the resonator is mounted on a substrate in a cavity then this will significantly alter the resonant frequency. A more accurate model to take into account cavity and substrate is the Itoh and Rudokas model [7] which, is shown below in Figure 18:

L2

L

L1

a

er6

er1

er2

er4

shield

shield

Region 2

Region 1

Region 4

Region 6

Figure 18 Itoh & Rudokas model of a dielectric resonator inside a metallic shielded cavity

This model can be simplified to the numerical solution of a pair of transcendental equations:

( ) ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−=

00

01

0120146

2o0

GHz)((mm)o

y291.0y

2.43+12.4048

y+2.4048=ak

2.4048 be to taken is xxak y

L height the calculate to entered isfrequency initial An

.a.150 = ak

ρ

εε

π

rr

f

Page 15: Oscillator Basics

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15 of 26

( ) ( )[ ]2221

1111

216

20

220

212

120

211

L.cothtanL.cothtan1 =

L Length Resonator

k.k =

-: is 6 and 4 regions to common constant npropagatio The

.kk

.kk

-: are 2 and 1 regions in constants nattenuatio The

1 ααβ

εβ

εα

εα

β

α

βα

ρ

ρ

ρ

−− +

−=

−=

r

r

r

COUPLING OF RESONATOR TO MICROSTRIP LINE [16]

For analysis of the resonator coupled to a micro-strip line, the transformation shown in the Figure 19 below is used. β (coupling coefficient) is used to provide an equivalent series resistance for the resonator:-

d

R

≡ L

C

Figure 19 Dielectric resonator coupled to a micro-strip line and the corresponding circuit diagram. The resistor L simulates the coupling of the L-C resonant circuit of the dielectric resonator.

Calculation of loaded Q:

( )

ββ

β

π

= 1QQ

+1QQ

*Zo*2 = R

21 = LC

L

UUL

2

−⎟⎟⎠

⎞⎜⎜⎝

⎛=

f

With the above equations it is possible to design VCO for a given Q for example if we want a minimum Q of 1000:

Ω

−−⎟⎟⎠

⎞⎜⎜⎝

⎛=

4K = 4*50*2 *Zo*2 = R

of resistor series a withresonator the replace can weCAD a on analysing For

4 = 110005000 = 1

QQ

+1Q

Q

5000 of Q unloaded a withResonator a use weIf

L

UUL

β

ββ

Trans-Tech have a CAD package [15] to calculate vari-ous design parameters using their dielectric resonators. We can use the CAD package to calculate a plot of the coupling coefficient β vs distance from the centre of the micro-strip line to the centre of the DRO puck. The plot of the analysis is shown below in Figure 20.

5

10

15

20

25

30

35

40

45

5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5

Coupling Coefficient

|B|

D (mm) Center to Center

Figure 20 Plot of coupling coefficient (β) with dis-tance from the centre of the puck to the centre of the microstrip line in mm

Page 16: Oscillator Basics

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16 of 26

Therefore, in our example, the puck would be placed at a distance of 7.15mm from the puck centre to the micro-strip line centre.

TRANSMISSION LINE RESONATOR [17] Over a narrow bandwidth L-C lumped components can be realised using short-circuit and open-circuit transmis-sion lines. If we analyse a transmission line terminated in a load ZL we can define the transformed impedance in terms of the characteristic line impedance and the elec-trical length of the transmission line. The diagram below (Figure 21) shows a transmission line loaded with ZL.

ZLT.L ZoZ(in) →

l l=0 Figure 21 Transmission line loaded with load ZL

[ ][ ]

( ) ( )( ) ( )

ljeelee

eeZoeeZleeZoeeZlZoinZ

eZoZleZoZl

eZoZleZoZlZoIVinZ

ZoZZoZ

VV

evevZo

evevIVinZ

ljlj

ljlj

ljljljlj

ljljljlj

ljlj

ljlj

L

L

ljlj

ljlj

.sin2)(.cos2)(

)()()()(.)(

..

