10 - RF Oscillators

10 - RF Oscillators

The information in this work has been obtained from sources believed to be reliable. The author does not guarantee the accuracy or completeness of any information presented herein, and shall not be responsible for any errors, omissions or damages as a result of the use of this information.

April 2012

 2006 by Fabian Kung Wai Lee

1

Main References • • • • • • • •

[1]* D.M. Pozar, “Microwave engineering”, 2nd Edition, 1998 John-Wiley & Sons. [2] J. Millman, C. C. Halkias, “Integrated electronics”, 1972, McGraw-Hill. [3] R. Ludwig, P. Bretchko, “RF circuit design - theory and applications”, 2000 Prentice-Hall. [4] B. Razavi, “RF microelectronics”, 1998 Prentice-Hall, TK6560. [5] J. R. Smith,”Modern communication circuits”,1998 McGraw-Hill. [6] P. H. Young, “Electronics communication techniques”, 5th edition, 2004 Prentice-Hall. [7] Gilmore R., Besser L.,”Practical RF circuit design for modern wireless systems”, Vol. 1 & 2, 2003, Artech House. [8] Ogata K., “Modern control engineering”, 4th edition, 2005, Prentice-Hall.

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1

Agenda • • • • • • • •

Positive feedback oscillator concepts. Negative resistance oscillator concepts (typically employed for RF oscillator). Equivalence between positive feedback and negative resistance oscillator theory. Oscillator start-up requirement and transient. Oscillator design - Making an amplifier circuit unstable. Constant |Γ1| circle. Fixed frequency oscillator design. Voltage-controlled oscillator design.

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1.0 Oscillation Concepts

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2

Introduction • • • •

• •

Oscillators are a class of circuits with 1 terminal or port, which produce a periodic electrical output upon power up. Most of us would have encountered oscillator circuits while studying for our basic electronics classes. Oscillators can be classified into two types: (A) Relaxation and (B) Harmonic oscillators. Relaxation oscillators (also called astable multivibrator), is a class of circuits with two unstable states. The circuit switches back-and-forth between these states. The output is generally square waves. Harmonic oscillators are capable of producing near sinusoidal output, and is based on positive feedback approach. Here we will focus on Harmonic Oscillators for RF systems. Harmonic oscillators are used as this class of circuits are capable of producing stable sinusoidal waveform with low phase noise. April 2012


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2.0 Overview of Feedback Oscillators

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3

Classical Positive Feedback Perspective on Oscillator (1) • •

Consider the classical feedback system with non-inverting amplifier, Assuming the feedback network and amplifier do not load each other, we can write the closed-loop transfer function as: Non-inverting amplifier

Si(s)

E(s)

+

So(s)

A(s) + High impedance

Positive Feedback

• •

Feedback network

High impedance

F(s)

So (s ) = 1− AA(s(s)F) (s ) (2.1a) Si

T (s ) = A(s )F (s ) (2.1b) Loop gain (the gain of the system around the feedback loop)

Writing (2.1a) as: S o (s ) = 1− AA(s()sF) (s ) S i (s ) We see that we could get non-zero output at So, with Si = 0, provided 1-A(s)F(s) = 0. Thus the system oscillates! April 2012


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Classical Positive Feedback Perspective on Oscillator (1) •

The condition for sustained oscillation, and for oscillation to startup from positive feedback perspective can be summarized as: For sustained oscillation For oscillation to startup

•

•

1 − A(s )F (s ) = 0

A(s )F (s ) > 1

Barkhausen Criterion

arg( A(s )F (s )) = 0

(2.2a) (2.2b)

Take note that the oscillator is a non-linear circuit, initially upon power up, the condition of (2.2b) will prevail. As the magnitudes of voltages and currents in the circuit increase, the amplifier in the oscillator begins to saturate, reducing the gain, until the loop gain A(s)F(s) becomes one. A steady-state condition is reached when A(s)F(s) = 1. Note that this is a very simplistic view of oscillators. In reality oscillators are non-linear systems. The steady-state oscillatory condition corresponds to what is called a Limit Cycle. See texts on non-linear dynamical systems. April 2012


