Microwave Oscillators Design

Microwave Oscillators Design

Oscillators Classification Feedback Oscillators β Α Oscillation Condition: Gloop = A β(jω 0 ) = 1 Gloop(jω 0 ) = 1, Gloop(jω 0 )=2nπ Negative resistance oscillators Most used at microwave frequencies Passive Network Z p Z A Active Network ( ω), ( ω) Z = R + j X Z = R + j X A A A p p p Oscillation Condition : ( ω ) ( ω ) R = R, X + X = 0 A p A 0 p 0

Characteristic Parameters Generated power at output Shape of the signal generated Amplitude/Phase stability Sensitivity of the oscillation frequency to the circuit components and to the bias sources Dependence of amplitude and frequency on the external loads (load pulling) Initial start-up

Main parameters definition Stability of the oscillation frequency: it is referred as Indirect Stability Coefficient S F : dφ d X A( ω) + X p ( ω) Gloop SFX, = ω SF, Φ = ωo o dω dω ω= ω 0 A change Φ of the Gloop phase (or X of overall reactance) produces a variation of the oscillation radian frequency given by: ω = ω0 φ S F, Φ Harmonic distortion: it is an index of how close is the produced waveshape to the ideal sinusoid. It defines numerically the amplitude of the harmonic referred to the fundamental (ω o ). Phase and Amplitude Noise: it defines the random fluctuations of amplitude and phase of the generated signal. While the noise amplitude can be removed (or limited) with a suitable circuitry, the phase noise can not be removed from the output signal and is the most important drawback introduced by the oscillators in communication systems ω= ω 0

Graphical representation of indirect stability Gloop 2π S F,Φ φ 0 0 φ ω = ω 1+ S F, Φ ω 0 ω0 φ = ω S 0 F, Φ The larger is S F, the lower is the frequency shift (ω 0 -ω 0 ) for a phase drift φ ω 0 ω 0 f

General configuration of a microwave oscillator Z A, Γ A Z B, Γ B Network A Network B R 0 Z in, Γ in Active Device Z out, Γ out In order to produce a sinusoidal signal, it is necessary that a negative resistance is observed at input and output of the active device (that is an input/output reflection coefficient with magnitude larger that one). As a consequence, the active device must be potentially instable (K<1), so, given Γ A and Γ B, we must have (at the oscillation frequency): Γ > 1, Γ > 1 in out

Increase the instability of active devices To increment the instability we must decrease the K parameter of the device. This can be obtained by changing the reference terminal or by introducing positive feedback: L C a

Oscillation Conditions (regime) Γ out Γ B Γ A Γ in Z A Z B A B Γin ( jω0) Γ A ( jω0) = 1 ( Z Z ) ( Y Y ) + = 0 or + = 0 in A in A Γ ( jω ) Γ ( jω ) = ( Zout ZB ) ( Yout YB ) out 0 B 0 1 + = 0 or + = 0 These condition apply in steady-state, i.e. when the output voltage amplitude is stabilized; it can be shown that in this condition only one of the equations above is required (the other is implied).

Oscillation Conditions (start-up) To guarantee the start of the oscillation the poles must be in the right-hand plane, then: Γ ( jω ) Γ ( jω ) > ( Z Z ) ( Y Y ) in 0 A 0 1 Re + < 0 or Re + < 0 in A in A Γ ( jω ) Γ ( jω ) > ( Z Z ) ( Y Y ) out 0 B 0 1 Re + < 0 or Re + < 0 out B out B In this case both conditions must be verified in order to guarantee the oscillation start-up. After the start-up the oscillation grows and the poles must move towards the left-hand plane (it is produced by the non-linearity of the active device). Once the poles reach the imaginary axis the oscillation remains with constant amplitude and the transient is concluded.

Frequency of oscillation The oscillation frequency is determined by the regime condition. Combining with the start-up requirement: ( jω0) ( jω0 ) 1, ( jω0) ( jω0) Γ ( jω ) Γ ( jω ) = 0 Γ Γ > Γ Γ > 1 in A out B ( in 0 A 0 ) To increase the oscillation stability, only the phase of Γ A should determine ω 0. This is obtained by imposing the slope of Γ A (ω 0 ) much large than that of Γ in (ω 0 ) (by using resonating components for which Γ A 1). Note that once ω 0 is imposed at section A, it is also verified (at regime) at section B. If the impedances are considered the equations become: ( ω ) ( ω ) ( ω0) B ( ω0) X ( jω ) + X ( jω ) = 0 R j 0 + R j 0 < 0, R j + R j < 0 in A out in 0 A 0

