Wiley Fundamentals Of RF Circuit Design With Low Noise Oscillators Jan 2001 ISBN 0471497932 pdf

  Fundamentals of RF Circuit Design with Low Noise Oscillators. Jeremy Everard Copyright © 2001 John Wiley & Sons Ltd

  ISBNs: 0-471-49793-2 (Hardback); 0-470-84175-3 (Electronic)

1 Transistor and Component

  Models at Low and High Frequencies

1.1 Introduction

  Equivalent circuit device models are critical for the accurate design and modelling of RF components including transistors, diodes, resistors, capacitors and inductors. This chapter will begin with the bipolar transistor starting with the basic T and then the model at low frequencies and then show how this can be extended for use at

  π

  high frequencies. These models should be as simple as possible to enable a clear understanding of the operation of the circuit and allow easy analysis. They should then be extendible to include the parasitic components to enable accurate optimisation. Note that knowledge of both the T and models enables regular

  π

  switching between them for easier circuit manipulation. It also offers improved insight.

  As an example S for a bipolar transistor, with an f of 5GHz, will be calculated _{21 T} and compared with the data sheet values at quiescent currents of 1 and 10mA. The effect of incorporating additional components such as the base spreading resistance and the emitter contact resistance will be shown demonstrating accuracies of a few per cent.

  The harmonic and third order intermodulation distortion will then be derived for common emitter and differential amplifiers showing the removal of even order terms during differential operation.

  2 Fundamentals of RF Circuit Design It should be noted that this chapter will use certain parameter definitions which will be explained as we progress. The full definitions will be shown in Chapter 2. Techniques for equivalent circuit component extraction are also included in Chapter 2.

1.2 Transistor Models at Low Frequencies

  1.2.1 ‘T’ Model

  Considerable insight can be gained by starting with the simplest T model as it most closely resembles the actual device as shown in Figure 1.1. Starting from a basic NPN transistor structure with a narrow base region, Figure 1.1a, the first step is to go to the model where the base emitter junction is replaced with a forward biased diode.

  The emitter current is set by the base emitter junction voltage The base collector junction current source is effectively in parallel with a reverse biased diode and this diode is therefore ignored for this simple model. Due to the thin base region, the collector current tracks the emitter current, differing only by the base current, where it will be assumed that the current gain, remains effectively

  β , constant.

  C C C i

  β

  i b

  β _b N

  i _b i _b

  P

  B B B

  N

  r _e E E E

  (a) (b) (c)

  Figure 1.1

  Low frequency ‘T’ model for a bipolar transistor Note that considerable insight into the large signal behaviour of bipolar Transistor and Component Models at Low and High Frequencies 3 distortion in a common emitter and differential amplifier. Here, however, we will concentrate on the low frequency small signal AC ‘T’ model which takes into account the DC bias current, which is shown in Figure 1.1c. Here r is the AC _e resistance of the forward biased base emitter junction. The transistor is therefore modelled by an emitter resistor r and a controlled _e current source. If a base current, i , is applied to the base of the device a collector _b current of i flows through the collector current source. The emitter current, I , is

  β _{b E}

  therefore (1+ )i . The AC resistance of r is obtained from the differential of the

  β _{b e}

  diode equation. The diode equation is:

  eV    

  I I exp 1 (1.1) = −

  E ES   kT

      _-13

  where I is the emitter saturation current which is typically around 10 , e is the _ES charge on the electron, V is the base emitter voltage, V , k is Boltzmann’s constant _be and T is the temperature in Kelvin. Some authors define the emitter current, I , as _E the collector current I . This just depends on the approximation applied to the _C original model and makes very little difference to the calculations. Throughout this book equation (1) will be used to define the emitter current.

  Note that the minus one in equation (1.1) can be ignored as I is so small. The _ES AC admittance of r is therefore: _e

  dI e eV   exp

  (1.2)

  = ES I   dV kT kT

   

  Therefore:

  dI e

  I

  (1.3)

  = dV kT

  The AC impedance is therefore:

  dV kT

  1 .