...)(

12

21121)(

..

..

....

....

..

..

..

..

β

βββ

ββ

ββββ

ββββ

ββ

ββ

ββ

ββ

=−

=+

⎥⎦

⎤⎢⎣

⎡++−−++

=

⎥⎥⎦

⎢⎢⎣

−−+

−++==∴

+−

==

+==

−−

−−

−+

−+

−+

−+

⎥⎦

⎤⎢⎣

⎡+−+−++

=∴

⎥⎦

⎤⎢⎣

⎡++

=∴

−−

−−

ljljljlj

ljljljlj

eZoeZleZoeZleZoeZleZoeZlZoinZ

lZoljZlljZolZlZoinZ

....

....

....

.....)(

.cos2..sin2.

.sin2..cos2..)(

ββββ

ββββ

ββββ

⎥⎦

⎤⎢⎣

⎡++

=

⎥⎥⎥⎥

⎢⎢⎢⎢

++

⎥⎥⎥⎥

⎢⎢⎢⎢

+

+

lZlZolZoZlZoinZ

lljZl

lljZo

ZoZlZo

llZo

lljZo

lljZl

llZl

Zo

.tan.

.tan..)(

.cos.sin.

.cos.sin.

.

.cos2.cos2.

.cos2.sin2.

.cos2.sin2.

.cos2.cos2.

.

.l2cosby through divide

ββ

ββ

ββ

ββ

ββ

ββ

ββ

β

This equation is the general expression for the imped-ance looking into a load ZL via a length of transmission line. If we now have the case where the transmission line is terminated with a short circuit we find the general expression simplifies ie let ZL = 0 then

Z in Zo Zl Zo lZo Zl l

( ) . . tan .. tan .

tan .

=++

⎣⎢

⎦⎥

ββ

β = jZ ( Short circuit)o l

We can now plot the impedance (Figure 22) of the shorted length of transmission line vs electrical length and we get the following graph, which shows how the transmission line equates to lumped capacitance and inductance with resonance’s in between. In general Z(in) = R(in) + jX(in) For S/CCT R(in) = 0 ; X(in) = Zotanβ.L Zotanβ.L is purely reactive varies between - ∞ & + ∞ as L varies

Page 17: Oscillator Basics

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17 of 26

l=0π3π/2 π/22π

4fo 3fo 2fo fo 0

θ = β.L

← f

0λg/4λg/23λg/4λg← λg

3 3 34 4

2 2

1 1

X = Z

= 2

= . = .v

= v

o

g

tan .β

β πλ

ϑ β ω

ω

l

ll

l⎛⎝⎜

⎞⎠⎟

Figure 22 Plot of impedance against length of a short circuited transmission line. The plot shows how the reactance of the transmission line varies between inductive and capacitive reactances with resonant frequency regions in between.

Each region of figure 40 is now described: (1) If θ between 0 & π/2 tanβ.L is positive ∴X is +ve ⇒ j(ω.L) - INDUCTIVE. (2) If π/2 < θ < π tanβ.L is -ve ∴ X is -ve ⇒ j(-1/ω.C) - CAPACITIVE. (3) If θ ≈ 0, π , 2π | X | goes to a minimum ie:- | X |

θ

≅L .C

(4) If θ ≅ π/2 , 3π/2 | X | goes to a maximum:- | X |

θ

≅L //C

Similarly, for a transmission line terminated by an open circuit we can repeat the analysis, but we dividing through by ZL. Note Zo/ZL tends to zero ie:-

ZLT.L Zo

ZL = ∞

V=Maxat O/cct

Z(in) →

[ ] circuit) Open ( .tan

1jZ =

.tan.

.tan.

