Electronic circuits handbook for design and application ( u tietze)

Figure 27.1a shows typical values of low-frequency input and output resistances ofthe voltage and the current amplifier in an integrated high-frequency amplifier where it isassumed that

High-Frequency Amplifiers

Today, in the high- and intermediate-frequency assemblies of telecommunication systems,amplifiers composed of discrete transistors are still used in addition to modern integratedamplifiers This is particularly the case in high-frequency power amplifiers employed intransmitters In low-frequency assemblies, on the other hand, only integrated amplifiersare used The use of discrete transistors is due to the status quo of semiconductor technol-ogy The development of new semiconductor processes with higher transit frequencies issoon followed by the production of discrete transistors, but the production of integratedcircuits on the basis of a new process does not usually occur until some years later Fur-thermore, the production of discrete transistors with particularly high transit frequenciesoften makes use of materials or processes which are not (or not yet) suitable for the produc-tion of integrated circuits in the scope of production engineering or for economic reasons.The high growth rate in radio communication systems has, however, boosted the devel-opment of semiconductor processes for high-frequency applications Integrated circuits

on the basis of compound semiconductors such as gallium-arsenide (GaAs) or germanium (SiGe) can be used up to the GHz range For applications up to approximately

silicon-3 GHz bipolar transistors are mainly used, which, in the case of GaAs or SiGe designs,

are known as hetero-junction bipolar transistors (HBT) Above 3 GHz, gallium-arsenide junction FETs or metal-semiconductor field effect transistors (MESFETs) are used.1The

transit frequencies range between 50 100 GHz.

27.1

Integrated High-Frequency Amplifiers

In principle, integrated high-frequency amplifiers use the same circuitry as low-frequency

or operational amplifiers A typical amplifier consists of a differential amplifier used as avoltage amplifier and common-collector circuits used as current amplifiers or impedanceconverters (see Fig 27.1a) The differential amplifier is often designed as a cascode dif-ferential amplifier to reduce its reverse transmission and its input capacitance (no Millereffect) Such circuits are described in Chap 4, Sect 4.1 Since the transit frequency of

high-frequency transistors (f T ≈ 50 100 GHz) is approximately 100 times higher than that of low-frequency transistors (f T ≈ 500 MHz 1 GHz), the bandwidth of the ampli-

fier increases by approximately the same factor This, however, presumes that the parasitics

of the bond wires and the connections within the integrated circuit can be reduced enough

so that the bandwidth is primarily determined by the transit frequency of the transistorsand is not limited by the connections This is a key problem in both the design and use ofhigh-frequency semiconductor processes

1The construction of an HBT corresponds to that of a conventional bipolar transistor Here, however,different material compositions are used for the base and emitter regions in order to enhance thecurrent gain at high frequencies The construction of a MESFET is shown in Fig 3.26b onpage 198

Power amplifier

Current amplifier (impedance converter)

I B,A

V b V b

I C,A

a Principle and design of an integrated amplifier

b Principle and design of a matched amplifier with one discrete transistor

con-2These are electrically short lines (see Sect 26.2) In this context the term ideal does not refer to

the losses; these are relatively high in integrated circuits due to the comparably thin metal coatingand the losses in the substrate

signal-carrying terminals must be matched to the characteristic impedance of the externallines to prevent any reflections In the ideal case, the circuit is dimensioned such that inputand output impedances, including the parasitic effects of bond wires, connecting limbsand the case, correspond to the characteristic impedance Otherwise, external components

or strip lines must be used for impedance matching (see Sect 26.3)

Figure 27.1a shows typical values of low-frequency input and output resistances ofthe voltage and the current amplifier in an integrated high-frequency amplifier where it isassumed that equivalent amplifiers are employed as signal source and load

Impedance Matching at the Input

For high frequencies, the input impedance of a differential amplifier is ohmic-capacitivedue to the capacitances of the transistor Generally, up to around 100 MHz, its value is

clearly higher than the usual characteristic impedance Z W = 50
.