  (1.4)

  = dI e I

_{-23 o}

  As k = 1.38 10 , T is room temperature (around 20

  C) = 293K and e is

• _-19 ×

  4 Fundamentals of RF Circuit Design

  dV

  25 r

  (1.5)

  = ≈ e dI I mA

  This means that the AC resistance of r is inversely proportional to the emitter _e current. This is a very useful formula and should therefore be committed to memory. The value of r for some typical values of currents is therefore: _e 1mA ≈

  25 Ω

  10mA ≈ 2 5 . Ω 25mA

  1 ≈ Ω

  It would now be useful to calculate the voltage gain and the input impedance of the transistor at low frequencies and then introduce the more common π model. If we take a common emitter amplifier as shown in Figure 1.2 then the input voltage across the base emitter is:

  1 i . r

  (1.6)

  = β in b e

• V

  ( )

  C β i _b i _b B

  R L E Figure 1.2

  A common emitter amplifier The input impedance is therefore: Transistor and Component Models at Low and High Frequencies 5

  1 r β

• V i

  25

  Z 1 r 1 (1.7) = = = β + = β in ( ) ( ) e i i

• in b ( ) e

  I b b mA

  The forward transconductance, g , is: _m

  I i β

  1 out b g

  (1.8)

  = = ≈ m

  β ( )

  1 r r in b e e

• V i

  Therefore:

  1 g

  (1.9)

  ≈ m r e

  and:

  V R out L g R

  (1.10)

  = − = − m L

  V r in e Note that the negative sign is due to the signal inversion.

  Thus the voltage gain increases with current and is therefore equal to the ratio of load impedance to r . Note also that the input impedance increases with current _e gain and decreases with increasing current.

  In common emitter amplifiers, an external emitter resistor, R , is often added to _e apply negative feedback. The voltage gain would then become:

  V R out L

  (1.11)

  =

• V r R

  in e e

  Note also that part or all of this external emitter resistor is often decoupled and this part would then not affect the AC gain but allows the biasing voltage and current to be set more accurately. For the higher RF/microwave frequencies it is often preferable to ground the emitter directly and this is discussed at the end of Chapter 3 under DC biasing.

  6 Fundamentals of RF Circuit Design

1.2.2 The π Transistor Model

  The ‘T’ model can now be transformed to the π model as shown in Figure 1.3. In the π model, which is a fully equivalent and therefore interchangeable circuit, the input impedance is now shown as ( β +1)r and the output current source remains _e the same. Another format for the π model could represent the current source as a voltage controlled current source of value g _m V . The input resistance is often called ₁ r π

  .

  C

i b

B C

  β i _b i b V β i o r g V b m ¹ ( β+1) r e ¹ B r e

  E E E Figure 1.3

  T to π model transformation At this point the base spreading resistance r should be included as this _bb’ incorporates the resistance of the long thin base region. This typically ranges from around 10 to 100 for low power discrete devices. The node interconnecting r π

  Ω and r is called b’. _bb’

1.3 Models at High Frequencies

  As the frequency of operation increases the model should include the reactances of both the internal device and the package as well as including charge storage and transit time effects. Over the RF range these aspects can be modelled effectively using resistors, capacitors and inductors. The hybrid π transistor model was therefore developed as shown in Figure 1.4. The forward biased base emitter junction and the reverse biased collector base junction both have capacitances and these are added to the model. The major components here are therefore the input capacitance C or C and the feedback capacitance C or C . Both sets of symbols _{b’e π b’c µ} are used as both appear in data sheets and books.

  Transistor and Component Models at Low and High Frequencies 7

  C b 'c

  I r b b'

  1

b

i b

  B C i rb 'e o r g V m ¹ V r b'e C b 'e β

1 E

  E

Figure 1.4 Hybrid model

  π

  A more complete model including the package characteristics is shown in

Figure 1.5. The typical package model parameters for a SOT 143 package is shown in Figure 1.6. It is, however, rather difficult to analyse the full model shown in

  Figures 1.5 and 1.6 although these types of model are very useful for computer aided optimisation.

  Figure 1.5

  Hybrid π model including package components

  8 Fundamentals of RF Circuit Design

Figure 1.6 . Typical model for the SOT143 package. Obtained from the SPICE model for a

  BFG505. Data in Philips RF Wideband Transistors CD, Product Selection 2000 Discrete Semiconductors.