Mapping Circles Smith Chart is useful also for oscillators design. Specifically, we can draw on the Γ s plane the locus of points where the value of Γ out has been assigned. This locus is a circle with the following center and radius: Γ S Γ out C s * 2 * 22 Γout s11 2 2 2 s = Γ out s 11 Γ s s S 12 21 Γ out = s22 + = ( 1 ΓS s11) cost r s s s Γ = Γ 12 21 out 2 2 2 out s 11

General design procedure (1) The design goal is to get Γ A e Γ B allowing the start of oscillation. The active device is characterized by the scattering parameters S at the oscillation frequency f 0. Usually Γ A =1 is imposed (in order to have the network A made of reactive components). Γ A (phase) and Γ B (magnitude and phase) are therefore the unknowns of the design Evaluation of Γ A Γ out > 1 is assigned (compatibly with the used active device). The relative mapping circle is drawn on the Smith Chart Mapping Circle Γ out >1 Plane of Γ s

General design procedure (2) The intersections of the mapping circle with the one delimiting the Smith Chart ( Γ s =1) are identified; if there are not intersections, a different value of Γ out must be assigned and another mapping circle drawn on the chart Intersections Plane of Γ S One of the intersection points is selected: Γ s =Γ A,opt ; network A is then synthesized as a reactive 1-port network presenting Γ A,opt at input

General design procedure (3) s21s12 Γ A, opt From Γ A,opt the ouput Γ is computed: Γ out = s22 + 1 s11 Γ A, opt Z out (o Y out ) is then evaluated from Γ out ; this impedance (admittance) has a negative real part Γ A,opt Γ out Γ B Rout (<0) Z out Z B R B (>0) X out X B Z B,opt (o Y B,opt ) is computed by imposing the following condition (start of oscillation): Z B,opt = R out /3-jX out or Y B,opt = G out /3-jB out This produces an overall impedance (admittance) real and negative

General design procedure (4) Z in and Y in are computed from Z B,opt (o Y B,opt ); one of the conditions (R in +R A,opt )<0 or (G in +G A,opt )<0 must be confirmed; otherwise a different value of Z B,opt (or Γ out ) must be selected and the design procedure repeated again Ζ A,opt Ζ in Ζ B,opt Network B Note: (R in +R A,opt )<0 is equivalent to: Γ in. Γ A,opt >1 Γ B,opt is finally evaluated from Z B,opt (o Y B,opt ) and the network B is synthesized by imposing the impedance transformation from the load

Notes on the design choices Besides the start of oscillations, the choice of (Γ A,opt, Γ B,opt ) must take into account all the other figure of merit of the oscillator (delivered power to load, harmonic distortion, load pulling...). Here are some general guidelines for the design: The loaded Q of the output network must be at least 10 (to suppress the harmonics) The reactance (or the susceptance) of one of the networks (A, B) should have a large derivative respect to the frequency for increasing the stability To limit the harmonic distortion, the active device should be biased in class A To increase the output power and efficiency the active device should be biased in class B or C

Design example: Oscillator at 6 GHz Biased Active device: MESFET GaAs DCVS ID=V2 V=Vds V Vds=3 Vgs=-0.3 S Parameters at 6 GHz: S 11 = 0.88458-73.509 S 21 = 3.1203 119.76 S 12 = 0.1152 48.699 S 22 = 0.56291-46.665 DCVS ID=V1 V=Vgs V 2 3 DC DC RF 1 RF & 2 PORT P=2 Z=50 Ohm PORT P=1 Z=50 Ohm 2 RF 3 DC BIASTEE ID=X1 DC RF & 1 1 3 BIASTEE ID=X2 SUBCKT ID=S1 NET="NE76000_CEL" Stability Coeffcient: 0.227

Evaluation of Γ A,opt and Γ B,opt Γ out =1.3 Γ A,opt We assign Γ out =1.3. From the intersections of the mapping circle with Γ s =1 we select Γ A,opt =1 62.28. Piano dei Γ S We compute Γ out = 1.318-164.16 From Γ out we obtain Y out = -0.0733+j0.0716 mho. The start of oscillation is then imposed: Y B,opt =0.0733/3-j0.0716= 0.0244-j0.0716 Γ B,opt =0.851 151.717. Verification of the condition on Γ in : Γ in, is computed from Γ B,opt : Γ in =1.046-62. Being Γ in >1 the condition is met.

