# Refinements to Incremental Transistor Model

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1 Refinements to Incremental Transistor Model This section presents modifications to the incremental models that account for non-ideal transistor behavior Incremental output port resistance

2 Incremental changes in the output port voltage cause incremental changes in the output port current r ce and r ds are commonly referred to as r o Typical values 20 kω kω

3 Input Port Resistances Accounts for the effects of any ohmic resistance that appears in series with the input port terminal Effects are particularly important in high frequency and high power devices Input port resistance can be taken into account by adding a resistance r x to the input port

4 Note: The incremental models do not take into account incremental capacitances which will be discussed in a later section.

5 Alternative BJT Representation Since v BE and i B are related by the input port characteristics it is possible to describe the output port in terms of v BE instead of i B g r m be δ ic δ vbe δ ic δ v δ i δ i BE B B β o

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7 Two Port Representation of Incremental Circuits

8 Time Dependent Circuit Behavior The role of time dependence and frequency response is critical in determining circuit behavior Many time-dependent properties can be modeled by the addition of capacitances to the PWL incremental model In very high speed circuits the addition of capacitances and inductances are necessary to model inter-connections between devices (parasitics)

9 Review of RC Circuits

10 Response can be determined by using the appropriate differential equations or Laplace transforms For a step function of the form v ( t) V u( t) u( t) IN o step function v OUT v ( t) V at t OUT ( t) 0 at t 0 o / v ( t) V ( 1 e ) OUT " Time Domain" o t RC In the time domain the V-I relationship for a capacitor is given by i( t) C d v ( t ) dt

11 In the frequency domain the impedence representation transforms differential equations into simple algebraic equations Voltage and current variables are expressed as complex numbers (phasors) which have both a magnitude and phase From Euler s formula cosθ + j sinθ e jθ jθ Me M cosθ + jm sinθ jθ Re( Me ) M cosθ jθ Im( Me ) M sinθ θ tan 1 jθ Re( Me ) jθ Im ( Me )

12 Acos( ω t + θ ) has frequency ω, amplitude A and phase angle θ. Ae j( ω t+ θ ) has frequency ω, amplitude and phase angle θ. A j( ω t+ θ ) jθ jω t jθ Ae Ae e Ae A θ Looking at a Capacitor i C dv dt Ie ( ) j( ω t+ θ ) j( ω t+ φ ) C dv dt Ve jω t jθ jω t jφ Ie e jω CV e e

13 jθ Ie jω CV e jφ I θ jω CV φ I jω CV V 1 jω C I Z C 1 jω C Similarly: Z R Z jω L R L If we apply an input sinusoid of the form jω t [ ] v v( t) Re Ve V cosωt IN : No phase angle

14 The RC circuit can now be analyzed using the complex impedance representation v OUT v IN Z C ZC + Z R v IN 1 1 jω C jω C + R v OUT v IN 1+ 1 jω RC v v OUT IN / 2 [ 1+ ( ωrc) ] v OUT tan 1 ωrc

15 v v OUT IN 1( 1 jωrc) ( 1+ jωrc)( 1 jωrc) 1 jωrc 1+ ω R C A( 1 jωrc)

16 The analysis can also be performed on the following circuit

17 In the frequency domain the response to a single frequency sinusoid is given by v OUT v IN Z R ZR + Z C v IN 1 R j C + R ω v OUT v IN jω RC 1+ jω RC v v OUT IN ωrc [ 1+ ( ωrc) ] 2 1 / 2

18 π vout tan 2 1 ωrc v OUT v IN jωrc 1+ jωrc ( 1 jωrc) ω R C ( 1 jωrc) 1+ ω + jωrc R C Aω R C + jaωrc

19 Bode Plot Representation For any linear circuit with a frequency dependent system function, both the magnitude and phase angle of the response are of great interest Often desirable to know these values over a wide frequency range Helpful to display information in graphical form called a Bode plot

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21 Common to express magnitude in decibels db 20log10 v v OUT IN

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24 Arbitrary System Functions Constructing a Bode plot of a complex circuit is greatly simplified if it can be expressed in the form H( jω ) jω jω jω 1+ ω 1 + ω K 2 4 A + jω jω jω 1 ω 1 + ω 1 + ω K Where ω1, ω2, ωn are the breakpoints of the system function and A is a constant The solitary jω in the numerator is not present in all system functions Each ωn in the numerator is called a zero. Each ωn in the denominator is called a pole. If the input frequency ω is well below the breakpoint frequency ωn the term associated with ωn will not affect the system transfer function jω 1+ 1 ω n jω o 1+ 0 for ω << ω ω n n

25 If the driving frequency is well above ωn the term associated with ωn will contribute a factor ω/ωn to the magnitude and an angle factor of 90 H j ω ω n ω ω + jω jω o 1 90 ω >> ω ω ω n n n n Given these guidelines the bode plot can be easily constructed. Beginning at a frequency far below the lowest breakpoint and increasing the frequency. As the frequency passes through each breakpoint its term will contribute a factor ω/ωn to the magnitude and 90 to the phase angle. If ωn is in the numerator the slope of the magnitude Bode plot shifts upward by a factor +20 db/decade. If ωn is in the denominator the slope will shift downward -20 db/decade. If ωn is in the numerator the angle will shift +90. If ωn is in the denominator the angle will shift -90.