  We should therefore revert to the model for the internal active device for analysis, as shown in Figure 1.4, and introduce some figures of merit for the device such as f and f . It will be shown that these figures of merit offer significant

  β T

  information but ignore other aspects. It is actually rather difficult to find single figures of merit which accurately quantify performance and therefore many are used in RF and microwave design work. However, it will be shown later how the S parameters can be obtained from knowledge of f . _T

  It is worth calculating the short circuit current gain h for this model shown in ₂₁ Figure 1.4. The full definitions for the h, y and S parameters are given in Chapter 2. h is the ratio of the current flowing out of port 2 into a short circuit load to the ₂₁ input current into port 1.

  c

  I

  (1.12)

  h 21 = b

  I The proportion of base current, i , flowing into the base resistance, r , is therefore: _{b b’e} Transistor and Component Models at Low and High Frequencies 9

  1 i

  ⋅ b r i b ' e b i

  (1.13)

  = =

  1

  1 j ω CR j ω C C

• rb ' e

  ' ' + + ( b e b c ) r b ' e

  where the input and feedback capacitors add in parallel to produce C and the r _b’e becomes R. The collector current is I = β i , where we assume that the current _{C rb’e} through the feedback capacitor can be neglected as I << β i . Therefore: _{Cb’c rb’e}

  h I fe β c

  (1.14)

  h 21 = = = i SCR

1 SCR

  1

  b

  Note that β and h are both symbols used to describe the low frequency current _fe gain.

  A plot of h versus frequency is shown in Figure 1.7. Here it can be seen that ₂₁ the gain is constant and then rolls off at 6dB per octave. The transition frequency f _T occurs when the modulus of the short circuit current gain is 1. Also shown on the graph, is a trace of h that would be measured in a typical device. This change in ₂₁ response is caused by the other parasitic elements in the device and package. f is _T therefore obtained by measuring h at a frequency of around f /10 and then _{21 T} extrapolating the curve to the unity gain point. The frequency from which this extrapolation occurs is usually given in data sheets. h

  fe

  f (3dB p oint)

  β

  M e asure f for h extrapo lation

  T 2 1

  A f w h e n h = 1

  T 2 1

  F re qu ency A ctual device m easurem ent

2 CR

  1

  (1.16) As f _T is defined as the point at which

  10 Fundamentals of RF Circuit Design The 3 dB point occurs when ω CR = 1. Therefore:

  f CR β

  π =

  1

  f =

  1 2 π β

  (1.15) and h ₂₁ can also be expressed as:

  21

• =

  β f f j h h fe

• β β
• = =

  (1.19) As:

  2

  2 − = 

     

     fe T h f f

  β

  ( )

  ( )

  1

  2 >> fe h

  (1.20)

  CR h f h f fe T fe

  β .

  = =

  (1.21)

  1

    

  (1.18)

  1    

  

21

  1

h =

  , then:

  2

  1

  1

     

  f f h f f j h

  T fe T fe

• β

  ( ) fe T h f f

  2

  2

  1 = 

     

  (1.17) Transistor and Component Models at Low and High Frequencies 11 Note also that:

  f T f

  (1.22)

  = β h fe

  As:

  h fe h

  (1.16)

  =

  21 f

• 1 j

  f β

  it can also be expressed in terms of f : _T

  h fe h

  (1.23)

  =

  21 h . f fe

• 1 j

  f T

  Take a typical example of a modern RF transistor with the following parameters:

  

f = 5 GHz and h = 100. The 3dB point for h when placed directly in a common

_{T fe 21} emitter circuit is f = 50MHz.

  β

  Further information can also be gained from knowledge of the operating current. For example, in many devices, the maximum value of f occurs at currents _T of around 10mA. For these devices (still assuming the same f and h ) r = 2.5 , _{T fe e} Ω therefore r 250 and hence C _{b’e T} ≈ Ω ≈ 10pF with the feedback component of this being around 0.5 to 1pF.

  For lower current devices operating at 1mA (typical for the BFT25) r is now _e around 25 , r around 2,500 and therefore C is a few pF with C 0.2pF.

