Finn Andersen, 2012 1

ECE2061 - Analogue Electronics Summary Notes Contents Equivalent Circuits ............................................................................................................ 5

Thevenin Equivalent ............................................................................................................... 5

Norton Equivalent .................................................................................................................. 5

Analogue Signal Amplification .......................................................................................... 6

Amplifier Frequency Response .............................................................................................. 6

Tolerance Modelling ......................................................................................................... 7

Ideal Amplifiers ................................................................................................................ 7

Differential Amplifier Model .................................................................................................. 7

Input and Output Resistances ................................................................................................ 7

Unity Gain Buffer .................................................................................................................... 8

Summing Amplifier ................................................................................................................. 8

Difference Amplifier ............................................................................................................... 8

Instrumentation Amplifier ..................................................................................................... 8

Integrator ............................................................................................................................... 9

Differentiator.......................................................................................................................... 9

Non-Ideal Operational Amplifiers ................................................................................... 10

Feedback .............................................................................................................................. 10

Classic Feedback Systems ..................................................................................................... 10

Non-Ideal Operational Amplifier .......................................................................................... 10

Non-zero output resistance ................................................................................................. 11

Finite Input Resistance ......................................................................................................... 11

Summary .............................................................................................................................. 11

Feedback Amplifier Categories: ........................................................................................... 12

Blackman’s  Theorem ............................................................................................................ 13

Series-Shunt Feedback Model .............................................................................................. 13

Shunt-Shunt Feedback Model .............................................................................................. 16

Series-Series Feedback Model ............................................................................................. 17

Shunt-Series Feedback Model .............................................................................................. 18

Successive Voltage and Current Injection ............................................................................ 19

Distortion Reduction through use of Feedback ................................................................... 21

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DC Error Sources and Output Range Limitations ................................................................. 22

Common-Mode Rejection and Input Resistance ................................................................. 23

Frequency Response and Bandwidth of Op-Amps .............................................................. 24

Slew Rate and Full Power Bandwidth .................................................................................. 27

Operational Amplifier Applications ................................................................................. 28

Cascaded Amplifiers ............................................................................................................. 28

Instrumentation Amplifier ................................................................................................... 28

Switched-Capacitor Circuits ................................................................................................. 29

Digital to Analogue Converters (DACs) ................................................................................ 30

DAC Circuits ...................................................................................................................... 31

Analogue to Digital Conversion ............................................................................................ 32

ADC Errors ........................................................................................................................ 33

AD Conversion Techniques .............................................................................................. 34

Oscillators ............................................................................................................................. 37

Circuits Using Positive Feedback .......................................................................................... 38

Solid State Electronics .................................................................................................... 40

Drift Currents and Mobility .................................................................................................. 41

Impurities in Semiconductors .............................................................................................. 42

Electron and Hole Concentrations in Doped Semiconductors............................................. 43

Mobility and Resistivity in Doped Semiconductors ............................................................. 44

Diffusion Current .................................................................................................................. 44

Total Current ........................................................................................................................ 45

Energy Band Model .............................................................................................................. 45

Integrated Circuit Fabrication .............................................................................................. 46

Solid-State Diodes .......................................................................................................... 47

I-V Characteristics of a Diode ............................................................................................... 48

Diode Biasing ........................................................................................................................ 49

Diode Reverse Bias ............................................................................................................... 50

Schottky Barrier Diode ......................................................................................................... 51

Diode Circuit Analysis ........................................................................................................... 51

Multiple Diode Circuits ......................................................................................................... 52

Diodes in Reverse Breakdown .............................................................................................. 53

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Half-Wave Rectifier Circuits ................................................................................................. 54

Full-Wave Rectifier Circuits .................................................................................................. 57

Full-Wave Bridge Rectification ............................................................................................. 58

Rectifier Summary ................................................................................................................ 58

Diode Dynamic Switching Behaviour ................................................................................... 58

Photo Diodes, Solar Cells, LEDs ............................................................................................ 59

Bipolar Junction Transistors ............................................................................................ 60

Physical Structure ................................................................................................................. 60

Transport Model for npn Transistor..................................................................................... 60

The pnp Transistor ............................................................................................................... 61

Equivalent Circuit Representations ...................................................................................... 62

The i-v Characteristics of BJT ................................................................................................ 62