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4

Classical Positive Feedback Perspective on Oscillator (2) •

Positive feedback system can also be achieved with inverting amplifier: Inverting amplifier

Si(s)

E(s)

+

-A(s)

So(s)

-

So (s ) = 1− AA(s(s)F) (s ) Si

Inversion F(s)

• •

To prevent multiple simultaneous oscillation, the Barkhausen criterion (2.2a) should only be fulfilled at one frequency. Usually the amplifier A is wideband, and it is the function of the feedback network F(s) to ‘select’ the oscillation frequency, thus the feedback network is usually made of reactive components, such as inductors and capacitors. April 2012


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Classical Positive Feedback Perspective on Oscillator (3) • •

In general the feedback network F(s) can be implemented as a Pi or T network, in the form of a transformer, or a hybrid of these. Consider the Pi network with all reactive elements. A simple analysis in [2] and [3] shows that to fulfill (2.2a), the reactance X1, X2 and X3 need to meet the following condition: So(s)

E(s)

+

-A(s)

X 3 = −( X 1 + X 2 ) (2.3)

-

If X3 represents inductor, then X1 and X2 should be capacitors.

X3

X1

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X2


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5

Classical Feedback Oscillators •

The following are examples of oscillators, based on the original circuit using vacuum tubes. +

+

+

-

-

-

Colpitt oscillator +

Hartley oscillator

-

Armstrong oscillator

Clapp oscillator

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Example of Tuned Feedback Oscillator (1) A 48 MHz Transistor Common -Emitter Colpitt Oscillator

2.0 1.5 1.0

R RB1 R=10 kOhm

R RC R=330 Ohm

C CD1 C=0.1 uF

VB

VL C Cc2 C=0.01 uF

pb_mot_2N3904_19921211 Q1 R RB2 R=10 kOhm

0.5 0.0 -0.5

VC

C Cc1 C=0.01 uF

VB, V VL, V

V_DC SRC1 Vdc=3.3 V

R RE R=220 Ohm

-1.0 R RL R=220 Ohm

-1.5

A(ω )F (ω )

C CE C=0.01 uF

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

time, usec

1 0 Si(s)

L C L1 C1 L=2.2 uH C=22.0 pF R=

C C2 C=22.0 pF

t E(s)

+

-A(s)

So(s)

-

F(s) April 2012


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6

Example of Tuned Feedback Oscillator (2) A 27 MHz Transistor Common-Base Colpitt Oscilator 600

R RC R=470 Ohm

R RB1 R=10 kOhm

VC

R RE R=100 Ohm

-400 L L1 L=1.0 uH R= C C2 C=100.0 pF

C C3 C=4.7 pF

R R1 R=1000 Ohm

-600 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

time, usec

E(s)

+

0 -200

pb_m ot_2N3904_19921211 Q1 VE

R RB2 R=4.7 kOhm

Si(s)

200

VL C C1 C=100.0 pF

C Cc2 C=0.1 uF

VB C Cc1 C=0.1 uF

400

C CD1 C=0.1 uF

VE, mV VL, mV

V_DC SRC1 Vdc=3.3 V

So(s)

A(s)

+

F(s)

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Example of Tuned Feedback Oscillator (3) V_DC SRC1 Vdc=3.3 V R RB1 R=10 kOhm