A rigorous impedance matching method involves inserting a terminating resistance

R = 2Z W = 100
between the two inputs of the differential amplifier (see Fig 27.2a);

0

2 I 0

a With terminating resistance

b With common-base circuits (I 0⬇ 520 µA for = 5Z W Ω)

this matches both inputs to Z W = 50
This method is simple, easy to accomplish with

a resistor in the integrated circuit and acts across a wide band A disadvantage is the poorpower coupling owing to the dissipation of the resistor and the large increase in the noise

figure (see Sect 27.1.2) Instead of placing a resistance R = 2Z Wbetween the two inputs,

each of the two inputs can be connected to ground via a resistance R = Z W However, thismeans that a galvanic coupling to signal sources with a DC voltage is no longer possible

as the inputs are connected to ground with low resistance The version with a resistance

R = 2Z W is thus preferred

As an alternative, common-base circuits can be used for the input stages (see Fig 27.2b);then, the input impedance corresponds approximately to the transconductance resistance

1/g m = V T /I0 of the transistors With a bias current I0 ≈ 520 mA, this resistance is

1/g m ≈ Z W = 50
In this case, the power coupling is optimal A disadvantage is the

comparably high noise figure (see Sect 27.1.2)

Both methods are suitable for frequencies in the MHz range only In the GHz range, theinfluence of the bond wires, the connecting limbs and the casing have a noticeable effect.The situation can be improved by using loss-free matching networks made up of reactivecomponents or strip lines that must be fitted externally This will provide an optimumpower coupling with a very low noise figure In practice, impedance matching focuses less

on optimum power transmission than it does on optimum noise figure, or a compromisebetween both optima This is described in more detail in Sect 27.1.2

Impedance Matching at the Output

Wideband matching of the output impedance of a common-collector circuit to the usual

characteristic impedance Z W = 50
can be achieved by influencing the output impedance

of the voltage amplifier while taking into consideration the impedance transformation in

a common-collector circuit For the qualitative aspects refer to Fig 2.105a on page 149and to the case shown in the left portion of Fig 2.106 where the output impedance of acommon-collector circuit has a wideband ohmic characteristic if the preceding amplifierstage has an ohmic-capacitive output impedance with a cut-off frequency that corresponds

to the cut-off frequency ω β = 2 πf βof the transistor Due to secondary effects this type of

matching can be achieved quantitatively only with the aid of circuit simulation Again, in

the GHz range, the influence of the bond wires, the connecting limb and the casing show

a disturbing effect In principle, impedance matching remains possible, but not with thewideband effect

If impedance matching is not possible by influencing the output impedance of thecommon-collector circuit, external matching networks with reactive components or striplines are used

27.1.2

Noise Figure

In Sect 2.3.4 we showed that the noise figure of a bipolar transistor with a given collector

current I C,Ais minimum if the effective source resistance between the base and the emitterterminal reaches its optimum value:

Here, R B is the base spreading resistance and β the current gain of the transistor For the collector currents I C ,A ≈ 0.1 1 mA, which are typical of integrated high-frequency circuits, the source resistance for β ≈ 100 is in the region R g opt ≈ 260 2600
With larger collector currents, R g opt can be further reduced, e.g to 50
at I C ,A= 23 mA and

R B = 10
, but the noise figure reaches only a local minimum as shown in Fig 2.52 on

page 92 This is caused by the base spreading resistance Very large transistors with verysmall base spreading resistances are used in low-frequency applications which enables theglobal minimum of the noise figure to be nearly reached even with small source resistances.However, in this case the transit frequency of the transistors drops rapidly; thus, in high-frequency applications, this method can be used in exceptional cases only

In impedance matching at the input side by means of a terminating resistance as

shown in Fig 27.2a, the effective source resistance has the value R g,eff = R g ||R/2 =

Z W /2 = 25
for each of the two transistors in the differential amplifier due to the parallel connection of the external resistances R g = Z W and the internal terminat-

ing resistance R = 2Z W It is thus clearly lower than the optimum source resistance

R g opt ≈ 260 2600
Furthermore, the noise of the terminating resistance causes the

noise figure to become relatively high With impedance matching at the input side by means

of a common-base circuit as shown in Fig 27.2b, the effective source resistance has the

value R g,eff = R g = Z W = 50
; here, too, the noise figure is comparably high.