  Ω Ω _{b’e T b’e ≈}

  Note, in fact, that these calculations for C are actually almost independent of _T and only dependent on I , r or g as the calculations can be done in a different

  h _{fe C e m}

  way. For example:

  CR

  1 2 f h 2 f (1.24) = π = π

  β fe T

  Therefore:

  12 Fundamentals of RF Circuit Design

  h g fe

  1

m

  C

  (1.25)

  = ≈ = 2 f h π + 1 r 2 π f r 2 π f

  T fe e T e T ( )

  Many of the parameters of a modern device can therefore be deduced just from f , _T , I and the feedback capacitance with the use of these fairly simple models.

  h _{fe c}

1.3.1 Miller Effect

  is a commonly used figure of merit and is quoted in most data sheets. It is now

  f _T worth discussing f in detail to find out what other information is available. _T

  1. What does it hide? Any output components as there is a short circuit on the output.

  2. What does it ignore? The effects of the load impedance and in particular the Miller effect. (It does include the effect of the feedback capacitor but only into a short circuit load.)

  It is important therefore to investigate the effect of the feedback capacitor when a load resistance R is placed at the output. Initially we will introduce a further _L simple model.

  If we take the simple model shown in Figure 1.8, which consists of an inverting voltage amplifier with a capacitive feedback network, then this can be identically modelled as a voltage amplifier with a larger input capacitor as shown in Figure 1.8b. The effect on the output can be ignored, in this case, because the amplifier has zero output impedance.

  Figure 1.8a

  Amplifier with feedback C Figure 1.8.b Amplifier with increased input C Transistor and Component Models at Low and High Frequencies 13

  V V

  V

  (1.26)

  = − c in out

  ( )

  If the voltage gain of the amplifier is -G then the voltage across the capacitor is therefore:

  V V GV

  V

1 G (1.27)

  = = + + c ( in in ) in ( )

  The current through the capacitor, I , is therefore I = V j ω _{c C c}

  C. The change in input

  admittance caused by this capacitor is therefore:

  V

  1 G j C ω ( ) ω c c in

• I V j C

  1 G j C (1.28) = = = + ( ) ω

  V V

  V in in in

  The capacitor in the feedback circuit can therefore be replaced by an input capacitor of value (1 + G)C. This is most easily illustrated with an example. Suppose a 1V sinewave was applied to the input of an amplifier with an inverting gain of 5. The output voltage would swing to –5V when the input was +1V therefore producing 6V (1 + G) across the capacitor. The current flowing into the capacitor is therefore six times higher than it would be if the same capacitor was on the input. The capacitor can therefore be transferred to the input by making it six times larger.

1.3.2 Generalised ‘Miller Effect’

  Note that it is worth generalising the ‘Miller effect’ by replacing the feedback component by an arbitrary impedance Z as shown in Figure 1.8c and then investigating the effect of making Z a resistor or inductor. This will also be useful when looking at broadband amplifiers in Chapter 3 where the feedback resistor can be used to set both the input and output impedance as well as the gain. It is also worth investigating the effect of changing the sign of the gain.

  Z

• G Z

  V _in

  V _{o u t} (1 + G ) V _{in o u t}

  V

• G

  14 Fundamentals of RF Circuit Design As before:

1 V

  (1.29)

  = Z ( ) in

• V G

  As:

  V Z I (1.30) =

  Z Z

  the new input impedance is now:

  V Z in

  (1.31)

  =

  1 I G Z ( )

• as shown in Figure 1.8d. If Z is now a resistor, R, then the input impedance becomes:

  R

  (1.32)

• 1 G

  ( )

  This information will be used later when discussing the design of broadband amplifiers.

  If Z is an inductor, L, then the input impedance becomes:

  j ω L

  (1.33)

1 G

• As before, if Z was a capacitor then the input impedance becomes:

  ( )

  j −

  (1.34)

  C G ( )

  ω

• 1

  If the gain is set to be positive and the feedback impedance is a resistor then the input impedance would be:

  Transistor and Component Models at Low and High Frequencies 15

  R

  (1.35)

  1 − G

  ( ) which produces a negative resistance when G is greater than one.