.

ie Zby bottom & top divide.tan..tan..)(

o

L

⎟⎟⎠

⎞⎜⎜⎝

⎥⎥⎥

⎢⎢⎢

+

+

⎥⎦

⎤⎢⎣

⎡++

=

β

β

ββ

ZllZl

ZlZo

ZllZo

ZlZl

Zo

lZlZolZoZlZoinZ

Again we can plot the impedance against electrical length of the transmission line (Figure 23) to see the equivalent lumped reactance and resonance points. In general Z(in) = R(in) + jX(in) For O/CCT R(in) = ∞ ; X(in) = Zocotβ.L Zocotβ.L is purely reactive varies between - ∞ & + ∞ as L

l=0π3π/2 π/22π

4fo 3fo 2fo fo 0

θ = β.L

← f

0λg/4λg/23λg/4λg← λg

3 34 4

2

1

2

1

X = Z

= 2

= . = .v

= v

o

g

cot .β

β πλ

ϑ β ω

ω

l

ll

l⎛⎝⎜

⎞⎠⎟

4

Figure 23 Plot of impedance against length of a open circuited transmission line. The plot shows how the reactance of the transmission line varies between in-ductive and capacitive reactance’s with resonant fre-quency regions in between.

Page 18: Oscillator Basics

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18 of 26

The previous graphs show that we can realise lumped components from transmission lines eg

DESIGN EXAMPLE OF INDUCTOR USING A TRANSMISSION LINE

The following section describes the process of designing a transmission line to have a specific inductance of 0.7nH at a frequency of 8.8GHz. The transmission line is to be etched on RT duroid substrate material, which has a relative permittivity of 2.94 and a substrate thick-ness of 0.25mm.

ll

for Solve 2 = where.tan

1j.Zo- = Zin

0.466pF = C C =

8.8GHz at 0.7nH of inductance of 38.8 = Reactance

g

f21 2

λπβ

β

π

⎟⎟⎠

⎞⎜⎜⎝

Ω

⎟⎠

⎞⎜⎝

L

Using the transmission line equation for an open-circuit stub we can calculate the electrical length required for an inductance of 0.7nH. Therefore a open-circuit stub of length 3.1mm will have an inductance of 0.7nH at 8.8GHz. As the equations show the resulting impedance is a func-tion of the characteristic of the line and generally we use a narrow high impedance line ~ 100Ω for an inductive impedance and a wide length of line ~ 20Ω, for a capaci-tive impedance. For completeness the empirical equa-tions for calculating line widths are given in the next section:-

3.1mm = 293

38.950arctan

= XZarctan

=

293 = 0214.02 =

21.4mm or 0.0214m = 53.2

3E8/8.8E9 =

therefore 2.94 is used be to material the ofty permittivi Relative

o

eg

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=

β

πβ

ελλ

l

ff

air

CALCULATION OF EFFECTIVE RELATIVE PERMITTIVITY [18]

The following section describes the empirical equations that are used to calculate the dimensions of the micro-

strip lines and characteristic impedance [8]. The first equation describes the effective relative permittivity which, differs from the specified value due the width of the micro-strip track.

( )( )

( ) )1.18/(17.18

1432.0/

)52/(/491+1 = a and

39.00.564 = b where

.1012

12

1 =

34

24

053.0

r

r

.rr

hWLnhW

hWhWLn

wh ba

eff

+⎟⎠⎞

⎜⎝⎛+⎥

⎤⎢⎣

++

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

⎟⎠⎞

⎜⎝⎛ +

−+

+ −

εε

εεε

Calculation of W/h (width of micro-strip/substrate thickness) for a given characteristic impedance and effective relative permitivity:

ro

rr

r

ro

2Z377 = B where

0.517-0.293+1)-Ln(B2

1+1)-Ln(2B-1-B2

hW

2 - 44 Z For

επ

εεε

π

ε

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+−

++

−=

rr

rr

2

ro

12.0226.0.11

60.