For impedance matching with reactive components or strip lines, the internal resistance

R g of the signal source can be matched to the input resistance r iof the transistor by means of

a loss-free and noise-free matching network If we disregard the base spreading resistance

R B , then r i = r BE For the effective source resistance R g,eff between the base and emitter

terminals this means that R g,eff = r BE For r BE = βV T /I C ,A and R g opt the following

relationship is obtained from (27.1) with R B= 0:

R g,eff = r BE = R g opt

Thus, with impedance matching, the effective source resistance is higher than the optimumsource resistance by a factor of√

β ≈ 10 This might make the noise figure lower than

that in the configurations with a terminating resistance or a common-base circuit, but it isstill clearly higher than the optimum noise figure

The optimum noise figure is only obtained when noise matching is performed instead

of power matching This means that the internal resistance R g = Z W of the signal source

is not matched to r i = r BE but to R g opt = r BE /√β Conversely, the input resistance of the

(noise) matched amplifier is no longer Z W but Z W√β This leads to the input reflectionfactor

Above f = f T /√

β ≈ f T /10 the optimum source resistance decreases, as can be seen from the equation for R g opt,RF in Sect 2.3.4 This does not mean that the matching

methods in Fig 27.2 can achieve a lower noise figure in this range Factor R g,eff /R g opt

does go down but the minimum noise figure increases as the equation for F opt ,RF inSect 2.3.4 shows We will not examine this range more closely as the noise model forbipolar transistors with a transit frequency above 10 GHz as used in Sect 2.3.4 will only

allow qualitative statements in this case The range f > f T /10 is then entirely in the GHz

range and some secondary effects, such as the correlation between the noise sources ofthe transistor, which were disregarded in Sect 2.3.4, become significant, and the optimumsource impedance is no longer real

Example: With the help of circuit simulation we have determined the noise figure of the

different circuit versions for an integrated amplifier with the transistor parameters in Fig 4.5

on page 278 Owing to the symmetry, we can restrict the calculations to one of the twoinput transistors; Fig 27.3 shows the corresponding circuits We use a transistor of size 10

and a bias current of I C ,A= 1 mA In the common-base circuit according to Fig 27.3c, wereduce the bias current to 520mA in order to achieve impedance matching to Z W = 50
.

d With matching network

(power matching or noise matching)

c With common-base circuit

Fig 27.3.Circuits for a noise figure comparison

The base spreading resistance is R B = 50
and the frequency f = 10 MHz From (27.1)

it follows that R g opt = 575
for I C ,A = 1 mA and R g opt = 867
for I C ,A= 520 mA

The circuit without matching in Fig 27.3a achieves an optimum noise figure F opt =

1.12 (0.5 dB) for R g = R g opt = 575
and F = 1.52 (1.8 dB) for R g = 50
The circuit with terminating resistance in Fig 27.3b results in the noise figure F = 2.66 (4.2 dB); the noise figure thus clearly increases A more favourable value is achieved with the common-base circuit in Fig 27.3c where F = 1.6 (2 dB) With power matching to

R g = Z W = 50
, according to Fig 27.3d, the value obtained is F = 1.25 (0.97 dB), which is only a factor of 1.1 (0.5 dB) above the optimum value The optimum noise figure

is achieved with noise matching

If power matching is essential in order to prevent reflections, the circuit with matchingnetwork and power matching according to Fig 27.3d leads to the lowest noise figure,followed by the common-base circuit in Fig 27.3c and then the circuit with terminatingresistance in Fig 27.3b Without power matching, the circuit with matching network andnoise matching according to Fig 27.3d is clearly superior to the circuit without matching

in Fig 27.3a for R g = 50
with regard to both the noise figure and the reflection factor.

27.2

High-Frequency Amplifiers with Discrete Transistors

Figure 27.1b shows the principle design of high-frequency amplifiers made up of discretetransistors It is clear that the circuit design differs fundamentally from that of the inte-grated amplifier shown in Fig 27.1a The actual amplifier consists of a bipolar transistor

in common-emitter configuration and circuitry for setting the operating point, which is

presented in Fig 27.1b, by the two current sources I B,A and I C ,A The practical ality will be further described below Instead of a bipolar transistor, a field effect transistorcan also be used Coupling capacitances are used in front of and behind the transistor toprevent the operating point from being influenced by the additional circuitry The networks

function-for impedance matching to the characteristic impedance of the signal lines include π

el-ements (Collins filters) with a series inductance and two shunt capacitances as shown inFig 27.1b

27.2.1

Generalised Discrete Transistor

The term discrete transistor should not be misunderstood in a limited sense because the

components used in practice often contain several transistors and additional resistancesand capacitances in order to simplify the process of setting the operating point We call

these components generalised discrete transistors.3

Figure 27.4a shows the graphic symbol and the most important versions of a alised discrete transistors without additional components for setting the operating point

gener-A Darlington circuit is often used to enhance the current gain at high frequencies.Figure 27.4b presents some typical designs with additions for setting the operatingpoint The version at the left can be used equally well for the Darlington circuits inFig 27.4a The resistances provide a voltage feedback which, at sufficiently high-resistive

