1. Frequency Analysis of BJT and MOSFET Amplifiers
UNIT-IV
Frequency Analysis of BJT and MOSFET Amplifiers

2. Outline Low Frequency and Miller Effect
High Frequency Analysis of CE Amplifier
High Frequency Analysis of MOSFET Amplifier
Cut off frequency –Unity gain
Single stage and Multi stage Amplifiers

3. Introduction
Frequency Response of an electric or electronics circuit allows us to see exactly how the output gain and the phase changes at a particular single frequency, or over a whole range of different frequencies depending upon the design characteristics of the circuit.

4. Introduction -contd…
Frequency response analysis of a circuit or system is shown by plotting its gain  against a frequency scale.
The circuits gain, (or loss) at each frequency point helps us to understand how well (or badly) the circuit can distinguish between signals of different frequencies.

5. Introduction-contd…
There are many different ways for the calculations of the frequency depending on the combination of components.
The -3dB frequency for resistance and capacitance (the most common in amplifier design) is determined by
fo = 1 / (2 Π R C)   
where fo is the -3dB frequency

6. Effect of Coupling Capacitors
Coupling capacitors are in series with the signal and are part of a high-pass filter network. They affect the low-frequency response of the amplifier.

7. Effect of Coupling Capacitors
The equivalent circuit for C1 is a high-pass filter:

8. Effect of Bypass Capacitors-contd…
A bypass capacitor causes reduced gain at low-frequencies and has a high-pass filter response. The resistors “seen” by the bypass capacitor include RE, re’, and the bias resistors.

9. Effect of Internal capacitances
The high-frequency response of an amplifier is determined by internal junction capacitances. These capacitances form low-pass filters with the external resistors.

10. Decibel
The decibel is a logarithmic ratio of two power levels and is used in electronics work in gain or attenuation measurements.
Decibels can be expressed as a voltage ratio when the voltages are measured in the same impedance.
To express voltage gain in decibels, the formula is
Av(dB) = 20 log Av

11. Typical Frequency response

12. Typical Frequency response
Gain is more commonly stated using a logarithmic scale, and the result is expressed in decibels (dB). For voltage gain, this takes the form
The upper and lower frequencies defining the bandwidth, called the corner
or cutoff frequencies.

13. Bandwidth
The range of frequencies with close to constant gain is known as the bandwidth.

14. Bandwidth-contd…
The bandwidth represents the amount or "width" of frequencies, or the "band of frequencies," that the amplifier is most effective in amplifying.
The bandwidth is not the same as the band of frequencies that is amplified. The bandwidth (BW) of an amplifier is the difference between the frequency limits of the amplifier.

15. Bandwidth-contd… BW= fc2 - fc1
For example, the band of frequencies for an amplifier may be from 10kilohertz (10 kHz) to 30 kilohertz (30 kHz).

16. Bandwidth-contd…
In this case, the bandwidth would be 20 kilohertz (20 kHz).
As another example, if an amplifier is designed to amplify frequencies between 15 hertz (15 Hz) and 20kilohertz (20 kHz), the bandwidth will be equal to 20 kilohertz minus 15 hertz.

17. High Frequency Roll-off of Amplifier
As frequency of operation increases, the gain of amplifier decreases. 17

18. Example: Video Signal High Bandwidth Low Bandwidth
Video signals without sufficient bandwidth become fuzzy as they fail to abruptly change the contrast of pictures from complete white into complete black. 18

19. Gain Roll-off: Simple Low-pass Filter
In this simple example, as frequency increases the impedance of C1 decreases and the voltage divider consists of C1 and R1 attenuates Vin to a greater extent at the output. 19

20. Millers Theorem
Miller’s theorem states that, for inverting amplifiers, the capacitance between the input and output is equivalent to separate input and output capacitances to ground.

21. Millers Theorem-contd…
Av is the absolute value of the gain. For the input capacitance, the gain has a large effect on the equivalent capacitance, which is an important consideration when using inverting amplifiers.

22. Miller’s Theorem-contd…
If Av is the gain from node 1 to 2, then a floating impedance ZF can be converted to two grounded impedances Z1 and Z2.

23. Miller Multiplication
With Miller’s theorem, we can separate the floating capacitor. However, the input capacitor is larger than the original floating capacitor. We call this Miller multiplication. 23

24. Low frequency Response Of CE Amplifier
In capacitively coupled amplifiers, the coupling and bypass capacitors affect the low frequency cutoff. These capacitors form a high-pass filter with circuit resistances. A typical BJT amplifier has three high-pass filters.
For example, the input coupling capacitor forms a high-pass filter with the input resistance of the amplifier:

25. Low frequency Response Of CE Amplifier- Input coupling capacitor

26. Low frequency Response Of CE Amplifier- Output coupling capacitor
The output RC circuit is composed of the series combination of the collector and load resistors with the output capacitor. The cutoff frequency due to the output circuit is

27. High-Frequency Bipolar Model
At high frequency, capacitive effects come into play. Cb represents the base charge, whereas C and Cje are the junction capacitances.