21 = n where

28

hW

2 - 44 Z For

εεεε

ε

Zo

ee

n

n

INTER-DIGITAL MICRO-STRIP CAPACITORS [19]

Normally resonators need to be lightly coupled in order to maintain a high Q, this can be done by using a filter arrangement or by using very small value capacitors. Normal chip capacitors can go as low as 0.1pF, but for smaller capacitance it is convenient to use transmission line inter-digital capacitors. Literature on the subject is very scarce so a basic design formula was used to get the initial dimensions and the final dimensions were optimised during RF simulations.

Page 19: Oscillator Basics

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19 of 26

The basic formula for the inter-digital capacitor is given by:-

fingers long 600um =

cm06.01)-0.83(2

0.05 = L)1(N*0.83

C

-:be willfingers the of length the thenfingers 2 are there that assume weif

and capacitor 0.05pF a want weif example For10um of widthfinger a and

5um of spacing finger a assumes formula This

pF in eCapacitanc C cm in fingers of Length = L fingers of Number = N Where

L).1(N 0.83 = C

F

F

F

==−

=

To further aid in the evaluation of a inter-digital capaci-tor the model was analysed in Libra RF CAD with a finger width and gaps of 0.1mm and number of fingers 2,3 & 4. The graph (Figure 24) shows the relationship between capacitance and finger length.

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3Finger Length mm

Cap

acita

nce

pF

Figure 24 Graph of a micro-strip inter-digital ca-pacitor vs capacitance. The plots were calculated by analysis on HP/Eesof libra.

Transmission lines may be used as single resonators capacitively coupled to the active device, but also they may be configured as a micro-strip band-pass filter. The basic principle involves using open circuit transmission lines of electrical length 180 degrees, which is equiva-

lent to a ‘tuned circuit’ parallel resonator. What tends to differ in the topographies are the ways in which the resonators are coupled together. The resonators can be end coupled or parallel coupled using the gaps between them as the low value coupling capacitors. It is also pos-sible to use inter-digital capacitors to generate coupling apacitors less than 1pF

aries inversely with the ap-

ion for calculating the capacitance of e varactor is :-

c

VARACTORS [21] Voltage variable capacitors or tuning diodes are best described as diode capacitors employing the junction capacitance of a reverse biased PN junction. The ca-pacitance of these devices vplied reverse bias voltage. The general equatth

exponent eCapacitanc = and 0.7V)(~ potential contact junction=

voltage, applied =V e;capacitanc diodeC where

)(

D

γφ

φ γ

=

+=

VCC D

J

selected is a Macom Tuning diode

sheet gives the following parameters for the iode:-

0 = .5;Gamma=0.75;Q @ 50MHz=4500

DESIGN EXAMPLE OF A VARACTOR DIODE The following section describes how information from a data sheet can be used to predict the capacitance of the varactor diode for a given reverse bias. For this example the varactor diodetype MA46H071. The datad C = 0.9-1.1pF @ 4V;cap ratio Cto/Ct25

75.0

12

0.7512-

JD

)7.0(E19.3 =

bias given a for ecapacitanc a calculate to therefore,

3.19pF 0.7)+(41E =

).(C = C give to rearrange )(

+=

=

++

=

VC

VV

CC

J

DJ

γγ φ

φ

This is obviously the ideal case as it does not take into account the case parasitics

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20 of 26

TUNING RATIOS

The tuning or capacitance ratio, TR, denotes the ratio of capacitance obtained with t o values of applied bias voltage. This ratio is given by the following:-

w

γ

φφ⎥⎤

⎢⎡

+12J

V+V =

)V(C)V(C = TR

⎦⎣ 21J

e

ng diode capacitors falls off at high frequencies be-

equencies be-ause of the back resistance of the reverse-biased diode.

The equivalent circuit of a tuning diode is often shown in the form given below in Figure 25.

where CJ(V1) = junction capacitance at V1;CJ(V2) = junction capacitance at V2 (V1>V2).

CIRCUIT Q The Q of the varactor can be very important, because thevaractor usually directly forms the tuned circuit and thoverall Q is dominated by the worst Q factor. The Q of tunicause of the series bulk resistance of the silicon used in the diode. The Q also falls off at low frc

Rp

Cj

Rs Ls Ls’

Cc

Figure 25 Equivalent circuit of a typical varctor diode together with case and lead parasitic components.