3This can be related to the CC operational amplifier which may also be regarded as a generaliseddiscrete transistor (see Sect 5.5 and Figs 5.82 to 5.87)

a Symbol and circuit configurations

b Circuit configurations with additional elements for setting the operating point

V b

BGA427

Fig 27.4.Generalised discrete transistor

dimensions, becomes virtually inefficient at high frequencies if the impedance of thecollector-base capacitance falls below the value of the feedback resistor The externalelement is an inductance which represents an open circuit at the operating frequency andconsequently causes a separation of the signal path and the DC path The version shown inthe centre of Fig 27.4b has an additional emitter resistance for current feedback; therefore,

it is particularly suitable for wideband amplifiers or amplifiers with a high demand in terms

of linearity

The version shown at the right of Fig 27.4b consists of a common-emitter circuit withvoltage feedback followed by a common-collector circuit Strictly speaking, this does notbelong to the group of discrete transistors since, like the integrated amplifier in Fig 27.1b, itcomprises a voltage amplifier (common-emitter circuit) and a current amplifier (common-collector circuit) Nevertheless, we have included it since it usually comes in a casing that istypical of discrete transistors The voltage feedback is often operated with two resistancesand one capacitance Only the resistance, which is directly connected between the base andthe collector, influences the operating point and is used for setting the collector voltage

at the operating point The capacitance is given dimensions such that it functions as ashort circuit at the operating frequency, thus allowing the parallel arrangement of the tworesistances to become effective

The versions shown in Fig 27.4 are regarded as low-integrated circuits and are termed

monolithic microwave integrated circuits (MMIC) They are made of silicon (Si-MMIC),

silicon-germanium (SiGe-MMIC) or gallium-arsenide (GaAs-MMIC) and are suitable forfrequencies of up to 20 GHz

27.2.2

Setting the Operating Point (Biasing)

Generally, the operating point is set in the same way as for low-frequency transistors.However, with high-frequency transistors, one attempts to make the resistances required

in order to set the operating point ineffective at the operating frequency otherwise theywill have an adverse effect on the gain and noise figure For this reason, the resistancesare combined with one or more inductances which can be considered short-circuited withregard to setting the operating point, and nearly open-circuited at the operating frequency

A description of how the operating point is set in a bipolar transistor is given below.The circuits described may equally well be used for field effect transistors

DC Current Feedback

If we apply the above-mentioned principle to the operating point adjustment with DCcurrent feedback as shown in Fig 2.75a on page 119, we obtain the circuit design shown

in Fig 27.5a in which high-frequency decoupling is achieved for the base and the collector

of the transistor by means of inductances L B and L Crespectively The collector resistancecan be omitted in this case Thus, there is no DC voltage drop in the collector circuit sothat this method is particularly suitable for low supply voltages In extreme situations, one

may remove R1 and R2 and connect the free contact of L Bdirectly to the supply voltage;

the transistor then operates with V BE,A = V CE ,A Due to the decoupled base, the noise of

resistors R1 and R2have only very little influence on the noise figure of the amplifier atthe operating frequency which is a particularly low-noise method for setting the operating

point This is especially the case if an additional capacitance C B is introduced which, at

and decoupling of the

base (low noise)

With current feedback and no decoupling of the base

With voltage feedback

Fig 27.5.Setting the operating point in high-frequency transistors

the operating frequency, acts almost as a short circuit Where a slight increase in the noisefigure is not critical, it may not be necessary to decouple the base and thus the circuitshown in Fig 27.5b may be used.

With an increase in frequency decoupling becomes more and more difficult sincethe characteristics of the inductors used to achieve the required inductance become lessfavourable In order to make the magnitude of the impedance as high as possible, aninductor with a resonant frequency that is as close as possible to the operating frequency

is used As a result, the resonant impedance is approximately reached which, however,decreases with an increasing resonant frequency as shown in Fig 28.4 on page 1406 Forthis reason, in the GHz range, the inductances are replaced by strip lines of the length

λ/4 These lines are short-circuited for small signals at the end opposite the transistor

by capacitance C B or by connecting them to the supply voltage The end closest to thetransistor then acts as an open circuit

Particularly problematic is the capacitance C Ewhich, at the operating frequency, mustperform as a short circuit Here, too, a capacitance with a resonant frequency as close aspossible to the operating frequency is used, whereby doing so results in impedances with

a magnitude close to that for the series resistance of the capacitance (typically 0.2
).