28. High-Frequency Model of Integrated Bipolar Transistor
Since an integrated bipolar circuit is fabricated on top of a substrate, another junction capacitance exists between the collector and substrate, namely CCS.

29. Example: Capacitance Identification

30. HYBRID Model of CE Amplifier

31. High-Frequency Model of Integrated circuits
Cce is small and can be neglected
rbc is large and considered open
Rbb small neglected
Rin=Rb||rπ

32. High-Frequency Model of Integrated circuits-contd…
Rcin=Rs||Rin
fH=ωH/2π

33. Transit Frequency
Transit frequency, fT, is defined as the frequency where the current gain from input to output drops to 1.

34. MOS Intrinsic Capacitances
For a MOS, there exist oxide capacitance from gate to channel, junction capacitances from source/drain to substrate, and overlap capacitance from gate to source/drain.

35. Gate Oxide Capacitance Partition and Full Model
The gate oxide capacitance is often partitioned between source and drain. In saturation, C2 ~ Cgate, and C1 ~ 0. They are in parallel with the overlap capacitance to form CGS and CGD.

36. Effect of coupling capacitor
With FETs, the input coupling capacitor is almost always smaller because of the high input resistance. The output capacitor may be smaller or larger depending on the drain and load resistor size.

37. Example: Capacitance Identification

38. Capacitive Coupling vs. Direct Coupling
Capacitive coupling, also known as AC coupling, passes AC signals from Y to X while blocking DC contents.
This technique allows independent bias conditions between stages. Direct coupling does not.

39. Gain Roll-off: Common Source
The capacitive load, CL, for gain roll-off since at high frequency, it will “steal” away some signal current and shunt it to ground.

40. Frequency Response of the CS Stage
At low frequency, the capacitor is effectively open and the gain is flat. As frequency increases, the capacitor tends to a short and the gain starts to decrease. A special frequency is ω=1/(RDCL), where the gain drops by 3dB.

41. Typical Frequency Response
Lower Corner
Upper Corner

42. Low-Frequency Response of the Common-Source Amplifier

43. Low-Frequency Response of the Common-Source Amplifier

44. Low-Frequency Response of the Common-Source Amplifier

45. Low-Frequency Response of the Common-Source Amplifier

46. Low-Frequency Response of the Common-Source Amplifier

47. High Frequency Circuit Analysis Procedure
Determine which capacitor impact the low-frequency region of the response and calculate the low-frequency pole (neglect transistor capacitance).
Calculate the midband gain by replacing the capacitors with short circuits (neglect transistor capacitance).
Include transistor capacitances.
Merge capacitors connected to AC grounds and omit those that play no role in the circuit.
Determine the high-frequency poles and zeros.
Plot the frequency response using Bode’s rules or exact analysis.

48. High frequency Analysis

49. High frequency Analysis

50. High frequency Analysis

51. Frequency Response of CS Stage with Bypassed Degeneration
To increase the midband gain, a capacitor Cb is placed in parallel with Rs.
The pole frequency must be well below the lowest signal frequency to avoid the effect of degeneration.

52. Unified Model for CE and CS Stages

53. Unified Model Using Miller’s Theorem

54. Example: Half Width CS Stage

55. Direct Analysis of CE and CS Stages
Direct analysis yields different pole locations and an extra zero.

56. Example: CE and CS Direct Analysis

57. Input Impedance of CE and CS Stages

58. Frequency response CS amplifier

59. Hybrid model –CE Amplifier
Cb'c
rbb'
Co l l e c t o r b'
rb' c
Base
rb'e
rce
gmVb'e
Cb'e
Em i t t e r

60. The resistance rbb' is rb'c Cb'c Hybrid π model (cont.) Cb'c rbb' rb'
Co l l e c t o r b'
rb' c
Base
rb'e
rce
gmVb'e
Cb'e
Em i t t e r
The resistance rbb' is
the base spreading
resistance.
The resistance
rb'c
and the capacitance
Cb'c
represent the dynamic
(differential) resistance and the capacitance of the reverse-biased
collector-base junction. 6

61. Hybrid π model
r bb'
i c C
b'c
r b'e V
b'e C
b'e
gm V b'e

62. Hybrid π model

63. Beta cutoff frequency

64. Beta cutoff frequency

65. Alpha cutoff frequency-T equivalent of Common base Amplifier

66. Alpha cutoff frequency-T equivalent of Common base Amplifier

67. Frequency Response
Overall frequency response is the combination of three lower critical frequencies due to coupling and bypass capacitors and two upper critical frequencies due to internal capacitances.

68. Gain–bandwidth product
The gain–bandwidth product (designated as GBWP, GBW, GBP or GB) for an amplifier is the product of the amplifier's bandwidth and the gain at which the bandwidth is measured.
Gain-bandwidth product defined as
GB = |AM|BW

69. Gain–bandwidth product-CE Amplifier
The common emitter amplifier, the gain-bandwidth product for the common emitter BJT amplifier configuration is substantially less.