Where Rp = Parallel resistance /back resistance of the

Rs = Bulk resistance of the silicon in the diode.

= Case Capacitance.

ormally the lead inductance and case capacitance can be ignored, which results in a simplified circuit shown in Figure 26.

diode. Ls’ = External lead inductance. Ls = Internal lead inductance. Cc N

Rp

Rs

Cj

Figure 26 Simplified model of a typical varactor di-ode with parasitic reactance removed.

The resulting Q for the above circuit is given by :-

ΩΩ

=

9

22

2

30x10 = & 1 = Rs Typically

pC)Rs

C.Rpf

t

Rp

Rs.R2(+2 fπ

Rp+

Therefore for a MA/COM MA46H071 we would expecthe following Q’s at different frequencies as shown in the table below:

f(GHz) Q 0.05 3500 2 88 6 30

The degradation of Q at microwave frequencies means that the varactor, has to be lightly coupled, or Q trans-formed in order not to load the resonant circuit, lowering the loaded Q with the resultant degradation in phase noise performance. The following graph (Figure 27) of the varactor diode frequency response shows that at low frequencies the Q is dominated by the parallel term ie Qp = 2πf.Rp.C and at high frequencies by the series term Qs = 1/(2πfRs.C).

0.1

11 100 10000 1000000 100000000 1E+

10

100

1000

10000

100000

10

Frequency (Hz)

Page 21: Oscillator Basics

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21 of 26

Figure 27 Plot of Q against frequency. The vertical

echanisms for the variation of capacitance over temperature are (i) contact potential and (ii) case capacitance. The contact potential will vary at -2.2mV/°C thus for the MACom diode we would expect the following tempera-ture drifts as shown in Table 2.

scale is Q and the horizontal scale is frequency in Hz.

TEMPERATURE VARIATION The two m

V Cj Cj+1 decC Diff ppm/degC1 2.1426636 2.1405863 0.0020773 2077.2922 1.5144951 1.5135702 0.0009249 924.865384 0.9993492 0.9989986 0.0003507 350.691746 0.7660103 0.7658217 0.0001886 188.590138 0.6297254 0.629606 0.0001194 119.40427

10 0.5392037 0.5391205 8.313E-05 83.13327512 0.4741742 0.4741126 6.16E-05 61.59596914 0.4249156 0.4248679 4.769E-05 47.68836416 0.3861476 0.3861094 3.815E-05 38.1479118 3170.3547394 0.3547081 3.13E-05 31.29720 0.32871 0.3286838 2.62E-05 26.199086

Table 2 Calculated data of the capacitance varia-tion with temperature for the MACom varactor di-ode.

TEMPERATURE COMPENSATION popular method of temperature comA pensation involves

the use of a forward bias diode. The voltage drop of a forward biased diode decreases as the temperature rises, therefore applying a changing voltage to the tuning di-ode. For the circuit to be effective the compensating diode must be thermally coupled to the varactor to be corrected. Figure 28 shows a method for temperature compensating a varactor diode.

R

Compensatin

Vi

Varactor

Figure 28 Schematic circuit diagram, for tempera-

Normally, however t of a feedback

uning di-

e eed to estimate the loaded Q of a resonator,

ith a varactor connected, in order to calculate the phase noise performance of the oscillator. It is useful to be able to simplify the e ivalent Q of a circuit, so the effect of the varactor Q can be evaluated. Some basic definitions of Q in the series and parallel form are:

g

n

ture compensation, of a varactor diode

the varactor is parloop, which controls the frequency of oscillation eg in aPLL system. In this case, the temperature effects are generally accounted for in the loop so that external com-pensation is not required.