However, with increasing resonant frequency, the resonance quality of the capacitancesincreases (see Fig 28.5 on page 1406), thus making the adjustment more and more difficult

As an alternative, an open-circuited strip line of length λ/4 could be used that acts as a

short circuit at the transistor end but, owing to the unavoidable radiation at the open-endedside (antenna effect), this method is not practical A short-circuited strip line must also

be rejected as it provides a short circuit for the DC current and thus short-circuits the

resistance R E Owing to these problems, the DC current feedback is used only in the MHzrange while in the GHz range the emitter terminal of the transistor must be connecteddirectly to ground

DC Voltage Feedback

Figure 27.5c shows the method of setting the operating point by means of DC voltagefeedback This is used in many monolithic microwave integrated circuits (see Fig 27.4b)

A collector resistance R Cis essential in order to render the feedback effective and to ensure

a stable operating point The collector is decoupled by the inductance L C so that, at theoperating frequency, the output is not loaded by the collector resistance The base can be

decoupled by adding series inductances to the resistances R1 and R2; however, this method

is not used in practice A disadvantage is an increase in the noise figure due to the noise

contributions from R1 and R2, but these can be kept low using high-resistive dimensioning.

Automatic Operating Point Control

Amplifiers, whether consisting of integrated circuits or discrete components, are oftenprovided with automatic control of the operating point as shown in Fig 27.6 Here, the

collector current of the high-frequency transistor T1 is measured from the voltage drop V RC

across the collector resistance R C and compared with a setpoint value V D1 Transistor T2 controls the voltage at the base of transistor T1 so that V RC ≈ V D1 ≈ 0.7 V

Fig 27.6.Automatic operating point control

Let us first look at the circuit in Fig 27.6a It follows that:

R C −R21



R2 R C is typically the case in practice; thus I C1,A ≈ 0.7 V/R C

The control circuit must have a pronounced low-pass characteristic of the first order

to ensure stability; capacitance C Bserves this purpose It is selected such that the cut-offfrequency

2πC B (R2|| r BE1 )

is below the operating frequency by a factor of at least 104

Figure 27.6b shows the control of the operating point for an integrated circuit where the

elements L C and C B must be provided externally The inductance L Bis usually replaced by

a resistance which slightly shifts the operating point Resistance R Cis usually an externalcomponent so that the bias current can be adjusted This adjustment is necessary as thebias current, which is optimum in terms of gain and noise figure, depends on the operating

frequency Furthermore, the ground connection of resistance R1is usually accessible fromthe outside so that the amplifier can be turned on and off by a switch

Impedance Matching for a Single-Stage Amplifier

Calculation of the matching networks for an amplifier with a generalised single transistor

is complex because the impedances at the input and output port depend on the circuitryconnected to the other port, respectively; this is due to the internal reactive feedback whichalso leads to a non-zero reverse transmission The calculation is usually based on the S

parameters of the transistor including the circuitry for setting the operating point.

Conditions for Impedance Matching

Figure 27.7 shows a transistor with matching networks and the corresponding reflectionfactors at various positions Since these points are fully matched, the reflection factors atthe signal source and the load are zero The matching network at the input side transforms

the reflection factor of the signal source from zero to r g at the transistor input where it

meets the input reflection factor r1of the transistor Similarly, the matching network at

the transistor output transforms the reflection factor of the load from zero to r L, which

meets the output reflection factor r2of the transistor For two-sided impedance matching,the respective reflection factors must be conjugate complex to one another:

Reflection Factors of the Transistor

The reflection factors r1 and r2 of the transistor depend on r L and r gdue to the reverse mission (see Fig 27.8) For the transistor, including the circuitry for setting the operatingpoint, the following is true:

 

a1 a2

Matching network

Fig 27.8.Calculating the reflection factors of a connected transistor

With a load with reflection factor r L connected to the output, the input reflection factor r1is

determined by inserting the condition a2 = b2r Lfrom Fig 27.8a and solving the equation

for r1 = b1/a1 Similarly, the output reflection factor r2 with a source with reflection

factor r g connected to the input is calculated by inserting the condition a1 = b1r gfrom

Fig 27.8b and solving the equation for r2 = b2/a2 This leads to:

r1 = S11+ S12 S21r L

r2 = S22+ S12S21r g

Without reverse transmission (S12 = 0), there is no interdependence and the reflection

factors are r1 = S11 und r2 = S22.