26 At the breakpoint the phase shift will be equal to ± 45 If the solitary factor jω appears in the numerator the Bode plot will begin with a +20 db/decade slope and a phase angle of +90 at low frequencies Problem: Construct a Bode plot for the following transfer function. H( jω ) vout ( t) v IN jω jω jω jω

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28 at ω ( ) 4 7 ( )( ) ω ω > vout v ( ω )( ω ) IN db

29 Problem: Construct a Bode plot for the following function. H( jω ) vout ( jω ) 100 v jω jω IN

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32 Superposition of Poles Many applications require a constant response over a range of frequencies called the midband. The limits of the midband region may not coincide with a single pole since multiple poles can contribute to the output response Consider a system function as follows H( jω ) jω jω jω jω ( 2 x10 )

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34 H ω ω H 100ω H 2 1 / / / 2 H 1 H H ω ω ω x db down from midband value ω H x10 4 rad / s

35 It is possible to approximate ω L and ω H with multiple poles using superposition of poles The system function is first put into the following form

36 Where A o is the midband magnitude and H L and H H are the low and high frequency contributions It can be shown that K ω ω ω ω H 1 2 n ω ω + ω + Kω L a b m

37 Sources of Capacitance in Electronic Circuits Capacitance plays a dominant role in shaping the time and frequency response Inductances are usually important only well above breakpoint frequencies of major circuit capacitances Some exceptions include: Power circuits using transformers RF circuits Oscillator circuits High speed digital circuits Filters Common sources of capacitance Discrete capacitors Interconnect (stray) capacitance Internal capacitance of devices

38 Stray Lead Capacitance Capacitance due to connections between devices Sometimes referred to as package capacitance Usually small < 10 pf

39 Stray Lead Inductance Stray lead inductance result from closed conduction paths Only important for high frequency signals or large signal paths

40 Internal PN Junction Capacitance PN junction forms the basis of Mos semiconductor devices The junction exhibits capacitance in both forward and reverse bias conditions This capacitance is usually much larger than stray capacitance and therefore dominates frequency dependent behavior

41 Internal PN Junction Capacitance As a result of junction capacitances we can develop a revised small signal model of the BJT. This model is commonly referred to as the Hybrid PI model. In most cases the base resistance r x can be ignored Important to note that the dependent source is dependent on the current through r π (i.e. i π ) not on the total base current i b It is possible to write g m v π in terms of i b if the frequency dependence of β o is taken into account g v g ( r i ) β i β i m π m π π ο π ο b

42 Internal PN Junction Capacitance

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44 g v m π β ( ω ) β ( ω ) Ι β ο b 1 1+ jω r ( C + C ) π π µ - β(ω) is equivalent to β ο below the pole formed by C π and C µ ω H 1 r ( C + C ) π π µ

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46 At Frequency ωt β ο ( ω ) β ο jω r ( C + C ) T π π µ 2 1 / ω T rπ ( Cπ + Cµ ) ω T rπ ( Cπ + Cµ ) ω T β ο gm r ( C + C ) ( C + C ) π π µ π µ C gm C ω π µ T

47 MOSFET Model Similar models can be developed for other devices such as the small signal MOSFET model below

48 Time and Frequency Response Presence of stray, internal and discrete circuit capacitance greatly affects the way electronic circuits respond to time-varying input signals Incremental step response of a transistor amplifier

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51 v g ( R r ) v g R v OUT m c o π m c π v π t v e s / ( r C ) in s for t > 0 r R r in A π v g R v g R v e t r C in s / OUT m c π m c s Small signal voltage decays to zero over time Total output voltage returns to the DC bias value In practice make C s large such that v v OUT s lim r C in s g R m c For the circuit in Fig. 9.24, V cc 5V, C s 1µF, R A 1MΩ, R C 3.3 kω, βf βο 200, V f 0.6V, η 1 v in square wave with V p 5mV

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55 r in C s 5.7 ms

56 Time and Frequency Response Complete description of a circuit must include the effects of internal capacitances C π and C µ C π and C µ are usually small compared to C s The time constant r in C s will be long compared to the time duration over which C π and C µ affect circuit response. Therefore C s can be treated as an incremental short circuit

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59 Using Miller s Theorem C C v v C C v v A OUT B OUT µ π µ π 1 1 For Transverse Capacitance C x Z Z V V j C j C V V j C V V C C V V C C V V A x B A A x B A x B A A x B A B x A B or Similarly ω ω ω

60 Miller s Theorem Equivalence principle that can be applied to any port of a linear circuit that is connected to another port by a transverse element

61 V I R V I V R I V V R R R V V V R V V I V R V R V V V V V R R R V V V R V V A A x B A A A A A B x A x A A B x B A A A A A x A B A A B x B x B A B x A B i.e.

62 Time and Frequency Response For the circuit configuration of figure 9.30 over the duration of the principal time constant the current through C µ is negligible compared to the current through g m v π if g m r π >> 1 i i g v g r i µ π m π m π π v g ( R r ) OUT m c o [ 1 ( )] C C + g R r A µ m c o C B C µ

63 Time and Frequency Response Taking the Thevenin equivalent of v in, R A, r x and r π results in the following circuit

64 The input loop has the same form as a simple RC circuit for an input step function v s u(t) v v e t r s C C A π π Th / ( + ) 1 v Th v s r π rπ r u ( + t ) x If r x << r π then r s r x v v e t r x C C A π π s / ( + ) 1 u( t) v g v R r g R r e t r C C OUT x A m π c o π m c o / ( + ) ( ) ( ) 1 vsu( t)

65 The principal time constant is r x (C π + C A ) The current source g m v π will not be shorted out by C B if: ( R r ) C << r ( C + C ) c o B x π A where (R c r o )C B is the time constant for C B

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