LOADED & UNLOADED Q [22,23,24] UNLOADED Q

The earlier section described how the Q of a tode varies over frequency and can be quite low (~ 30) atmicrowave frequencies. This will obviously have an effect on the loaded Q of a circuit where the individual components may have higher Q’s in the hundred’s. Wtherefore nw

qu

LR = Q External

LR = R.C =

Q circuit Parallel Unloaded

L

L

ooo

o

ωωω

We can take the specified Q values for inductors and capacitors from the data sheets and calculate the equlent series or parallel resistance that di

RL = Q External

R.C1 =

RL =

Q circuit Series Unloaded

oo ωω

ω

iva-stinguish the

omponent from an ‘ideal’ component to one with a finite Q. Once the resistance has been calculated, the circuit can be simplified down to a single component or a series/parallel combination of two circuits, to allow calculation of the unloaded circuit Q. The following example (shown in Figure 29) shows a simple L-C tuned circuit but with losses added.

c

L ~ 2.5uH

=100 @ 100MHzQ

C = 1pF

Q = 200 @ 100MHz

RIND=163KΩ

C Ω R =318K

Figure 29 Simple L-C circuit with component losses added

The equivalent parallel loss resistance for each compo-nent was calculated as follows-

Page 22: Oscillator Basics

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22 of 26

67 E108*E1*E100*2 L.

R Q

and R.C. circuit of Q Unloaded

K108 K318K163

318K*163K

//RR resistance equivalent Parallel

318K E1*E100*2

200R and K163

E5.2*E100*100*2R

C.QR and L..R

3126

o

o

CPLP

126CP

66LP

oCPoPL

===

=∴

=+

=

=

Ω==Ω=

=

==

πω

ω

π

π

ωωQ

A useful transformation from series equivalent resistive loss (Rs) to parallel equivalent resistive loss (Rp) is given as –

Xp Xs and Rs*)(Q Rp

10 Q For

Rs*)1(Q Rp

10 Q For

2

2

≈≈

>

+=

<

These transformations are only valid at one frequency, as they involve the component reactance, which is fre-quency dependant.

LOADED Q The loaded Q of a resonant circuit is dependent on three main factors:

(1) The source impedance (Rs). (2) The load impedance (RL). (3) The component Q.

The circuit used in the example of section 3.5.1 is to be loaded in a 50-ohm system as shown in Figure 30.

L ~ 2.5uH

Q =100 @ 100MHz

C = 1pF

Q = 200 @ 100MHz

RRES=108KΩ

RL=50Ω

Rs = 50Ω

Figure 30 Simple L-C resonant circuit loaded, with 50-ohm source and load impedances.

The addition of the source and load impedances will degrade the loaded Q of the circuit as they will effec-tively be in parallel with the high impedance resonant circuit as shown below in Figure 31.

L ~ 2.5uH

Q =100 @ 100MHz

C = 1pF

Q = 200 @ 100MHz

RRES=108KΩ

RL=50Ω

Rs = 50Ω

=

Requ = 24.99Ω

Figure 31 L-C resonant circuit reduced to one resis-tive loss component.

The loaded Q of the circuit of Figure 31 is:-

0.0159 2.5E*100E*2

24.99 Rp Q 6-6o

===πω L

This dramatic decrease in Q will give the simple L-C network a 3dB bandwidth of:

!! GHz6 0.0159

100MHz f ff

o

==∆∴∆

=Q

To improve the loaded Q, given a restraining source and load impedance, we could alter the value of Xp. This however, results in either very high inductors, or very low capacitors. If we are restrained from altering the value of Xp we can either use a tapped L or C transformer or coupling L or C.

Page 23: Oscillator Basics

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23 of 26

Q TRANSFORMATION

The circuits shown in Figure 32 show the two methods of transforming the Q of a circuit, by the use of impedance transformers.

Rs RL

2

C2C11Rs Rs' ⎟

⎠⎞

⎜⎝⎛ +=

Tapped C circuit

Rs RL

2

n1nRs Rs' ⎟⎠⎞

⎜⎝⎛=

Tapped L circuit

n1 n

Figure 32 Impedance transformation circuits (Tapped L & C). These circuits can be used to increase the ef-fective source & or load impedances in order to im-prove the loaded Q of a circuit.

