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Chapter 13 Small-Signal Modeling and Linear Amplification

Chapter 13 Small-Signal Modeling and Linear Amplification. Microelectronic Circuit Design Richard C. Jaeger Travis N. Blalock. Chap 13 - 1. Chapter Goals. Understanding of concepts related to: Transistors as linear amplifiers dc and ac equivalent circuits

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Chapter 13 Small-Signal Modeling and Linear Amplification

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  1. Chapter 13Small-Signal Modeling and Linear Amplification Microelectronic Circuit Design Richard C. Jaeger Travis N. Blalock Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 1

  2. Chapter Goals Understanding of concepts related to: • Transistors as linear amplifiers • dc and ac equivalent circuits • Use of coupling and bypass capacitors and inductors to modify dc and ac equivalent circuits • Small-signal voltages and currents • Small-signal models for diodes and transistors • Identification of common-source and common-emitter amplifiers • Amplifier characteristics such as voltage gain, input and output resistances and linear signal range • Rule-of-thumb estimates for voltage gain of common-emitter and common-source amplifiers. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 2

  3. Introduction to Amplifiers • BJT is used as an amplifier when biased in the forward-active region • FET can be used as amplifier if operated in the pinch-off or saturation region • In these regions, transistors can provide high voltage, current and power gains • Bias is provided to stabilize the operating point in a desired operation region • Q-point also determines • Small-signal parameters of transistor • Voltage gain, input resistance, output resistance • Maximum input and output signal amplitudes • Power consumption Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 3

  4. BJT Amplifier BJT is biased in active region by dc voltage source VBE. Q-point is set at (IC, VCE) = (1.5 mA, 5 V) with IB = 15 mA. Total base-emitter voltage is: Collector-emitter voltage is: This is the load line equation. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 4

  5. BJT Amplifier (cont.) If changes in operating currents and voltages are small enough, then iC and vCE waveforms are undistorted replicas of input signal. Small voltage change at base causes large voltage change at collector. Voltage gain is given by: Minus sign indicates 1800 phase shift between input and output signals. 8 mV peak change in vBE gives 5 mA change in iB and 0.5 mA change in iC. 0.5 mA change in iC produces a 1.65 V change in vCE . Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 5

  6. MOSFET Amplifier MOSFET is biased in active region by dc voltage source VGS. Q-point is set at (ID, VDS) = (1.56 mA, 4.8 V) with VGS = 3.5 V. Total gate-source voltage is: 1 V p-p change in vGS gives 1.25 mA p-p change in iD and 4 V p-p change in vDS. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 6

  7. Coupling and Bypass Capacitors C1 and C3 are large-valued coupling capacitors or dc blocking capacitors whose reactance at the signal frequency is designed to be negligible. C2 is a bypass capacitor that provides a low impedance path for ac current from emitter to ground, thereby removing RE (required for good Q-point stability) from the circuit when ac signals are considered. • AC coupling through capacitors is used to inject ac input signal and extract output signal without disturbing Q-point • Capacitors provide negligible impedance at frequencies of interest and provide open circuits at dc. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 7

  8. dc and ac Analysis • DC analysis: • Find dc equivalent circuit by replacing all capacitors by open circuits and inductors by short circuits. • Find Q-point from dc equivalent circuit by using appropriate large-signal transistor model. • AC analysis: • Find ac equivalent circuit by replacing all capacitors by short circuits, inductors by open circuits, dc voltage sources by ground connections and dc current sources by open circuits. • Replace transistor by small-signal model • Use small-signal ac equivalent to analyze ac characteristics of amplifier. • Combine end results of dc and ac analysis to yield total voltages and currents in the network. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 8

  9. dc Equivalent for BJT Amplifier • All capacitors in original amplifier circuits are replaced by open circuits, disconnecting vI , RI , and R3 from circuit. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 9

  10. ac Equivalent for BJT Amplifier Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 10

  11. DC and AC Equivalentsfor MOSFET Amplifier dc equivalent Full circuit ac equivalent Simplified ac equivalent Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 11

  12. Small-Signal Operation of Diode • The slope of the diode characteristic at the Q-point is called the diode conductance and is given by: • gd is small but non-zero for ID = 0 because slope of diode equation is nonzero at the origin. • Diode resistance is given by: Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 12