Operating Regions ................................................................................................................ 63

Transport Model Simplifications .......................................................................................... 63

Non-Ideal BJT Behaviour ...................................................................................................... 64

Practical Bias Circuits for BJT ................................................................................................ 65

Tolerances in Bias Circuits .................................................................................................... 66

Field-Effect Transistors ................................................................................................... 67

MOS Capacitor...................................................................................................................... 67

The NMOS Transistor ........................................................................................................... 68

PMOS Transistors ................................................................................................................. 73

MOSFET Circuit Symbols ...................................................................................................... 74

Capacitances in MOS Transistors ......................................................................................... 75

MOS Transistor Scaling ......................................................................................................... 75

Biasing NMOSFET ................................................................................................................. 76

The Junction Field-Effect Transistor (JFET) .......................................................................... 78

Small Signal Modelling and Linear Amplification ............................................................. 80

BJT Amplifier ......................................................................................................................... 80

MOSFET Amplifier ................................................................................................................ 81

Coupling and Bypass Capacitors........................................................................................... 81

Circuit Analysis using DC and AC Equivalent Circuits ........................................................... 82

Small-Signal Modelling ......................................................................................................... 83

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Small-Signal Models for BJT ................................................................................................. 84

Common-Emitter Amplifier .................................................................................................. 86

Limits and Model Simplifications ......................................................................................... 87

Small-Signal Models for FETs ............................................................................................... 87

Summary of BJT and FET Differences ................................................................................... 89

Common-Source Amplifier ................................................................................................... 90

Input and Output Resistances .............................................................................................. 90

Common-Emitter and Common-Source Summary .............................................................. 91

Amplifier Power and Signal Range ....................................................................................... 91

Single Transistor Amplifiers ............................................................................................ 93

Inverting Amplifiers - Common Emitter/Source .................................................................. 94

Follower Circuits - Common-Collector/Drain ....................................................................... 96

Non-inverting Amplifiers – Common-Base/Gate ................................................................. 98

Amplifier Comparison .......................................................................................................... 99

Coupling and Bypass Capacitor Design .............................................................................. 101

Multi-stage AC-Coupled Amplifiers .................................................................................... 102

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Equivalent Circuits

Thevenin Equivalent Finding voltage:

x 𝑣 defined as output voltage with no load

𝑣 − 𝑣𝑅 + 𝛽𝑖 =

𝑣𝑅                                            𝑖 =

𝑣 − 𝑣𝑅

Substitute and re-arrange:

𝑣 =(𝛽 + 1)𝑅

(𝛽 + 1)𝑅 + 𝑅 𝑣        𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒  𝑣𝑎𝑙𝑢𝑒𝑠  𝑎𝑛𝑑  𝑣 = 𝑣

Finding Resistance:

1. Cut out independent voltage/current sources. If dependent ones exist, add in test voltage:

Norton Equivalent

1. Apply KCL at output node.

𝑣𝑖

1. Thevenin equivalent resistance equal to:

2. Do KCL at output node, solve for 𝑖

3. Apply short circuit across output, current flowing here is Norton equivalent current.

4. Apply KCL to solve for in

No current flows because of resistor

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Analogue Signal Amplification For input signal:

After a linear amplifier, output has same frequency but different amplitude and phase:

Gain (phasor notation):

Amplifier model:

Power gain:

𝐴 =𝑉 𝐼𝑉 𝐼 = |𝐴 ||𝐴 |

Convert to decibel scale:

Amplifier Frequency Response Amplifiers can be designed to amplify specific frequency ranges.

Negative gain is equivalent to 180° phase shift

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Tolerance Modelling Components have manufacturing tolerances, e.g Resistors with ±5% tolerance.

x Leads  to  ‘nominal’  and  ‘worst  case’  circuit  situations.  

Maximimise/minimize appropriate values to achieve max/min vo.

Ideal Amplifiers

Differential Amplifier Model Amplifies the voltage difference of two inputs.

𝑣 = 𝐴𝑣𝑅

𝑅 + 𝑅                                𝑣 = 𝑣𝑅

𝑅 + 𝑅

Input and Output Resistances For basic amplifier,

𝐼𝑛𝑣𝑒𝑟𝑡𝑖𝑛𝑔:  𝑅 =𝑣𝑖 = 𝑅                                          𝑁𝑜𝑛 − 𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑛𝑔: 𝑅 =

𝑣𝑖 = ∞  𝑠𝑖𝑛𝑐𝑒  𝑖 = 0

Apply test current, turn off independent sources, determine voltage.

