seminar report stabilization methods fixed

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OSKARI LEPPÄAHO, 202080STABILIZATION METHODSSeminar report

ELE-6256 Active RF circuits2.11.2011



1 INTRODUCTION TO AMPLIFIER STABILITY

Radio frequency (RF) amplifier circuit is usually made around one RF transistor. Most

of these transistors in amplifier circuits are potentially unstable at least in some bias and

frequency points. Potential instability means that the amplifier has a tendency to oscil-

late when improper loading is applied. The other two possible characterizations are un-

conditionally stable and unstable. Unconditional stability means that the amplifier will

not oscillate no matter what loading is applied. Unstable means that the amplifier oscil-

lates regardless of the loading. Term loading consists in this context not only the load

impedance but also the source impedance. [1. p. 470][2. p. 542-543]

Oscillation is not desirable for an amplifier circuit because it prevents or distracts the

circuit’s ability to form an amplified replica of the input signal. If the oscillation is pre-

sent in the operating frequency band all or at least the majority of the information

modulated to the input signal is lost and only high amplitude wave is seen in the output.

If the oscillation is outside the operating frequency band an excessive amount of power

is lost and the modulated signal is lost in situations when, for example, a BJT-transistor

is on the saturation region and cannot amplify the input signal as it should. If the oscilla-

tion is present at much higher frequency than the operating frequency, it may not be

seen in the output if the measuring equipment is non-ideal and not meant for measuring

the oscillation frequency in question. Measuring equipment can also act as a load damp-

ening the oscillation.

1.1 Improper loading of potentially unstable amplifier

To see if loading is improper for a potentially unstable amplifier, a reflection coefficient

approach can be used. First, we need to define scattering matrix. Simple amplifiers are

usually modeled as two-ports, meaning the equivalent circuit has an input and an output

port. The resulting model is valid for a small signal and class-A large signal amplifiers.

[3. p. 35] Scattering parameters are the parameters characterizing linear functionality of

the two-port. They depend on frequency and bias point, and are usually given in a form

of a scattering matrix. [2. p. 174]

= (1)

Now it’s possible to define input and output reflection coefficients. [4. p. 272]

Γ = +Γ

1 − Γ= − ∆Γ

1 − SΓ (2)

Γ = +

Γ

1 − Γ=

− ∆Γ

1 − SΓ (3)



1

Γ is the input reflection coefficient, Γ is the output reflection coefficient, Γ is the

load reflection coefficient, Γ is the source reflection coefficient and ∆= −. Source and load reflection coefficients are defined as [2. p. 58]

Γ =

− +

Γ = − +

(4)

(5)

where Z is the source impedance, Zis the load impedance and Z is the reference im-

pedance.

A potentially unstable two-port oscillates when the absolute value of either Γ or Γ

and loop gain on the same side is greater than one. When the oscillation is started and

approaches steady state, value of the loop gain saturates unity. If input side is consid-

ered, the loop gain is the product of source and input reflection coefficients.

ΓΓ = 1 (6)To get a clearer picture of what this means Γ can be substituted with equation 4 and

Γ = − + (7)

Now a new form for equation 6 can be written.

ΓΓ = − + ∙

− + = 1 (8)

− − + = + + + (9)

2 + 2 = 0 (10)

+

= 0 (11)

is the input impedance of a two-port. Equation 11 means both real and imaginary

parts of the impedances must be opposite numbers to each other. If analysis of equation

6 is continued interesting results are found out by substituting equation 2.

Γ =1

Γ=

1 − SΓ − ∆Γ, − ∆Γ ≠ 0 (12)

Solving Γ from the equation leads to

Γ =1 − Γ − ∆Γ

, − ∆Γ ≠ 0 (13)

Now from equation 3 it’s easy to see

Γ =1

Γ (14)

ΓΓ = 1 (15)

This means if there is oscillation in either of the ports the other will oscillate also. So

one can start thinking of filling the oscillation requirements of either port and after it’s

done the whole circuit will or will not oscillate depending on choices. [4. p. 277, 280-

281]

There are two mentioned cases where the statement above isn’t valid: the points where

denominator in equation 12 and 13 is zero. These points are



2

Γ = 0

Γ = 0 (16)

By definition reflection coefficient is

Γ =

(17)

where is the reflected wave from a port and

is the incident wave to the port.