R RC R=330 Ohm

A 16 MHz Transistor Common-Emitter Crystal Oscillator

C CD1 C=0.1 uF

VC

VB C Cc1 C=0.1 uF

C C1 C=22.0 pF

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VL C Cc2 C=0.1 uF

R RL R=220 Ohm

pb_mot_2N3904_19921211 Q1 R RB2 R=10 kOhm

R RE R=220 Ohm

C CE C=0.1 uF

sx_stk_CX-1HG-SM_A_19930601 XTL1 Fres=16 MHz

C C2 C=22.0 pF


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7

Limitation of Feedback Oscillator •

•

•

•

At high frequency, the assumption that the amplifier and feedback network do not load each other is not valid. In general the amplifier’s input impedance decreases with frequency, and it’s output impedance is not zero. Thus the actual loop gain is not A(s)F(s) and equation (2.2) breakdowns. Determining the loop gain of the feedback oscillator is cumbersome at high frequency. Moreover there could be multiple feedback paths due to parasitic inductance and capacitance. It can be difficult to distinguish between the amplifier and the feedback paths, owing to the coupling between components and conductive structures on the printed circuit board (PCB) or substrate. Generally it is difficult to physically implement a feedback oscillator once the operating frequency is higher than 500MHz.

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3.0 Negative Resistance Oscillators

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8

Introduction (1) • • • • •

• •

An alternative approach is needed to get a circuit to oscillate reliably. We can view an oscillator as an amplifier that produces an output when there is no input. Thus it is an unstable amplifier that becomes an oscillator! For example let’s consider a conditionally stable amplifier. Here instead of choosing load or source impedance in the stable regions of the Smith Chart, we purposely choose the load or source impedance in the unstable impedance regions. This will result in either |Γ1 | > 1 or |Γ2 | > 1. The resulting amplifier circuit will be called the Destabilized Amplifier. As seen in Chapter 7, having a reflection coefficient magnitude for Γ1 or Γ2 greater than one implies the corresponding port resistance R1 or R2 is negative, hence the name for this type of oscillator. April 2012


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Introduction (2) •

•

•

For instance by choosing the load impedance ZL at the unstable region, we could ensure that |Γ1 | > 1. We then choose the source impedance properly so that |Γ1 Γs | > 1 and oscillation will start up (refer back to Chapter 7 on stability theory). Once oscillation starts, an oscillating voltage will appear at both the input and output ports of a 2-port network. So it does not matter whether we enforce |Γ1 Γs | > 1 or |Γ2 ΓL | > 1, enforcing either one will cause oscillation to occur (It can be shown later that when |Γ1 Γs | > 1 at the input port, |Γ2 ΓL | > 1 at the output port and vice versa). The key to fixed frequency oscillator design is ensuring that the criteria |Γ1 Γs | > 1 only happens at one frequency (or a range of intended frequencies), so that no simultaneous oscillations occur at other frequencies.

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9

Recap - Wave Propagation Stability Perspective (1) •

From our discussion of stability from wave propagation in Chapter 7… Zs or Γs Source b1

bsΓs 2Γ13 bsΓs 3Γ14

April 2012

Port 2

2-port Network a1

Z1 or Γ1

bsΓ1 bsΓs Γ12

Port 1

a1 = bs + bs Γ1Γs + bs Γ12Γs 2 + ... bs ⇒ a1 = 1 − Γ1Γs b1 = bs Γ1 + bs Γ12Γs + bs Γ13Γs 2 + ... ⇒ b1 =

bs bsΓs Γ1

b ⇒ 1= bs

Γ1 1 − Γ1Γs

Compare with equation (2.1a)

bsΓs 2Γ12 bsΓs 3Γ13

bs Γ1 1 − Γ1Γs

So A(s ) ( s) = Si 1− A(s )F (s )

Similar mathematical form


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Recap - Wave Propagation Stability Perspective (2) •

•

• •

We see that the infinite series that constitute the steady-state incident (a1) and reflected (b1) waves at Port 1 will only converge provided |Γ sΓ1| < 1. These sinusoidal waves correspond to the voltage and current at the Port 1. If the waves are unbounded it means the corresponding sinusoidal voltage and current at the Port 1 will grow larger as time progresses, indicating oscillation start-up condition. Therefore oscillation will occur when |Γ sΓ1 | > 1. Similar argument can be applied to port 2 since the signals at Port 1 and 2 are related to each other in a two-port network, and we see that the condition for oscillation at Port 2 is |ΓLΓ2 | > 1.