𝑣 = 𝑖 (𝑅 + 𝑅 )      𝑠𝑖𝑛𝑐𝑒  𝑖 = 𝑖

𝑣 = 0  𝑠𝑜  𝑖 = 0  𝑠𝑜  𝑣 = 0  

𝑎𝑛𝑑  𝑅 = 0

𝐴 =𝑣𝑣 =

𝑅𝑅 + 𝑅

𝑅𝑅 + 𝑅

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Unity Gain Buffer

Summing Amplifier Sums input signals

Difference Amplifier

Instrumentation Amplifier

Has a gain of 1 (no change in voltage)

Does not draw current from input terminal, does not provide load to input circuit.

𝑉 = −𝑅𝑅 (𝑣 − 𝑣 )

𝑅 = 𝑅 + 𝑅

𝑣 = −𝑅𝑅 1 +

𝑅𝑅 (𝑣 − 𝑣 )

Virtual ground at R1. R1 can be changed to change gain instead of altering resistor pairs.

Infinite input resistance

Zero output resistance

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Integrator

Differentiator

‘Integrates’  the  input  voltage.  

x Capacitor initially starts out as an effective short circuit x Gains resistance as it charges x Infinite  resistance  when  fully  charged,  output  voltage  reaches  ‘saturated’  level,  

determined by supply rails.

‘If alternating signal is applied:

x Capacitor is charged and discharged x Results in sawtooth waveform output x Waveform output dependent on RC time constant

𝑖 = −𝑣𝑅            𝑖 = 𝐶 𝑑𝑣𝑑𝑡

𝑖 = 𝑖 , 𝑠𝑜:

𝑣 = −𝑅𝐶 𝑑𝑣𝑑𝑡

Capacitor and resistor reversed.

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Non-Ideal Operational Amplifiers

Feedback Effects:

x Gain stability – Reduces sensitivity of gain to variations in values of transistor parameters and circuit elements

x Input and Output Impedances – Feedback can determine input and output resistances of amplifier

x Bandwidth – (range of frequencies that the amplifier is most effective at amplifying), can be extended using feedback

x Nonlinear Distortion – Reduces effects of nonlinear distortion

Classic Feedback Systems

Non-Ideal Operational Amplifier Various forms of error arise in practical operational amplifiers due to non-ideal behaviour:

x Finite open loop gain – causes gain error x Nonzero output resistance x Finite input resistance x DC error sources x Output voltage and current limits

Finite Open-Loop Gain

𝑉 = 𝑉 − 𝑉            𝑉 = 𝛽𝑉            𝑉 = 𝐴𝑉 1𝛽 = 𝐴 = 𝑖𝑑𝑒𝑎𝑙  𝑔𝑎𝑖𝑛

𝛽𝐴 = 𝒍𝒐𝒐𝒑  𝒈𝒂𝒊𝒏 = 𝑇     Closed-loop gain:

𝑣 =𝑅

𝑅 + 𝑅 𝑣 = 𝛽𝑣

𝛽 =𝑅

𝑅 + 𝑅      (𝑏𝑜𝑡ℎ  𝑎𝑚𝑝𝑙𝑖𝑓𝑖𝑒𝑟𝑠)

𝑣 = 𝐴𝑣 = 𝐴(𝑣 − 𝛽𝑣 )

𝐹𝑜𝑟  𝐴𝛽 ≫ 1  (𝑖𝑑𝑒𝑎𝑙  𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛)

𝐴 → 𝐴 =1𝛽 =

𝑅𝑅 + 1  (𝑛𝑜𝑛 − 𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑛𝑔)

𝑣 =𝑣

1 + 𝑇 → 0        (𝑛𝑜𝑛 − 𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑛𝑔)

𝐴 → 𝐴 = −𝑅𝑅 = −

1𝛽 + 1  (𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑛𝑔)

𝑣 = −𝑅𝑅

𝛽1 + 𝐴𝛽 𝑣 → −

𝑅𝑅

𝑣𝐴  (𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑛𝑔)

Feedback Factor Open-loop gain