  13. Small-Signal Operation of Diode (cont.) Subtracting ID from both sides of the equation, For id to be a linear function of signal voltage vd , This represents the requirement for small-signal operation of the diode. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 13

  14. Current-Controlled Attenuator Magnitude of ac voltage vo developed across diode can be controlled by value of dc bias current applied to diode. From ac equivalent circuit, From dc equivalent circuit ID =I, For RI = 1 kW, IS = 10-15 A, If I = 0, vo = vi, magnitude of vi is limited to only 5 mV. If I = 100 mA, input signal is attenuated by a factor of 5, and vi can have a magnitude of 25 mV. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 14

  15. Small-Signal Model of BJT Using 2-port y-parameter network, The port variables can represent either time-varying part of total voltages and currents or small changes in them away from Q-point values. bo is the small-signal common-emitter current gain of the BJT. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 15

  16. Hybrid-Pi Model of BJT Transconductance: • The hybrid-pi small-signal model is the intrinsic representation of the BJT. • Small-signal parameters are controlled by the Q-point and are independent of geometry of the BJT Input resistance: Output resistance: Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 16

  17. Small-Signal Current Gain and Amplification Factor of BJT Amplification factor is given by: For VCE << VA, mF represents maximum voltage gain individual BJT can provide and doesn’t change with operating point. bo > bF for iC < IM , and bo < bF for iC > IM , however, bF and bo are assumed to be equal. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 17

  18. Equivalent Forms of Small-Signal Model for BJT • Voltage -controlled current source gmvbe can be transformed into current-controlled current source, • Basic relationship ic = bib is useful in both dc and ac analysis when BJT is in forward-active region. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 18

  19. Small-Signal Operation of BJT For linearity, ic should be proportional to vbe with Change in ic that corresponds to small-signal operation is: Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 19

  20. Small-Signal Model for pnp BJT • For pnp transistor • Signal current injected into base causes decrease in total collector current which is equivalent to increase in signal current entering collector. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 20

  21. Small-Signal Analysis of Complete C-E Amplifier: ac Equivalent • Ac equivalent circuit is constructed by assuming that all capacitances have zero impedance at signal frequency and dc voltage sources are ac ground. • Assume that Q-point is already known. Microelectronic Circuit Design, 3E McGraw-Hill Chap13 - 21

  22. Small-Signal Analysis of Complete C-E Amplifier: Small-Signal Equivalent Input applied to Base Output appears at Collector Emitter is common (through RE) to both input and output signal - Common-Emitter (CE) Amplifier. is the terminal voltage gain of the CE amplifier. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 22

  23. Common-Emitter (CE): Terminal Voltage Gain Solving for ib and substituting, For and Using alternate small-signal model form and test source vb to drive the base terminal of the transistor, neglecting ro, Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 23

  24. Common-Emitter (CE): Input Resistance and Signal Source Voltage Gain is the impedance in the emitter side of the transistor reflected to the base side. Combining equations, the overall voltage gain can be written as: Rewriting the previous equation, we can find the impedance looking into the base terminal: Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 24

  25. C-E Amplifier Voltage Gain Example with RE = 0 • Problem: Calculate voltage gain • Given data:bF = 100, VA = 75 V, Q-point is (1.45 mA, 3.41 V), R1 = 10 kW, R2 = 30 kW,R3 = 100 kW, RC = 4.3 kW, RI = 1kW. • Assumptions: Transistor is in active region, bO = bF. Signals are low enough to be considered small signals. • Analysis: Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 25

  26. Small-Signal Model Simplification • For max gain • For maximum gain we set R3 >> RC and load resistor RC << ro. If we assume ICRC = VCC with 0 <  < 1 Typically,  = 1/3, since common design allocates one-third power supply across RC. To further account for other approximations leading to this result, we use: • Also, if the load resistor approaches ro (RC and R3 infinite), voltage gain is limited by amplification factor, mf of BJT itself. • For large RE, voltage gain can be aproximated as: Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 26

  27. C-E Amplifier Output Resistance • Output resistance is the total equivalent resistance looking into the output of the amplifier at coupling capacitor C3.Input source is set to 0 and a test source vx is applied at output. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 27

