Overview
This lecture covers the properties and assumptions of ideal operational amplifiers (op-amps), explains the inverting op-amp configuration, and walks through a design example using this configuration.
Op-Amp Fundamentals
- Output voltage for an op-amp is given by ( V_{out} = A \cdot V_D ), where ( V_D ) is the voltage difference between positive and negative terminals.
- An op-amp acts as a difference amplifier, amplifying the difference between its inputs.
- Op-amps are powered (e.g., ±12V supplies) and can't output voltages beyond their power rails.
Limitations of Real and Ideal Op-Amps
- Real op-amps can't output beyond their power supply voltages (e.g., max ±12V output).
- Input resistance is very high (ideally infinite), so input currents are zero.
- The maximum allowable input difference (( V_D )) is limited; for large gain (100,000), ( V_D ) maxes out at about ±120 μV if output is at its rail.
Ideal Op-Amp Assumptions
- Input resistance is infinite (( I_+ = I_- = 0 ) A).
- The voltage at the positive and negative terminals is equal (( V_+ = V_- )) in negative feedback.
- The op-amp can source or sink any required output current (idealization).
Inverting Op-Amp Configuration and Analysis
- The circuit consists of two resistors (( R_1 ) and ( R_2 )), input voltage ( V_{in} ), and output ( V_{out} ).
- Goal: Find a formula relating ( V_{out} ) to ( V_{in} ).
- Using nodal analysis: ( V_{out} = -\frac{R_2}{R_1} V_{in} ).
- The gain is ( -\frac{R_2}{R_1} ), which can invert and amplify or attenuate the input signal.
Example Calculation
- Given ( R_1 = 1,k\Omega ) and ( R_2 = 5,k\Omega ), supply ±12 V.
- Formula: ( V_{out} = -5 \cdot V_{in} )
- Calculated outputs:
- ( V_{in} = -1,V \Rightarrow V_{out} = 5,V )
- ( V_{in} = 1,V \Rightarrow V_{out} = -5,V )
- ( V_{in} = 2,V \Rightarrow V_{out} = -10,V )
- ( V_{in} = 3,V \Rightarrow V_{out} = -15,V ) (Exceeds rail; actual output is -12 V)
- Op-amp output saturates at ±12 V (power rail limits).
Understanding Saturation ("Railing")
- When output demand exceeds supply voltage, the op-amp "rails" or "saturates" at the supply limit.
- For this example, ( V_{in} ) must be between -2.4 V and 2.4 V for the output to remain unsaturated.
Key Terms & Definitions
- Op-amp (Operational Amplifier) — Amplifies the difference between its input terminals.
- Difference Amplifier — A circuit that outputs a voltage proportional to the difference between two inputs.
- Input Resistance — Resistance seen at the input terminals; assumed infinite for ideal op-amps.
- Gain — The ratio of output to input voltage, set by resistor values in the op-amp configuration.
- Saturation/Railing — Output voltage is limited by the op-amp's supply voltages.
Action Items / Next Steps
- Keep a dedicated sheet to document each op-amp configuration and its formula.
- Be prepared to identify op-amp configurations and apply the correct formulas for homework and exams.