Calculating Impedance Matching

If we insert the conditions (27.3) into (27.4) and (27.5), the reflection factors r g and r Lofthe matched condition are obtained through elaborate calculations [27.1]:

Stability at the Operating Frequency

To ensure that the amplifier is stable, the following must apply:

Without reverse transmission (S12 = 0), the k factor is k → ∞ In this case the

secondary conditions require that|S11| < 1 and |S22| < 1, i.e the real parts of the input

and output impedances of the transistor, including the circuitry for setting the operatingpoint, must be greater than zero Therefore, a transistor without reverse transmission can

be matched at both sides if the real parts of the impedances are greater than zero If

reverse transmission exists (S12 = 0), the secondary conditions are more stringent andthus positive real parts of the input and output impedance are no longer sufficient In this

case, however, the condition k > 1 is more crucial than the secondary conditions, i.e the secondary conditions are usually met but the condition k > 1 is not.

Calculating Matching Networks

If the conditions (27.8) and (27.9) are met, the matching networks can be determined from

(27.6) and (27.7) with the help of the reflection factors r g,m and r L,m First, the input andoutput impedances of the transistor, whose operating point is set for the matched condition,are calculated:

If conditions (27.8) and (27.9) are not met, a straight-forward procedure is not available

In this case a mismatch at the input or output must be accepted A problem arises in finding

suitable reflection factors r g and r Lfor which the mismatch is as small as possible whilethe operation of the system is sufficiently stable [27.1] describes a procedure on the basis

of stability circles which is not discussed in more detail here A relatively easy procedure

is to connect additional load resistances to the input or output of the transistor so that the

S parameters meet the conditions of (27.8) and (27.9) However, it depends on the givenapplication whether this yields a better overall result than a possible slight mismatch

Stability Across the Entire Frequency Range

The stability conditions (27.8) and (27.9) ensure stability only at the operating frequencyfor which the matching networks are determined However, in no way does this guaranteethat the amplifier will be stable at all frequencies This can be investigated by means of atest setup or by simulating the small-signal frequency response across the entire frequencyrange from zero up to and beyond the transit frequency of the transistor When measuringthe small-signal frequency response with a network analyser it should be noted that, in this

case, the amplifier is connected to wide-band circuitry with R g = Z W and R L = Z W Inthe actual application, the amplifier may only have narrow-band matching that can causeinstability at frequencies other than the operating frequency, i.e the stability at the networkanalyser does not necessarily indicate stable operating conditions in the actual application

Power Gain

For impedance matching on both sides with reactive, i.e loss-free, matching networks, the

maximum available power gain (MAG) [27.1]

can be determined from (27.8) with the stability factor k > 1 This and other power gains

are described in Sect 27.4 in more detail

Example: The task is to design a high-frequency amplifier with transistor type BFR93 matched at both sides for an operating frequency (centre frequency) f C = 1.88 GHz The supply voltage is to be 3.3 V We use automatic control of the operating point according

to Fig 27.6a with a bias current of I C = 5 mA For this bias current we obtain a minimumnoise figure as stated in the data sheet.4

Figure 27.9 shows the dimensioned components of a circuit for setting the operatingpoint The following aspects were taken into consideration:

– Since the input impedance of the transistor is very low (Re{S11 } < 0 → Re {Z i } <

50
), the inductive decoupling of the base is omitted; therefore, the inductance L Bof

Fig 27.6a is replaced by a resistor R B = 1 k
.

– An inductor with L C = 33 nH and a parallel resonant frequency of approximately

1.9 GHz (C ≈ 0, 2 pF) is used for the inductive decoupling of the collector.

– A resistor R LC = 100
is placed in series with L C so that at frequencies below the

operating frequency it causes losses which increase the k factor in the frequency range

100 MHz 1.8 GHz (see Fig 27.10) This reduces the tendency to oscillate in this

frequency range

– For capacitive blocking at the operating frequency, the capacitors C B1 and C C1, whose

series resonant frequency is approximately 1.9 GHz, are used (C = 4.7 pF, size 0604:

L ≈ 1.5 nH).