From 16, 17 and 18 it’s easy to see that with this special case all the reflected waves

must be zero. The S parameters are defined as

= (18)

and they are all zero now since there is no reflected waves. So the ports of a two-port

are galvanically isolated from each other. If only the input reflection coefficient or the

output reflection coefficient is zero, the connection is made only in one way and there is

no feedback loop. In these situations the circuit won’t oscillate without additional cir-cuits. [2, p. 58, 174]

Final issue in reasons of potentially unstable amplifier oscillating is the choice, which

makes either input or output reflection coefficient’s absolute value to go over unity. If

the amplifier is potentially unstable there is always at least one load with one specific

frequency that makes the amplifier to oscillate. Usually there are wide varieties of loads

in different frequencies, which make the circuit oscillate. As seen from equations 2 and

3, load reflection coefficient affects to input reflection coefficient and source reflection

coefficient affects to output reflection coefficient. [1. p. 478]

Now, think one potentially unstable frequency with which some load impedances makethe circuit oscillate (|Γ| > 1). The unstable area can be drawn into the load Smith

Chart plane as seen in Figure 1. Unstable area forms a circle called stability circle and

depending on the value of || the stable area is either the inside or the outside of the

circle. If || < 1 and stability circle doesn’t encompass the chart center, stable area is

the outside of the stability circle and inside if || > 1 . If the stability circle encom-

passes the chart center, stable areas go vice versa. [4. p. 272-276]



3

Figure 1: Output stability circles in input impedance plane. On the left

|| 1 and on the right || 1. Picture from [4. p. 275]



4 4

2 STABILIZATION METHODS

Unconditionally stable circuits are stable at all situations so they don’t need to be stabi-

lized. Unstable circuits are always unstable no matter of the load, so no stabilization

method will help. So every stabilization method focuses on potentially unstable circuits

that are manipulated so that they will never “see” the impedances that make them oscil-

late. So, a potentially unstable circuit becomes unconditionally stable.

2.1 Achieving unconditional stability

Stability circle approach is one way to make a potentially unstable circuit unconditional-

ly stable. By looking Figure 1 it is easy to see that if the stability circle would be outside

the Smith chart when || < 1, circuit would be unconditionally stable. For || > 1,

there is not any case where the circuit would be unconditionally stable.

Take an example of output stability circle as in Figure 2 and assume || < 1 so the

unstable area is inside the stability circle. Some inductive loads, at a particular frequen-

cy, are causing the absolute value of input reflection coefficient to go over unity. This

can be prevented by adding a series resistor to the output so amplifier will not see fully

inductive loads anymore. In fact all the possible impedances seen by the amplifier are

restricted to the inside of a dotted line circle seen in figure 2.

To calculate a numerical value for the resistor, assume 50 Ω as the reference impedance.

Now the resistance value is = 0.6 ∙ 50Ω = 30Ω (19)

One could also think of stabilizing with a capacitive component but, since the capacitive

and inductive reactances cancel each other, the result would be just moving the stability

circle towards higher inductive loads instead of moving it away from the Smiths chart.

Another way to stabilize would be use of shunt resistor and admittance Smiths chart. [1.

p. 480-483]



5

2.2 Designing stabilization network for different kinds ofamplifiers

Sometimes stabilization using resistive components can cause problems to the amplifiercircuit. This can be thought through with two example circuits: low noise amplifier

(LNA) and power amplifier.

The main idea of low noise amplifier is to get the best possible amplification of the sig-

nal with minimum amount of noise added to it. The noise effect of a resistor can be

modeled into some extent with a resistive attenuator although attenuator has some dif-

ferent properties like ability to not change the input or output impedance. Examine three

different situations described in Figure 3.

Figure 2: Output stability circle in load plane. Also all the possible impedances, seen

by the amplifier after the stabilizing resistor is placed, are marked.



6

In basic situation LNA has following noise properties.

(20)

(21)

(22)

Where is the output noise power, is amplifier gain, is Boltzmann constant,

is room temperature, is equivalent noise temperature of amplifier, is measuring

bandwidth of the noise power, is total gain and is noise factor. Now, when adding

an attenuator having attenuation of and gain of on input of LNA, equations look

like this.

1

(23)

(24)

1

1

1

1

(25)

Where is the equivalent noise temperature of the attenuator and is the noise factor

of the whole circuit. Then compare this to a situation where the attenuator is on output

of the LNA.

Figure 3: Three different situations whose noise behavior is analyzed.



7

= + + 1

(26)

= (27)

= = + +1

= 1 + +1

= + − 1

(28)

When comparing equations 24 and 27 it is easy to see that gain isn’t depending on

which side of an amplifier the attenuator is connected. The available noise power is not

staying the same as seen from equations 23 and 26. The equivalent temperatures are

different. In input connected attenuator the part that differs is +

and in output

connection the same part is

+

. Now

is smaller than one and

is larger

than one so we can write.

+1 ≤ + ≤ +

1 (29)

From equation 29 is easy to see that equivalent temperature of the circuit with attenuator

connected in input is larger than the one with attenuator connected to output. As the

available noise power grows with the equivalent temperature, the noise power is also

bigger.