Linear simulation Rete A Γ A,opt Γ B,opt Rete B TLIN ID=TL1 Z0=50 Ohm EL=Elin Deg F0=6 GHz ( A, opt ) Elin = 180 Γ 2 = 58.86 SUBCKT ID=S1 NET="NE76000POL" 1 2 Γ out TLIN ID=TL2 Z0=50 Ohm EL=29.9624 Deg F0=6 GHz LOAD ID=Z1 Z 50 Oh Y tot =Y out +Y B,opt

Magnitude of Γ out 4 3.5 3 2.5 2 1.5 1 0.5 GammaOut 6 GHz 1.321 S(1,1) Small Signal Input 0 5.5 5.6 5.7 5.8 5.9 6 6.1 6.2 6.3 6.4 6.5 Frequency (GHz)

Total admittance at output Y tot 0.2 0.15 Ammettenza totale (uscita del FET) Re(Y(1,1)) Oscill_smallsignal Im(Y(1,1)) Oscill_smallsignal 0.1 0.05 6 GHz -0.00038028 0-0.05-0.1 6 GHz -0.049366 5.5 5.6 5.7 5.8 5.9 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7 Frequency (GHz)

Large Signal Analysis in MWO (Harmonic Balance) Connect a source V s (ω p ) to an arbitrary node of the oscillator circuit. Assume the following source impedance: 0 for ω = ω Z ( ω) = for p ω ω If the circuit oscillates and V s coincides with V x (magnitude and phase at ω p ) the current of the source (at ω p ) must be zero. The steady-state solution can be then found following these steps: p Connect the probe source (OSCAPROBE) to a convenient node Numerically search the frequency ω p and the phasor amplitude V s (ω p ) that determine I(ω p ) =0 (analysis is performed with harmonic balance, assuming the fundamental tone ω p with n harmonics)

Oscillation frequency: 6.377 GHz Output Power at fundamental 9.11 dbm Example

Noise in oscillators The existence of noise in electric circuits determines fluctuations of the instantaneous phase of the generated signal. Power Spectrum of phase fluctuations This fluctuation can be assumed as a modulation of the sinusoid produced by noise; the resultant spectral density L(f) depends on the power spectrum of the fluctuations ( Φ 2 ). It is found that L(f) has the same shape of Φ 2 (for small modulation angle)

Phasorial description of noise + oscillation φ(t) V 0 n ( t) V, V V s u u V u n(t) 0 n c (t) n s (t) n(t) is a phasor with random magnitude and phase. n c (t) and n s (t) are the in phase and quadrature components. Half of the overall power density is associated to each of them n () t n () t n () t φ( t) = sin Vu Vu V0 1 s s s The spectrum of the phase fluctuations is proportional to the spectrum of added noise: Sφ ( f ) N( f ) = 2 P o

Typical Noise Spectrum N(f) 10 db/decade Noise 1/f White Noise f c f

Noise generated inside the feedback loop n(t) A + V u (t) β(ω) Noise is not simply summed to the output signal (it is instead injected into the feedback loop). The noise frequency components close to the oscillation frequency are amplified by the positive feedback. Ultimately, a broadening of the ideal spectral line characterizing the ideal oscillator is produced; the width of the spectrum depends on both the noise and the Gloop.

Leeson s Model for the Phase Noise This model relates the power spectrum of the phase fluctuations with the noise spectrum and the indirect stability of the oscillator: S φ f S 0 F 2 N( f ) 2 2 ( f ) N(f ) is the power spectrum of the noise; f is the deviation with respect the oscillation frequency f 0 ; S F is the indirect stability. For what said before, the spectrum of the signal generated by the oscillator around f 0 has the same shape of S φ. The model is accurate for small values of f

S φ ( f) distant from f 0 The Leeson s model holds true for the noise components inside the band of Gloop. This band (B) is the frequency interval within of which the oscillation condition are about satisfied. B is mainly determined by the selectivity of the feedback network (β). We can relate the value of B to the quality factor Q 0 of the feedback network through the well know relationship: B f = 0 Q0 Outside the band B the noise is simply added to the signal and the spectrum results proportional to the noise spectrum (white, density N).

Typical phase noise spectrum S φ 30 db/decade White noise + 1/f S φ f S 0 F 2 N( f ) 2 2 ( f ) S φ N( f ) 20 db/decade White noise White noise f c f 0 /2Q 0 f

Spectrum of oscillation voltage The spectrum of V u (t) around f 0 is proportional to S φ (f ) The spectrum of V u (t) is symmetric if only phase noise is present The"Carrier to noise ratio (CNR) is defined as: CNR( f ) = P0 S f f = P ( + ) S ( f ) V0 0 φ 0 CNR is the main figure of merit of an oscillator concerning the phase noise; it is typically specified (in db) for small deviations with respect f 0 (from few Hz to several MHz). Typical values in microwave amplifiers span from 80 dbc to over 120 dbc (for f = 100 KHz).