1.3.3 Hybrid π Model

  It is now worth applying the Miller effect to the hybrid π model where for convenience we make the current source voltage dependent as shown in Figure

  1.9. C _{b 'c}

  I r _{b b'}

  1 b i _b B

  C r β i rb 'e o r g V m ¹ V b'e C b 'e

  1 E E

Figure 1.9 Hybrid model for calculation of Miller capacitance

  π

  Firstly apply the Miller technique to this model. As before it is necessary to calculate the input impedance caused by C . The current flowing into the collector _b’c load, R , is: _L

  I g

  V I c = m

  1

• (1.36)

  1

  where the current through C is: _b’c (1.37)

  I V

  V C . j ω 1 = 1 − b ' c

  ( )

  The feedforward current I through the feedback capacitor C is usually small _{1 b’c} compared to the current g _m V and therefore: ₁

  16 Fundamentals of RF Circuit Design Therefore:

  I V g R

  V C . j (1.39) = ω

• 1

  (

  1 m L 1 ) b ' c

  V 1 g R C . j (1.40) = ω

• I

  1 1 ( m L ) b ' c

  The input admittance caused by C is: _b’c

  I

  

1 g R C . j (1.41)

= ω
• 1

  ( m L ) b ' c

  V

  1

  which is equivalent to replacing C with a shunt capacitor in parallel with C to _{b’c b’e} produce a total capacitance:

  C

1 g R C C (1.42)

= + +

  T ( m L ) b ' c b ' e

  The model generated using the Miller effect is shown in Figure 1.10. Note that this model is an approximation in this case, as it is only effective for calculating the forward transmission and the input impedance. It is not useful for calculating the output impedance or the reverse transmission or stability. This is only because of the approximation used when deriving the output voltage. If the load is zero then the current gain would be as derived when h was calculated. ₂₁

  i r _{b b b '} R s B

  C i o r g V β rb 'e m ¹ V C _T in r _{b e} E

  E

Figure 1.10 Hybrid model incorporating Miller capacitor

  π

  It is now worth calculating the voltage gain for this new model into a load R to _L observe the break point as this capacitance degrades the frequency response. This will then be converted to S parameters using techniques discussed in Chapter 2 on

• =
• = ω
• = ω
• =
•    

• +

  +    
• =

  ' ' ' ' '

  1

  (1.45) As:

  a b c a b c b

  1

  (1.46)

  

( )

in s bb e b e b s bb T s bb e b e b

  V R r r R r r C j R r r r

       

  V      

   

  

' '

' ' ' ' '

  1

  1

  1 ω

  (1.47) As:

  V L m out − =

  (1.48) the voltage gain is therefore:

   

  V C j r R r R r r r

  V    

  1

  Transistor and Component Models at Low and High Frequencies 17

  in s bb T e b e b

  T e b e b

  V R r C j r r C j r r

  V      

       

  ' ' ' '

  '

  1

  ( )( ) in T e b s bb s bb e b e b

  1 ω ω

  (1.43)

  ( )( )

in

T e b s bb e b e b

  

V

C j r R r r r    

   

  ' ' ' '

  1

  (1.44) Expanding the denominator:

1 V R g

  18 Fundamentals of RF Circuit Design

      V  r 

  1 ' out b e g R

  (1.49)

  = −   ( m L )

    ( )

• r R r V r r R bb ' s b ' e

      1 j C

• +

  in b ' e bb ' s

  ω T

   r r R 

  b ' e bb ' s  

  Note that:

     

  1

  

1

  (1.50) =

  r R r

  1

  ( )

    T 1 j C

• bb ' s b ' e j ω C R

  ω T

    r r R

  b ' e bb ' s  

  where:

  bb ' s b ' e ( )

• r R r

  R

  (1.51)

  = r r R

  b ' e bb ' s

  This is effectively r in series with R all in parallel with r which is the effective _{bb’ S b’e} Thévenin equivalent, total source resistance seen by the capacitor. The first two brackets of equation (1.49) show the DC voltage gain and the third bracket describes the roll-off where:

  C

1 g R C C (1.52)

= + +

  T ( m L ) b ' c b ' e

  The numerator of the third bracket produces the 3dB point when the imaginary part is equal to one.