If we require a Q of 10 then this will equate to a parallel equivalent resistance of:

18pFC2 and 1.055pFC1 have could We

pF1C2C1C2*C1 and C2*18 C1 Therfore

18 1-50

18K C2C1 1-

RsRs'

C2C11Rs Rs'

r,transforme tapped capacitor a using 18K to impedance source our transform to need weTherefore

18.37K Rs for solve 081Rs

108K*Rs 15707

108K RL nscalculatio previous From RLRsRL*Rs 15707

15.7K E6.2*100E*2*10 Q.Xp Rp

2

66

==

=+

=

=⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛∴⎟

⎠⎞

⎜⎝⎛ +=

Ω

Ω=Ω+Ω

=

Ω=+

=

Ω=== −

K

π

The final circuit designed to give a Q of 10 is shown in Figure 33.

Rs = 50Ω

R L = 108KΩ

1pF ~ 1.055pFpF181.055pF*18pF C1//C2

16K3 1.055

18150 Rs'

C2C11Rs Rs'

2

2

+=

Ω=⎟⎠⎞

⎜⎝⎛ +=

⎟⎠⎞

⎜⎝⎛ +=

L = 2.5uH

C2=18pF

C1= 1.055pF

Figure 33 L-C circuit with a capacitor tapped im-pedance transformer, to give a loaded Q of 10, when loaded with a source impedance of 50 ohms.

Equally we could use a coupling capacitor between the source impedance and resonant circuit such that the re-sistance will equal 16KΩ.

Page 24: Oscillator Basics

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24 of 26

0.1pF E16*100E*2

1 C

16K~ 50- 16K

100MHz at reactance capacitor coupling Required

36coupling ==∴

ΩΩΩ=

π

The addition of a coupling capacitor to the circuit is shown in Figure 34.

Rs = 50Ω

RL = 108KΩ

L = 2.5uH

C=1pF

Cc=0.1pF

Figure 34 Addition of a coupling capacitor to the simple L-C to increase the loaded Q to ~10

The required coupling capacitor is very small at 0.1pF and is probably impracticable at 100MHz. However this size of capacitor can be realised at microwave frequen-cies by the use of a microstrip gap or a inter-digital ca-pacitor (as described in section 0).

INSERTION LOSS OF RESONATOR The insertion loss of a resonator is important in oscilla-tor design as there needs to be enough loop gain to allow oscillation. A high insertion loss resonator may require two stages of amplification around the loop that will add to the size, power consumption and complexity of the oscillator. The insertion loss of the resonator is a func-tion of loaded and unloaded Q ie:-

Q unloaded Q and Q loaded Q where

QQ

1 20log- (dB) loss Insertion

UL

U

L

==

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

DESIGN EXAMPLE FOR A VARACTOR CONTROLLED RESONATOR

Consider the varactor resonator shown below in Figure 35. The capacitor combination can be simplified to a single capacitor that then forms a parallel resonant cir-

cuit with the inductor. In this example, we assume a source impedance of 50ohms.

Cdiode ~ 1pF Q =30 @ 2GHz

L ~ 7.6nH

Q =150 @ 1GHz C ~5pF

Q = 100 @ 5GHzRIND=7163Ω

RC=0.06Ω

Rcdiode= 2.65Ω

Figure 35 Schematic circuit diagram of a varactor controlled resonator for use at 2GHz. The equiva-lent loss resistances have been calculated using the equations of section 3.5.1

This circuit of Figure 35 can be simplified to that shown in Figure 36. The loss resistances of the capacitor arm can be added and converted to a parallel loss resistance that can be added to the loss of the inductor. The equivalent capacitor now equals 0.833pF ie 1pF // 5pF.

