# 24.1.4 Gain of Amplifiers

### Derivation of Gain of Inverting Amplifiers

• The virtual Earth approximation is:
• An op-amp has a very large open-loop voltage gain
• V+ is at earth potential (0 V)
• If the amplifier is not to saturate, V input must be almost at earth potential too
• In an inverting amplifier circuit, the current in the input resistor is Iin and the current in the feedback resistor is If
• If V = 0 V (Earth potential) then the equations for the currents are:

• The input resistance Rin of the op-amp is very high, so virtually no current enters or leaves the inverting input. This means:

If  = − Iin

• Or:

• Hence the gain (amplification) of the inverting amplifier A is given by:

### Voltage Gain of Inverting and Non-inverting Amplifiers

• The voltage gain for inverting amplifiers is given by:

• Where:
• A = gain
• Vout = output voltage (V)
• Vin = input voltage (V)
• Rf = feedback resistance (Ω)
• Rin = input resistance (Ω)
• The voltage gain for non-inverting amplifiers is given by:

• Where:
• Rf = feedback resistance (resistance of resistor connected to the amplifier)
• Both resistors form a potential divider

#### Worked example: Calculating resistance

Step 1:            Voltage gain equation for non-inverting amplifier

Step 2:            Rearrange for resistance R

Step 3:            Substitute in values

Rf = feedback resistance = 20 kΩ = 20 × 103 Ω

A = 45

### Author: Katie

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.
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