• r R r

  ( bb ' s ) b ' e

  2 f C 1 (1.53) π [ ] =

  3 dB T r r R

  b ' e bb ' s

  The full equation is:

• +

  r R r

  (

  

bb ' s ) b ' e

2 f 1 g R C C 1 (1.54)

  π [ ] = + + 3 dB ( ) m L b ' c b ' e Transistor and Component Models at Low and High Frequencies 19 Therefore from equation (1.53):

  1 f

  (1.55)

  =

  r R r ( bb ' s ) b ' e

• 3 dB

2 C

  π [ ] T r r R b ' e bb ' s + + r r R

  b ' e bb ' s f

  (1.56)

  = 3 dB

  2 π

  [ ] T bb ' s b ' e ( )

• C r R r

  As:

  C

1 g R C C (1.52)

= + +

  T [ ( m L ) b ' c b ' e ] r r R

  b ' e bb ' s f

  (1.57)

  = 3 dB

  2 π 1 g R C C r R r [ m L b ' c b ' e ] bb ' s b ' e + + +

  ( ) ( )

  Note that using equation (1.51) for R:

  1 f

  (1.58)

  =

  dB

  1 π g R C C R

  [ m L b ' c b ' e ] ( )

• 2
• 3

  S Parameter Equations

1.4 This equation describing the voltage gain can now be converted to 50 Ω S

  parameters by making R = R = 50 Ω and calculating S as the value of V when V _{S L 21 out in} is set to 2V. The equation and explanation for this are given in Chapter 2. Equation (1.49) therefore becomes:

       r 

  1 b ' e

  S 2 g .

  50

  (1.59)

  =   21 m

  ( )  

  50 r r r 50 ( bb ' ) b ' e b ' e bb ' + +

• r

     

• +

  1 j C

  ω

   

  r r b ' e bb '

• T
• 50

   

  20 Fundamentals of RF Circuit Design

1.5 Example Calculations of S

  21 It is now worth inserting some typical values, similar to those used when h was ₂₁ investigated, to obtain S . Further it will be interesting to note the added effect ₂₁ caused by the feedback capacitor. Take two typical examples of modern RF

  transistors both with an f of 5GHz where one transistor is designed to operate at _T 10mA and the other at 1mA. The calculations will then be compared with theory in graphical form.

1.5.1 Medium Current RF Transistor – 10mA Assume that f = 5GHz, h = 100, I =10mA, and the feedback capacitor, C ≈ 2pF.

  _{T fe c b’c} Therefore r 2 .

  5 and r 250 . Let r 10 for a typical 10mA = Ω ≈ Ω ≈ Ω e b 'e bb '

  device. The 3dB point for h (short circuit current gain) when placed directly in a ₂₁ common emitter circuit is f = f /h . Therefore f = 50MHz.

  β T fe β

  As:

1 CR

  (1.24)

  = f

  2 π β

  −

  9

  18

  10 = × As h is the short circuit current gain C and C are effectively in parallel. _{21 b’c b’e}

  then CR 3 .

  Therefore:

  6 pF (1.61) = = b ' e b ' c

• C C C 12 .

  As

  C 1 pF C 12 .

  6

  1 11 . 6 pF (1.62) ≈ = − = b ' c b ' e

  However, C for the measurement of S parameters includes the Miller effect _T because the load impedance is not zero. Thus:

   R 

L Transistor and Component Models at Low and High Frequencies 21

  C

  1

  20

  1 11 .

  6 32 . 6 pF (1.64) = = + +

  T ( )

  As:

  r r R

  b ' e bb ' s f (1.65)

  =

  2 π C r R r [ ] T bb ' s b ' e

• 3 dB

  ( )

  then:

  250

  10

  50

  f = 101MHz (1.66) =

  −

  12

  2 32 .

  6

  10

  10 50 250 π ×

• 3 dB

  [ ] ( )

  The value of S at low frequencies is therefore: ₂₁

   r  b ' e

  S 2 g R

  (1.67)

  = 21 m L ) (

    r r + +

  50 b ' e bb '

    250

    S

  2

  20 32 . 25 (1.68) = =

  21 ( ) 250

  10

  50

•  

  A further modification which will give a more accurate answer is to modify r to _e include the dynamic diode resistance as before but now to include an internal emitter fixed resistance of around 1 Ω which is fairly typical for this type of device. This would then make r = 3.5 Ω and r = 350 Ω. For the same f and the same _{e b’e T} feedback capacitance the calculations can be modified to obtain:

  9 pF As C 1 pF and C

  9

  1 8 pF (1.69) = = ≈ = − = b ' e b ' c b ' c b ' e

• C C C

   R  L

  (1.70)

  C =

1 C C =

  1 14 .