70. Multistage Amplifiers
Multi-stage amplifiers are amplifier circuits cascaded to increased gain. We can express gain in decibels(dB).
Two or more amplifiers can be connected to increase the gain of an ac signal. The overall gain can be calculated by simply multiplying each gain together.
A’v = Av1Av2Av3 ……
Fig 6-32

71. Multistage Amplifiers
Matching of inputs and outputs is necessary to ensure that the maximum amount of signal can be transferred between the amplifier, and any other circuit or device preceding or following it.
This is usually the case when the gain of a single amplifier is insufficient for a given purpose.

72. Multistage Amplifiers -contd…
Then several stages of amplification are used which involves feeding the output of one amplifier into the input of another

73. Multistage Amplifiers-contd…
This will occur if the output impedance of the first amplifier is a much lower value than the input impedance of the second amplifier. 
When connecting voltage amplifiers in cascade, the input signal to the second stage should ideally be 100% of the output voltage of stage 1, i.e. have as high a voltage amplitude as possible.

74. Multistage Amplifiers-contd…
This allows most of the voltage available at the output terminal (point A) to be developed across the input impedance of the second amplifier rather than across the first amplifier´s output impedance.

75. Multistage Amplifiers-contd…
The second amplifier is a current amplifier however, it will be necessary that as much current as possible flows into its input terminals.
The input impedance of the second amplifier must be low.
In the case of power amplifiers, the maximum power is transferred from output to input if both impedances are equal.

76. Start: Two-Stage Voltage Amplifier
Use two-port models to explore whether the combination “works”
CE1
CE2
CE1,2
Results of new 2-port: Rin = Rin1, Rout = Rout2

77. MULTISTAGE Amplifier

78. Multistage amplifiers-contd…
Gain
Input Impedance
Output Impedance

79. Multistage amplifiers-contd…
For multistage amplifiers, the individual stages have an effect on the overall response.
With different cutoff frequencies, the dominant lower cutoff frequency is equal to the highest fcl; the dominant upper critical frequency is equal to lowest fcu.
When the critical frequencies for multistage amplifiers are equal, the lower critical frequency is higher than any one as given by

80. Multistage amplifiers-contd…
Lower critical frequency is higher than any one as given by
and the upper critical frequency is given by

81. Multistage amplifiers

82. Small signal parameters of multi stage Amplifier

83. Multistage amplifiers-contd…
Low output resistance from EF provides a low source resistance for CE amplifier so good matching of output of EF to input of CE amplifier
High frequency response (3dB frequency) for Cascade Amplifier is improved over CE amplifier.

84. CE-CC
The cascade of a Common Emitter amplifier stage followed by a Common Collector amplifier stage can provide a good overall voltage amplifier 

85. CE-CC-contd…
The Common Emitter input resistance is relatively high and Common Collector output resistance is relatively low.
The voltage follower second stage, Q2, contributes no increase in voltage gain but provides a near voltage-source (low resistance) output so that the gain is nearly independent of load resistance. 

86. CE-CC-contd…
The high input resistance of the Common Emitter stage, Q1, makes the input voltage nearly independent of input-source resistance. Multiple Common Emitter stages can be cascaded with emitter follower stages inserted between them to reduce the attenuation due to inter-stage loading.

87. CE-CE
Each stage is separately biased and coupled to adjacent stages via DC blocking capacitors.
Inserting coupling capacitors between stages blocks the DC operating bias level of one stage from affecting the DC operating point of the next.

88. Frequency Response of Multistage Amplifier

89. Gain Bandwidth Product
The gain bandwidth product (GBW) for an amplifier is the product of the open loop gain (constant for a given amplifier) and its 3 dB bandwidth. This quantity is commonly specified for operational amplifiers, and allows circuit designers to determine the maximum gain that can be extracted from the device for a given frequency (or bandwidth) and vice versa.

90. Gain Bandwidth Product-contd…
When adding LC circuits to the input and output of an amplifier the gain raises and the bandwidth decreases, but the product remains constant.
Measure of the gain-frequency product of an amplifier; unity gain bandwidth is the frequency at which the open-loop gain becomes unity, based on 6 decibels per octave crossing.

91. Gain Bandwidth Product-contd…
Gain of FET A=gmR0
ω= 1/R0Ceq
Gain Bandwidth Product = A. ω
= (gmR0) (1/R0Ceq)
=gm Ceq
If Ceq= Cgs
Then GBW =gmCgs

92. Cascode Connection
This example is a CE–CB combination. This arrangement provides high input impedance but a low voltage gain.
The low voltage gain of the input stage reduces the Miller input capacitance, making this combination suitable for high-frequency applications.

93. Darlington Connection
The Darlington circuit provides very high current gain, equal to the product of the individual current gains:
D = 1 2
The practical significance is that the circuit provides a very high input impedance.

94. Summary
The circuits gain, (or loss) at each frequency point helps us to understand how well (or badly) the circuit can distinguish between signals of different frequencies.
For multistage amplifiers, the individual stages have an effect on the overall response.