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10

Oscillation from Negative Resistance Perspective (1) • • • •

Generally it is more useful to work with impedance (or admittance) when designing actual circuit. Furthermore for practical purpose the transmission lines connecting ZL and Zs to the destabilized amplifier are considered very short (length → 0). In this case the impedance Zo is ambiguous (since there is no transmission line). To avoid this ambiguity, let us ignore the transmission line and examine the condition for oscillation phenomena in terms of terminal impedance. Very short Tline

Zs

Zo

Z1

Destabilized Amp. and Load

Z ≅ Z s Z ≅ Z1

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Oscillation from Negative Resistance Perspective (2) •

We consider Port 1 as shown, with the source network and input of the amplifier being modeled by impedance or series networks. Zs

Z1

jXs

Amplifier with load ZL

jX1

Source Rs Network

V R1

Z2 Vamp

Port 2

Port 1

•

ZL

Using circuit theory the voltage at Port 1 can be written as: V= April 2012

R1 + jX 1 Z1 ⋅ Vs = Vs R1 + Rs + j ( X 1 + X s ) Z s + Z1  2006 by Fabian Kung Wai Lee

(3.1) 22

11


Furthermore we assume the source network Zs is a series RC network and the equivalent circuit looking into the amplifier Port 1 is a series RL network. Zs

Z1

L1

Cs Rs

•

V R1

Vs

Z2 ZL

Vamp

Using Laplace Transform, (3.1) is written as: R1 + sL1 (3.2a) V (s ) = ⋅Vs (s ) R1 + Rs + sL1 + sC1 s (3.2b) where s = σ + jω April 2012


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The expression for V(s) can be written in the “standard” form according to Control Theory [8]: 2 V (s ) = 1 ⋅ 2 s(RR1 ++ RsL1 ) 1 = sC2 s (R1 + sL1 )ωn2 (3.3a) Vs L1 s + s ( L ) + L C s + 2ωnδs + ω n 1

s

1

where

•

δ=

R1 + Rs 2

L1

Cs

1 s

= Damping Factor

ωn =

1 L1C s

= Natural Frequency

The transfer function V(s)/Vs(s) is thus a 2nd order system with two poles p1, p2 given by: 2

p1, 2 = −δωn ± ωn δ − 1

• •

(3.3b)

(3.4)

Observe that if (R1 + Rs) < 0 the damping factor δ is negative. This is true if R1 is negative, and |R1| > Rs. R1 can be made negative by modifying the amplifier circuit (e.g. adding local positive feedback), producing the sum R1 + Rs < 0. April 2012


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12

Oscillation from Negative Resistance Perspective (5) • •

Assuming |δ| 1 freq 410.0MHz freq 410.0MHz

K -0.516 S(1,1) 3.067 / -47.641

S(1,2) 0.251 / 62.636

S(2,1) 6.149 / 176.803

S(2,2) 1.157 / -21.427

ΓL Plane

Γs Plane

Unstable Regions April 2012


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Precautions • • •

The requirement Rs= (1/3)|R1| is a rule of thumb to provide the excess gain to start up oscillation. Rs that is too large (near |R1| ) runs the risk of oscillator fails to start up due to component characteristic deviation. While Rs that is too small (smaller than (1/3)|R1|) causes too much nonlinearity in the circuit, this will result in large harmonic distortion of the output waveform. Clipping, a sign of too much nonlinearity

V2

t

V2 Rs too small For more discussion about the Rs = (1/3)|R1| rule, and on the sufficient condition for oscillation, see [6], which list further requirements. April 2012


t

Rs too large 42

21

Aid for Oscillator Design - Constant |Γ Γ1| Circle (1)

•

In choosing a suitable ΓL to make |ΓL | > 1, we would like to know the range of ΓL that would result in a specific |Γ1 |. It turns out that if we fix |Γ1 |, the range of load reflection coefficient that result in this value falls on a circle in the Smith chart for ΓL . The radius and center of this circle can be derived from:

•

Assuming ρ = |Γ1 |:

• •

S −Γ D Γ1 = 11 L 1 − S 22ΓL

Tcenter =

April 2012

By fixing |Γ1 | and changing ΓL .