  28. Sample Analysis of C-E Amplifier Analysis: To find the Q-point, dc equivalent circuit is constructed. • Problem: Find voltage gain, input and output resistances. • Given data:bF = 65, VA = 50 V • Assumptions: Active-region operation, VBE = 0.7 V, small signal operating conditions. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 28

  29. Sample Analysis of C-E Amplifier (cont.) Next we construct the ac equivalent and simplify it. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 29

  30. Small-Signal Model for the MOSFET Using 2-port y-parameter network, The port variables can represent either time-varying part of total voltages and currents or small changes in them away from Q-point values. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 30

  31. Small-Signal Parameters of MOSFET Transconductance: Output resistance: • Since gate is insulated from channel by gate-oxide input resistance of transistor is infinite. • Small-signal parameters are controlled by the Q-point. • For same operating point, MOSFET has higher transconductance and lower output resistance that BJT. Amplification factor for lVDS<<1: Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 31

  32. Small-Signal Operation of MOSFET For linearity, id should be proportional to vgs: Since the MOSFET can be biased with (VGS - VTN) equal to several volts, it can handle much larger values of vgs than corresponding the values of vbe for the BJT. Change indrain current that corresponds to small-signal operation is: Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 32

  33. Body Effect in Four-terminal MOSFET Drain current depends on threshold voltage which in turn depends on vSB. Back-gate transconductance is: 0 <  < 3 is called back-gate tranconductance parameter. Bulk terminal is a reverse-biased diode. Hence, no conductance from bulk terminal to other terminals. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 33

  34. Small-Signal Model for PMOS Transistor • For PMOS transistor • Positive signal voltage vgg reduces source-gate voltage of the PMOS transistor causing decrease in total current exiting drain, equivalent to an increase in the signal current entering the drain. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 34

  35. Summary of FET and BJT Small-Signal Models Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 35

  36. Small-Signal Analysis of Complete C-S Amplifier: ac Equivalent • ac equivalent circuit is constructed by assuming that all capacitances have zero impedance at signal frequency and dc voltage sources represent ac grounds. • Assume that Q-point is already known. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 36

  37. Small-Signal Analysis of Complete CS Amplifier: Small-Signal Equivalent Noting similarity to CE case, Terminal voltage gain between gate and drain is found as: With r infinite, RiG is also infinite, therefore overall voltage gain from source vi to output voltage across RLis: Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 37

  38. C-S Amplifier Voltage Gain: Example • Problem: Calculate voltage gain • Given data:Kn = 0.5 mA/V2, VTN = 1V, l= 0.0133 V-1, Q-point is (1.45 mA, 3.86 V), R1 = 430 kW, R2 = 560 kW,R3 = 100 kW, RD = 4.3 kW, RI = 1 kW. • Assumptions: Transistor is in active region. Signals are low enough to be considered small signals. • Analysis: Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 38

  39. Small-Signal Model Simplification • If we assume RI << RG • Generally R3 >> RD and load resistor << ro. Hence, total load resistance on drain is RD. For this case, common design allocates half the power supply for voltage drop across RD and (VGS - VTN ) = 1V • Also, if load resistor approaches ro, (RD and R3 infinite), voltage gain is limited by amplification factor, mf of the MOSFET itself. This implies that total signal voltage at input appears across gate-source terminals. Microelectronic Circuit Design, 3E McGraw-Hill Chap13 - 39

  40. C-S Amplifier Input Resistance • Input resistance of C-S amplifier is much larger than that of corresponding C-E amplifier. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 40

  41. C-S Amplifier Output Resistance • Output resistance is calculated in a manner similar to that of CE amplifier with r infinite. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 41

  42. Sample Analysis of C-S Amplifier Analysis: dc equivalent circuit is constructed and analyzed • Problem: Find voltage gain, input and output resistances. • Given data:Kn = 500 mA/V2, VTN = 1V, l = 0.0167 V-1 Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 42

  43. Sample Analysis of C-S Amplifier (cont.) Next we construct the ac equivalent and simplify it. Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 43

  44. Summary CE and CS Characteristics Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 44

  45. Signal Range Constraints Minimum output voltage set by active region constraints of Q, maximum set by drop across RC. For a sine wave with peak VM, we can express the limits as: Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 45

  46. End of Chapter 13 Microelectronic Circuit Design, 3E McGraw-Hill Chap 13 - 46

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