L ~ 7.6nH

Q =150 @ 1GHz

RIND=7163Ω

Rcdiode= 2.71Ω(series)

3343Ω (parallel)

Q of capcitor+diode

~35

Figure 36 Simplified varactor controlled resonator for use at 2GHz

( )

( ) Ω===

=+

==−

331971.2*35 *Q R

35 0.062.65

E833.0*2E*.21

(series) RsXs Q

-:loss parallel tolosscapacitor series of Conversion

22P

129

sR

π

Page 25: Oscillator Basics

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25 of 26

Now we can calculate the equivalent loss resistance and the unloaded Q of the circuit:

23.7 E6.7*2E*2

2268

XpRp circuit the of Q Unloaded

2268 3319 // 7163 is circuit resonant the across resistance loss Equivalent

99 ==

=

Ω=ΩΩ

−π

We can see that the low Q of the inductor is going to dominate the unloaded Q of the parallel circuit. Now, if we load the circuit with 50-ohm source and load imped-ances, (as shown in Figure 37) we can calculate the loaded Q of the circuit.

Cdiode ~ 0.833pF

L ~ 7.6nH

RRES=2268Ω RL = 50Ω RS = 50Ω

Figure 37 Resonant varactor circuit loaded with 50ohm source and load impedances.

The loaded Q of the circuit will be the parallel combina-tion of the equivalent parallel resistance of the resonant circuit with the source and load impedances ie-

0.26 E6.7*2E*2

24.73 XpRp of Q loaded a give willThis

24.73 Rp 2268

1501

501

Rp1

9-9 ==

Ω=∴++=

π The circuit was analysed on the CAD to confirm the Q calculations and is shown in Figure 38.

0.2 2.2 4.2 6.2 8Frequency (GHz)

Graph 1

-10

-8

-6

-4

-2

0 DB(|S[2,1]|) *Varactor

Figure 38 Varactor resonator circuit loaded, with 50-ohm source and load impedances. The Q was graphi-cally measured at ~ 0.28.

The loaded Q is lower than the unloaded Q due to the damping effect of the low value source impedance. An oscillator with a resonant circuit with a Q of 0.24 will be very unsatisfactory, so a means of increasing the loaded Q is required. We cannot do much about the tuned circuit, but we can modify the source and load impedances either by the used of a C/L tapped trans-former or by the use of coupling capacitors. For this example we shall consider the use of coupling capacitors on the varactor circuit. Figure 39 shows the implemen-tation of coupling capacitors.

Cdiode ~ 0.833pF

L ~ 7.6nH

RRES=2268Ω

RL = 50Ω

RS = 50Ω Coupling

C Coupling

C

Figure 39 Varactor tuned circuit, with coupling ca-

we decide that we require a loaded Q of say 10, then

pacitors, added between 50- ohm source and load impedances.

Ifwe can calculate the value of the source resistors, that when placed in parallel with the tuned circuit, will give the required value of Q ie

Page 26: Oscillator Basics

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26 of 26

0.4pF (198) * 2E * 2

1 = capacitor series of Value

198 Rp 2268

111Rp1

95.5=10*7.6E*2E*2 = Rp .

10 of Q a give to resistance parallel Total

9

9-9

=

Ω=∴++=

Ω∴=

=

π

π

RLRs

RpQLX

This value of series coupling capacitor is very small but can be realised at microwave frequencies by the use of a inter-digital microstrip capacitor. The coupling capaci-tors were added to the CAD model and analysed to con-firm a Q of ~ 10, the plot is shown in Figure 40. Predicted insertion loss:

2.85dB- 2310-110log

QQ-110log (dB) loss

U

L =⎟⎠⎞

⎜⎝⎛=⎟⎟

⎞⎜⎜⎝

⎛=

1.5 2 2.5Frequency (GHz)

Loaded Q

-10

-8

-6

-4

-2

0DB(|S[2,1]|) *Varactor

Figure 40 Varactor resonator circuit loaded, with 50-ohm source, load impedances and coupling ca-pacitors. The Q was graphically measured at ~ 10,


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