7 - Amplifier subsystems

2016-05-19 10:42:00 +0000, 2 years and 3 months ago

7.01 - bandwidth

The bandwidth of an amplifier is the range of frequencies within the power gain that does not fall below half of its maximum value, since;

the bandwidth is also the range of frequencies within which the voltage gain does not fall below 1/√2 (0.7) of its maximum value.

The gain of a capacitor coupled with an amplifier decreases at the lower frequencies due to the increasing resistance of the capacitor, and at the upper frequencies due to stray capacitance in the circuit.

A typical voltage gain - frequency curve is shown below, with a LOG scale used on the x axis; to accommodate its large range.

100|             _____________
   |          ..'             '.
   |        ./                  `\
 80|      ./                      `\      x = frequency / Hz
70 +----./--------------------------`\    y = voltage gain, power
   |  ./|                            |\
 60| /  |                  ^         | \
   |/   |                  |         |  \
   |    |       BANDWIDTH  |         |   \
 40|    |                  |         |    \
   |    |   <--------------|--->     |     \
   |    |                  |         |      \
 20|    |                  |         |       |
   |    |                  |         |       |
   |    |                  V         |       |
  0+----+--+-------+-------+-------+-+-------+
   10    10^2    10^3    10^4    10^5      10^6

To estimate the bandwidth of an amplifier, either voltage of power, do:

V/Pmax / (1/√2) ~= 0.7 V/Pmax

Then draw a line on the Y axis parallel to the X axis. Where the 0.7 line intersects with the bandwidth curve, draw two X lines, parallel to the Y axis down, as shown in the above diagram. The frequency between the two X lines that now intersect the X axii origin, is the bandwidth.

7.02 - inverting amplifier

Below is a diagram of an inverting amplifier. The power supply connections are omitted, but the circuit it assumed to be operating from a dual-rail supply.

             Rf
        +---####---+
        |          |
   R1   |P  '\     |
--####--+---|-\    |
Vin         |  >---+--Vout
       +----|+/           
       |    ./            
OV     |                0V
-------+------------------

Since the open-loop gain is very large, if the output voltage is less than the power supply, then the voltage difference between the two input terminals is very small.
The positive (non-inverting) terminal is connected to the 0V rail, and so the voltage at the negative (inverting) input is virtually 0V, this is known as a Virtual Earth Point, P.
The input voltage, Vin, appears across the resistor, R1, so a current of Vout/R1 passes through R1.

This therefore means that the input resistance of the circuit is equal to R1.
It is assumed that because the input impedance (internal resistance) is so large that no current passes through the input terminals, therefore the only path for current passing through R1 must go through Rf , because the negative input terminal is also a virtual earth point; the output voltage, Vout, also appears across Rf. Therefore a current of Vout/Rf to pass through Rf.

This means that the voltage gain is determined only by the two resistors, R1 and Rf, the negative sign on the above equation, indicates that the amplifier is inverting.
The voltage gain of the op-amp has not been reduced, but the input signal has been reduced by cancelling part of it out, thus reducing the overall circuit voltage gain (closed loop-gain).
At high frequencies however the voltage gain will decrease in line with the frequency of the op-amp. The bandwidth of the amplifier will depend on the product of the closed loop gain of the circuit and the bandwidth.

The product of the voltage gain and bandwidth is assumed to be 10^6.

Therefore an amplifier with a closed-loop gain of 100, the bandwidth will be 10kHz.
The below waveform shows the input and output waveforms, with the output voltage being amplified and with a phase shift of 180 degrees, notice the peaks and troughs are the same, whilst the wavelength is still the same.

| x = time
| y = voltage  _.--._
|\           /`      `\
| `\       /`          `\   input 
|   `.___.`              \
|      
|   _.--._    
|  /      \   
| /        \  
|/          \    
|--------------------------- 
|             \          /
|              \        /   output
|               \      /
|                `'--'`

7.03 - summing amplifiers

This variation of an amplifier has several resistors, instead of one. The circuit diagram is shown below:

V3--####--+
     R3   |    Rf
          +---####--+
V2--####--+         |
     R2   |  '\     |
V1--####--+--|-\    |
     R1      |  >---+----Vout
          +--|+/
          |  ./
          |
0V--------+-----------0V

The negative terminal of the op-amp is Virtual Earth Point, and so the current at this point, is the sum of all the currents passing through the individual input resistors.

Therefore this current is equal to the current passing through Rf as a result of Vout

If R1 = R2 = R3 = R; then

if V1 = V2 V3 = V; then

Vout = -Rf * V * [1/R1 + 1/R2 + 1/R3]

This circuit can be used to make a simple Digital to Analogue converter (DAC) as is shown in the diagram below:

D3--####--+
     R3   |             R3 = Rf,
          |             R2 = 2Rf
D2--####--+             R1 = 4Rf,
     R2   |    Rf       R0 = 8Rf
          +---####--+
          |         |
D1--####--+         |
     R1   |         |
          |  '\     |
D0--####--+--|-\    |
     R0      |  >---+----Vout
          +--|+/
          |  ./
          |
0V--------+-----------0V

The logic inputs D3 - D0 all have the same input voltage, with D3 being the most significant bit of data, since it has the lowest series resistor. The output voltage will therefore be directly related to the digital number applied to the inputs, but it will be a negative voltage, since the inputs go to the inverting input.

7.04 - non-inverting amplifier

The input goes to the non-inverting input, the circuit uses negative feedback to reduce the overall voltage gain of the circuit by cancelling out part of the input signal. The advantage of sending the input to the non-inverting input is that it provides an input impedance for the circuit equal to the input impedance of the op-amp itself, which can be as high as 10^12 ohms.

        '\
Vin-----|+\       Vout
        |  >--+------
     +--|-/   |      
     |  ./    #      
     |        # Rf   
     |        #      
     +--------+      
              |      
              #      
              # R1     
              #     
              | 
0V------------+----0V

7.05 - difference amplifier

         Rf
     +--###---+        R1 = R2
     |        |        R3 = Rf
 R1  |  '\    |
-###-+--|-\   |   Vout
        |  >--+------
-###-+--|+/  
 R2  |  ./       
     #
     #
     # R3
     |
0V---+--------+----0V
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