− ρ 2 S 22* + D*S11 2

D − ρ S 22 2

2

(4.1a)

Radius = ρ

S12 S 21 2

D − ρ 2 S 22

2

(4.1b)


43

Aid for Oscillator Design - Constant |Γ Γ1| Circle (2) •

•

The Constant |Γ1 | Circle is extremely useful in helping us to choose a suitable load reflection coefficient. Usually we would choose ΓL that would result in |Γ1 | = 1.5 or larger. Similarly Constant |Γ2 | Circle can also be plotted for the source reflection coefficient. The expressions for center and radius is similar to the case for Constant |Γ1 | Circle except we interchange s11 and s22, ΓL and Γs . See Ref [1] and [2] for details of derivation.

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Example 4.1 – CB Fixed Frequency Oscillator Design •

•

•

In this example, the design of a fixed frequency oscillator operating at 410MHz will be demonstrated using BFR92A transistor in SOT23 package. The transistor will be biased in Common-Base configuration. It is assumed that a 50Ω load will be connected to the output of the oscillator. The schematic of the basic amplifier circuit is as shown in the following slide. The design is performed using Agilent’s ADS software, but the author would like to stress that virtually any RF CAD package is suitable for this exercise.

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Example 4.1 Cont... •

Step 1 and 2 - DC biasing circuit design and S-parameter extraction. S-PAR AME T ER S

DC

S t ab F ac t

DC D C1

V_DC SR C1 Vdc =4.5 V

R Rb1 R=10 kO hm

L LC L=330.0 nH R=

S_Param SP1 Start=410. 0 MH z Stop=410.0 MHz Step=2. 0 MH z

Port 2 - Output

C Cb C =1.0 nF

LB is chosen carefully so that the unstable regions in both ΓL and Γs planes are large April 2012 enough.

R Rb2 R=4.7 kOhm

Term Term 2 Num=2 Z=50 Ohm

C Cc1 C =1. 0 nF

SStabCircle

S_StabCircle S_StabCircle1 source_s tabc ir=s_st ab_c irc le(S ,51)

pb_phl_BF R 92A_19921214 Q1 L LE L=220.0 nH R=

LSt abCircle

L_St abCircle L_St abCircle1 load_s tabcir=l_s tab_c irc le(S, 51)

C Cc2 C =1. 0 nF L LB L=12. 0 nH R=

StabFact StabFact 1 K=st ab_f act (S )

Term Term 1 Num=1 Z=50 O hm

Port 1

Amplifier

Port 2

R Re R =100 Ohm

Port 1 - Input  2006 by Fabian Kung Wai Lee

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Example 4.1 Cont... freq 410.0MHz freq 410.0MHz

K -0.987 S(1,1) 1.118 / 165.6...

S(1,2) 0.162 / 166.9...

S(2,1) 2.068 / -12.723

S(2,2) 1.154 / -3.535

Unstable Regions

Load impedance here will result in |Γ1| > 1 April 2012

Source impedance here will result in |Γ2| > 1


47

Example 4.1 Cont... •

Step 3 and 4 - Choosing suitable ΓL that cause |Γ1 | > 1 at 410MHz. We plot a few constant |Γ1 | circles on the ΓL plane to assist us in choosing a suitable load reflection coefficient. LSC

This point is chosen because it is on real line and easily matched.

|Γ1 |=1.5 |Γ1 |=2.0

ΓL = 0.5