When looking on noise factor equations 25 and 28, the two different cases differ only in

equivalent temperatures. As it is in the numerator, the same analogy as in noise power

goes and the value of noise figure is bigger when an attenuator is connected to an input

of an amplifier. Based on this, it is more reasonable to put the stabilizing resistor in the

output port of the LNA to achieve the best possible noise figure. [2. p. 491-499]

In power amplifiers the noise figure is not so important but the output power is crucial.

The power to the output is power gain times input power. = (30)

Where is the output power, is the power gain and is the input power. The

input power of source in matched system is

=

4, (31)

where is the source generator voltage and is the reference impedance. If the input

is stabilized for example with

input power becomes smaller and so does the output

power also.

=425. (32)

=

4

25

(33)



8

Power lost in stabilization resistor.

=450 (34)

If the same stabilization is put into output side input power stays the same as in equation

31 but output power to the load is decreased.

= 425 (35)

Power lost in stabilization resistor is now

= 450. (36)

From 33 and 35 we see that power gain isn’t different for input and output stabilization

if same valued resistor is used, both ports are stable at the starting point and the reflect-

ed power due to mismatch will not go to another port. These strong assumptions state

that every situation needs to be thought through and the results may change depending

on situation. One noticeable effect of stabilization resistor’s place is its power loss.

From 34 and 36 is seen that power loss of output stabilization resistor is the same of

input times . For power amplifiers is quite large and so is the difference between

power losses. This means unnecessary power loss and efficiency decrease is made if the

stabilization resistor is connected to output. [5. p. 61-63]

2.3 Using frequency selective loading

One way to face the stability problems is frequency selective loading. This is the bestway if instabilities are not in the operating frequency band. This minimizes the change

of circuit properties at operating frequency. [6. p. 17]

For stability problems in higher than operating frequency band for example a shunt re-

sistor with a capacitor can be used. Now capacitor is so small, that it is almost an open

circuit at operating frequency. The illustration is in figure 4.



9

For stability problems in lower than operating frequency band for example a shunt resis-

tor with an inductor can be used. Now inductor must be an appropriate RF choke and so

it is an open circuit for operating frequency. The illustration is in figure 5.

2.4 Resistive feedback as a stabilizer

Resistive feedback as stabilizer means usually either base-collector resistor or emitter

resistor. One way to see how these resistors stabilize the circuit is to look at a simple

BJT small signal model and how the voltage gain varies with different feedbacks. To do

that three example circuits like in Figure 6 are generated. The voltage gain of basic

small signal model is

, , (37)

where is the output voltage over load resistor , is the input voltage of the

circuit and is now over BJT small signal model equivalent resistor and is the small

signal forward current gain. Compare this to voltage gain of model with emitter resistor

, 1, (38)

where

is the emitter resistor and input voltage is now over

and

, and to voltage

gain of model with collector-base resistor

Figure 4: High frequency output stabilization circuit.

Figure 5: Low frequency output stabilization circuit.



10

, , (39)

where is the collector-base feedback resistance. [7.]

By examining the results got in equations 37, 38 and 39 it is easy to see that the absolute

value of voltage gain decreases if emitter or collector-base resistance is applied to cir-

cuit. In emitter resistor case the denominator of basic gain is increased by 1 and in collector-base resistor case something positive is added to the negative basic gain

and then it is multiplied with something less than one. The voltage gain drop decreases

the possibility of an amplifier to oscillate and thus, if the drop is enough, stabilizes the

amplifier.

Figure 6: Three models used to demonstrate the effect of a feedback resistor.



11

REFERENCES

1. RF circuit design – theory and applications, R. Ludwig, P. Bretchko. 2000. United States

of America, Prentice-Hall, Inc. 642 p. ISBN 0-13-095323-7

2. Microwave engineering, David M. Pozar. Third edition. 2005. United States of America,

John Wiley & Sons, Inc. 700 p. ISBN 978-0-471-44878-5

3. Design of Class-E radio frequency power amplifier, S. Al-Shahrani. Dissertation for the

degree of Doctor of Philosophy in electrical engineering. 2001. Blacksburg, Virginia,

United States of America, Faculty of the Virginia Polytechnic Institute and State Uni-

versity. 121 p.

4. Foundations of oscillator circuit design, G. Gonzales. 2007. United States of America,

Artech House, Inc. 422 p.

5. Electronics - A systems approach, N. Storey. 3rd

edition. 2006. England, Pearson Educa-

tion Limited. 645 p. ISBN 0-13-129396-6

6. Practical RF Amplifier Design Using the Available Gain Procedure and the AdvancedDesign System EM/Circuit Co-Simulation Capability, Ken Payne. White paper. 2009.

United States of America, Agilent Technologies, Inc. 61 p. 5990-3356EN

7. BJT:n piensignaalimalli, luentokalvot. ELE-2100 Puolijohdekomponenttien sovellukset.

2011, TTY.

seminar report stabilization methods fixed

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