4The data sheet also specifies that the maximum transit frequency is reached with I C = 20 mA

so that I C = 5 mA is not optimum However, one should be careful, since the transit frequency

is measured with the output short-circuited, allowing only limited conclusions to be drawn asregards to the power gain that can be achieved with impedance matching on both sides In another

design, conducted in parallel to this one, for I C= 20 mA a power gain was achieved that was a

mere 0.2 dB greater, a value which does not warrant the bias quiescent current, especially since

the noise figure increases significantly

2.8 V 2.1 V

0.73 V

0.66 V

1 kΩ 6.8 kΩ 2.7 kΩ 4.Z pF

Fig 27.10.k factor for the

circuit shown in Fig 27.9

– An additional capacitor C C2 with a higher capacitance is placed in parallel to C C1 inorder to improve the capacitive blocking effect at low frequencies

– Capacitor C B2determines the cut-off frequency of the operating point control and fore has a relatively high capacitance

there-The S parameters of the transistor with operating point setting are determined by circuitsimulation:5

S11 = − 0.3223 + j 0.2527 , S12 = 0.1428 + j 0.1833

S21 = 1.178 + j 1.3254 , S22 = 0.09015 − j 0.249

5In this simulation, the high-frequency equivalent circuits of resistors and capacitors were takeninto consideration Nevertheless, the results of the simulation cannot be used for a real circuitdesign since the simulation model for transistor BFR93 provided by the manufacturer is notaccurate enough for this frequency range In practice, the S parameters of the transistor, includingthe network for setting the operating point, must be measured with a network analyser In this

example we use the S parameters from the simulation so that it can repeated with PSpice.

With (27.8) it follows that k = 1.05 > 1, i.e impedance matching on both sides is possible The power gain to be expected is obtained with (27.12): MAG = 5.57 ≈ 7.5 dB.

Equations (27.6) and (27.7) lead to:

r g,m = − 0.6475 − j 0.402 , r L,m = 0.3791 + j 0.6

Then, using (27.10) and (27.11) we can calculate the input and output impedances of thetransistor with operating point setting in the matched condition:

Z 1,m = (7.3 + j 14)
, Z2,m = (33 − j 80)

For both impedances, the real part is smaller than Z W = 50
so that matching requires a

step-up transformation according to Fig 26.21a on page 1342

For matching at the input side we obtain from (26.25) with R = 7.3
and X = 14
:

X1 = ± 20.7
, X2 = ∓ 17.7
− 14

We select the high-pass filter characteristic (X1 > 0, X2< 0) according to Fig 26.22b on page 1343, because then the series capacitance C2can simultaneously serve as a couplingcapacitor From

X1 = 20.7
, X2 = − 31.7

it follows with (26.26) that:

L 1,i = 1.75 nH , C2,i = 2.65 pF

The additional index i refers to the input side matching.

For matching at the output side we obtain from (26.25) with R = 33
and X =

− 80
:

X1 = ± 70
, X2 = ∓ 24
+ 80

We now select the low-pass filter characteristic (X1 < 0, X2 > 0) according to Fig 26.22a

on page 1343 so that the overall characteristic is that of a band-pass filter From

capac-of 1.9 GHz which, at the operating frequency f C = 1.88 GHz, it acts as a short-circuit and

thus has no influence on the matching effect

Figure 27.11 shows the amplifier with the two matching networks The elements ofthe matching networks are ideal; at this stage the design is not ready for practical use It isnecessary to check at which points inductors and capacitors can be connected and wherestrip lines may be advantageous or are mandatory for functionality of the elements This isnot discussed any further; please refer to the notes on impedance matching in multi-stageamplifiers in the next section

Finally we present the results achieved The upper part of Fig 27.12 shows themagnitudes of the S parameters in the matched amplifier at the operating frequency

f C = 1.88 GHz One can see that matching covers a relatively narrow frequency band.

If the requirements|S11| < 0.1 and |S22| < 0.1 hold for the reflection factors, then the

bandwidth is approximately 53 MHz Matching at the input covers a narrower band than

2.8 V 2.1 V

0.73 V

0.66 V

1 kΩ 6.8 kΩ 2.7 kΩ 4.7 pF

Fig 27.11.Amplifier with matching networks

at the output since the transformation factor for the real part of the impedance is higher:

7.3
→ 50
at the input compared to 33
→ 50
at the output In the centre of

Fig 27.12 the magnitudes of the S parameters are plotted over a wider range This showsthat the output is also nearly matched (|S22| ≈ 0.1) in the range around 600 MHz Theposition of this range depends on the capacitance of the coupling capacitor at the output,which can be used for adjustment This can be useful when the amplifier is followed by

a mixer for conversion to a low intermediate frequency A suitable choice of the couplingcapacitor can also provide a sufficient matching for the intermediate frequency This indi-cates that high-frequency circuit engineering often takes advantage of secondary effects.The bottom diagram of Fig 27.12 shows the gain in decibel At the operating frequency it

reaches its maximum, which we have calculated with (27.12): MAG ≈ 7.5 dB The gain

is comparatively low as the transistor type BFR93 has a transit frequency of only 5 GHzand is operated in our example at its performance limit Modern circuits for the frequencyrange around 2 GHz use transistors with transit frequencies of about 25 GHz, resulting in

gains of 20 25 dB.

