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Exercise 3.5.1 inverting amplifier

1. Derive the voltage gain $A_{\rm V}= {{U_{\rm O}}\over{U_{\rm I}}}$ for the inverting amplifier.
Use the procedure that was used for the non-inverting amplifier.

  • What is required?
  • Number of variables?
  • Number of necessary equations?
  • Set up the known equations
  • Derivation of the voltage gain

Take into account that for the differential gain $A_\rm D$ of the ideal OPV applies: $A_\rm D \rightarrow \infty$. And the following also applies: $1/A_\rm D \rightarrow 0$
But the following doesn't always apply: ${{C}\over{U_x \cdot A_\rm D}} \rightarrow 0$, for an unknown constant $C$ and a voltage $U_x$!

Solution for "What is required?"

$A_{\rm V} = \frac{U_{\rm O}}{U_{\rm I}}$

Solution for "Number of variables?"

  • 5 voltages: $U_{\rm I}$, $U_{\rm 1}$, $U_{\rm D}$, $U_2$, $U_{\rm O}$
  • 5 currents: $I_1$, $I_{\rm m}$, $I_{\rm p}$, $I_2$, $I_{\rm o}$
  • --> 10 variables

Solution for "Number of necessary equations?"

9, since one equation is to be determined

Solution for "Set up the known equations"

  • Fundamental equation: (1) $U_{\rm O} = A_{\rm D} \cdot U_{\rm D}$
  • Golden rules:
    • $R_{\rm D} \rightarrow \infty$, and thus (2) + (3) $I_{\rm m} = I_{\rm p} = 0$
    • $R_{\rm O} = 0$
    • (4) $A_{\rm D} \rightarrow \infty$ and with (1) $U_{\rm D} = \frac{U_{\rm O}}{A_{\rm D}} \rightarrow 0$
  • Mesh equations
    • Mesh 1: (5) $-U_{\rm I} + U_1 - U_{\rm D} = 0$
    • Mesh 2: (6) $U_{\rm D} + U_2 + U_{\rm O} = 0$
  • Node equation: (7) $I_1 - I_2 + 0 = 0$
  • Resistors:
    • (8) $R_1 = \frac{U_1}{I_1}$
    • (9) $R_2 = \frac{U_2}{I_2}$

Solution for "Derivation of the voltage gain"

\begin{align*} A_V &= \frac{U_{\rm O}}{U_{\rm I}} \quad | \quad \text{using (5) and (6)} \\ A_V &= \frac{- U_{\rm D} - U_2}{U_1 - U_{\rm D}} \quad | \quad \text{using (8) and (9)} \\ A_V &= \frac{- U_{\rm D} - R_2 \cdot I_2}{R_1 \cdot I_1 - U_{\rm D}} \quad | \quad \text{using (1) }\\ A_V &= \frac{- R_2 \cdot I_2}{R_1 \cdot I_1} \quad | \quad \text{using (7)} \\ A_V &= -\frac{R_2 }{R_1 } \end{align*}

2. Which type of amplifier circuit (inverting or non-inverting amplifier) has the lower input resistance? Why?