  

3

  1 + + + + 8 = 23 . 3 pF T b ' c b ' e ( ) r e

   

  As:

  22 Fundamentals of RF Circuit Design then:

  350 10 50

  f = 133MHz (1.71) =

  2 23 3 . E 12 10 50 350 π −

• 3 dB

  [ ] ( )

  The low frequency S is therefore: ₂₁

   r  b ' e

  S 2 . g .

  50

  (1.72)

  = 21 ( m )

   

  50 r r b ' e bb ' + +

    350

    S 2 .

  14 .

  3 24 . 4 (1.73) = =

  10

  50  

• 21 ( )
• 350

  This produces an even more accurate answer. These values are fairly typical for a transistor of this kind, e.g. the BFR92A. This is illustrated in Figure 1.11 where the dotted line is the calculation and the discrete points are measured S parameter data.

  Note that the value for the base spreading resistance and the emitter contact resistance can be obtained from the SPICE model for the device where the base series resistance is RB and the emitter series resistance is RE. ₁₀₀

10 S

  21 ₁

  0.1 _{3 4 10 100 1} . . _{10 1 10} Frequency, MHz Transistor and Component Models at Low and High Frequencies 23

1.5.2 Lower Current Device - 1mA

  If we now take a lower current device with the parameters f = 5GHz, h = 100, _{T fe}

  I =1mA, and the feedback capacitor, C 0.2pF , r _{c b’c ≈ = Ω ≈ Ω} 25 and r 2500 . e b ' e

  Let r for a typical 1mA device. The 3dB point for h (short circuit _bb’ ≈ 100Ω ²¹ current gain) when placed directly in a common emitter circuit is:

  f T f

  (1.22)

  = β hfe

  Therefore f = 50MHz

  β h fe f h . f

  (1.21)

  = = T fe β

  2 π CR

  Note also that:

1 CR

  (1.74)

  = 2 f

  π β

  9 −

  18

  10 = × As h is the short circuit current gain, C and C are effectively in parallel. _{21 b’c b’e}

  Therefore CR 3 .

  Therefore:

  26 pF (1.75) = = b ' e b ' c As C . 2 pF

• C C C 1 .

  ≈ b ' c

  C 1 .

  26 .

  2 1 . 06 pF (1.76) = − = b ' e

  However, C for the measurement of S parameters includes the Miller effect _T because the load impedance is not zero. Therefore:

  24 Fundamentals of RF Circuit Design

  C

  1 2 .

  2 1 .

  06 1 . 66 pF (1.78) = = + +

  T ( )

  As:

  r r R

  b ' e bb ' s f

  (1.65)

  =

  2 π C r R r [ ] T bb ' s b ' e

• 3 dB

  ( )

  then:

  2500 100

  50

  f = 678MHz (1.79) =

  3 dB −

  12

  2 1 .

  66 10 100 50 2500 π ×

• The low frequency value for S is therefore:
₂₁

  ( ) [ ]

   r  b ' e

  S 2 g .

  50

  (1.72)

  = 21 m ( )

    r r + +

  50 b ' e bb '

    2500

    S

  2

  2 3 . 77 (1.80) = =

  21 ( ) 2500 100

  50

•  

  As before an internal emitter fixed resistance can be incorporated. For these lower current smaller devices a figure of around 8 Ω is fairly typical. This would then make r = 33 Ω and r = 3300 Ω. For the same f and the same feedback capacitance _{e b’e T} the calculations can be modified to obtain:

  96 pF (1.81) = = b ' e b ' c As: C . 2 pF

• C C C .

  ≈ b ' c C . 96 . 2 . 76 pF (1.82)

  = − = b ' e

   R  L

  (1.83)

  C =

1 C C =

  1 1 .

52 .

2 . + + + + 76 = 1 . 26 pF T b ' c b ' e ( ) r e

    Transistor and Component Models at Low and High Frequencies 25 As:

  r r R

  b ' e bb ' s f (1.65)

  = 3 dB

  2 π C r R r [ ] + T bb ' s b ' e

  ( )

  then:

  3300 100

  50

  f = 880MHz =

  −

  12

  2 1 .

  26 10 100 50 3300 π ×

• 3 dB

  ( ) [ ]

  The low frequency S is therefore: ₂₁

   r  ' b e

  S 2 g .