27.2.4

Impedance Matching in Multi-stage Amplifiers

Matching in multi-stage amplifiers is done in the same way as in single-stage amplifiers.Each stage is matched at both sides and then arranged in series, where the matchingnetworks between the stages can often be simplified by combining the elements In mostcases, however, this is not the optimum procedure In practice, it is used only if, forconstruction purposes, the stages are so far apart that the connections between the stages can

no longer be considered as electrically short lines as is especially the case in the GHz range

In all other cases the output of each stage is matched directly to the input of thenext stage The calculation of this type of impedance matching is complicated since an

2 1.5

1.5 1

1 0.5

0.5 0

Fig 27.12.S parameters of the amplifier in Fig 27.11

amplifier with n stages, including n+ 1 matching networks (input side, output side and

n − 1 networks between the stages), are interdependent owing to the reverse transmission

of the transistors The procedure is divided into two steps:

– In the first step, structures must be selected that, in principle, allow impedance matching

on the basis of the S parameters of the individual transistors This must include all wiringthat is required for construction, i.e the PC board layout of the amplifier must be roughlyoutlined

– In the second step, the values of the elements in the individual structures must be

de-termined by means of a simulation program For this purpose, iterative optimisation

methods (optimisers) are used to find the ideal dimensions with regard to the

crite-ria specified by the user Often these critecrite-ria include maximising|S21| observing the

secondary conditions|S11| < 0.1 and |S22 | < 0.1 in the specified frequency range.

If the reverse transmission of the transistors is not very high, the first run may alreadyprovide a satisfactory result Otherwise the structures must be varied before further runsare carried out These may become necessary solely because the established element valuescannot be achieved or arranged on the predetermined layout of the PC board

In practice, this procedure is also used for single-stage amplifiers The ideal matchingnetworks can, of course, be calculated directly by following the procedure described inthe previous section, but practical operation on the basis of the properties of the realcomponents and the PC board layout require additional computer-aided optimisation

Impedance Matching with Series Inductance

For high-frequency bipolar transistors with a transit frequency above 10 GHz, the tances of the actual transistor are so low that the input and output capacitances are formed

capaci-by the parasitic capacitance of the case The equivalent circuit for these transistors with case

capacitances C BE and C CE and case inductances L B , L C and L Eis shown in Fig 27.13a

where the relationships are C BE > C CE > C C and L B ≈ L C > L E The equivalentcircuit can be simplified owing to the component dimensions When using the simplifiedequivalent circuit for a multi-stage amplifier as shown in Fig 27.13b, the circuitry betweeneach of the stages represents a Collins filter The capacitances of the filter are formed bythe capacitances of the transistor and the inductances of the filter by the series connec-tion of the case inductances and an external inductance Therefore, if the dimensions arefavourable, matching between the stages can be achieved with a series inductance Simi-larly, the parasitic elements of the transistors at the input and output of the amplifier can

be integrated into a Collins filter

27.2.5

Neutralisation

The main obstacle in impedance matching is the reverse transmission of the transistors

which reduces the stability factor k and prevents matching on both sides if k < 1 For

a transistor without reverse transmission S12 = 0 and k → ∞ holds, so both sides can

be matched provided the real parts of the input and output impedances are positive, i.e

|S11| < 1 and |S22 | < 1 A transistor without reverse transmission operates unilaterally

which means that signal transmission takes place only in the forward direction

Circuits for Neutralisation

The reverse transmission is caused by the collector-base capacitance C C in bipolar

tran-sistors and by the gate-drain capacitance C GDin FETs It can be eliminated by connecting

a neutralisation capacitance C n of the same value between the base and a point in thecircuit that carries the inverted small-signal voltage of the collector Such a point is created

by using an inductor with centre tap for decoupling the collector and connecting this tap

a Simplified equivalent circuit of a bipolar transistor in common-emitter configuration

b Simplified equivalent circuit of a two-stage amplifier with matching circuitry

Simplified equivalent circuit

Collins filter 3 Collins filter 2

in-arrangement which can then be neutralised by cross-coupling with two capacitances C n1

and C n2(see Fig 27.15) Neutralisation of a differential amplifier according to Fig 27.16

is based on the same principle