Operational Amplifiers

• Any input applied to the inverting terminal will appear inverted at the output
• The differential voltage input $v_d$ (voltage across $R_{i}$) is given by $v_{2}-v_{1}$
  • Therefore, the output voltage $v_{o}$ which is equal to the voltage of the VCVS, is given by $Av_d = A(v_{2}-v_{1})$, where $A$ is the open-loop gain
  • When there is feedback, the ratio $\frac{v_{o}}{v_{d}}$ is called the closed-loop gain
• Practically, $v_o \le |V_{\text{CC}}|$, which means there are three modes of operation:
  1. Positive saturation $(v_o = V_{\text{CC}})$
  2. Linear range $(-V_{\text{CC}} \le v_o = Av_{d} \le V_{\text{CC}})$
  3. Negative saturation $(v_{o} = -V_{\text{CC}})$
• We always assume that the op amps are linear and $v_o \in [-V_{\text{CC}}, V_{\text{CC}}]$

Pin Configuration

Circuit Symbol

Circuit Symbol With Power

• By KCL, $i_o = i_{1} + i_{2} + i_{+} + i_{-}$

Non-Ideal Op Amp

Example 05.2.

(Example 5.1 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 203])

Example 05.3.

(Practice Problem 5.1 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 204])

Ideal Op Amp

• This means:
  • The op amp has zero current in both input terminals (i.e., they are open circuits)
  • The voltage across the terminals, $v_d$, equals 0, which means $v_1 = v_2$

Example 05.5.

(Practice Problem 5.2 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 205])

Op Amp Configurations

Inverting Amplifier

• Reverses the polarity of the input signal while amplifying it

Example 05.6.

(Example 5.3 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 206])

Example 05.7.

(Practice Problem 5.3 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 206])

Example 05.8.

(Example 5.4 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 207])

Example 05.9.

(Practice Problem 5.4 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 207])

Noninverting Amplifier

• Provides a positive voltage gain

• Special case: the gain becomes 1 if $R_f = 0$ (short circuit) or $R_1 = \infty$ (open circuit) or both
  • This is known as a voltage follower (or unity gain amplifier) because $v_o = v_i$
  • Useful to isolate two cascading stages of a circuit

Example 05.10.

(Example 5.5 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 208])

Example 05.11.

(Practice Problem 5.5 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 209])

Summing Amplifier

• Variation of the inverting amplifier that outputs a weighted sum of inputs

Example 05.12.

(Example 5.6 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 210])

Example 05.13.

(Practice Problem 5.6 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 211])

Differential Amplifier

• Amplifies the difference between two inputs

• If $\frac{R_{1}}{R_{f}} = \frac{R_{3}}{R_{4}}$:

• If $R_2 = R_1$ and $R_3 = R_4$, the gain is 1 and the output is a subtractor:

Example 05.14 (Instrumentation Amplifier Example).

(Example 5.8 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 213])

Example 05.15 (Instrumental Amplifier Problem).

(Practice Problem 5.8 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 214])

Cascaded Op Amp Circuits

• A cascade is when the output of one is the output of the next, with each connection called a stage
• The overall gain is the product of the gains of the stages

Example 05.16.

(Practice Problem 5.9 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 216])

Example 05.17.

(Practice Problem 5.10 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 218])