E84 Home Work

1. Inverting amplifier

   Find the input resistance $R_{in}$, output resistance $R_{out}$ and open-voltage gain $G_v$ of the inverting amplifier. (Read and understand the derivation shown in the lecture notes.)

2. Non-inverting amplifier

   Find the input resistance $R_{in}$, output resistance $R_{out}$ and open-voltage gain $G_v$ of the non-inverting amplifier, and compare your results with those given at the bottom of this page.

   

3. Algebraic summer:

   

   Express the output voltage $v_{out}$ as a weighted sum of the four input voltages $v_1,\cdots,v_4$:

   $$v_{out}=\sum_{i=1}^4 k_i v_i$$
   Find the four coefficients (may be either positive or negative).

   Hint: Define $V=V^+ \approx V^-$, and apply KCL to $V^-$ and $V^+$ to get two equations. Solve one of them for $V$, substitute it into the other equation, and then write $v_{out}$ as a function of all four input voltages.

   The algebraic summer can be generalized to have $m$ inputs $v_i$ all connected through $R_i$ ( $i=1,\cdots,m$) to the inverting input of the op-amp, and $n$ inputs $u_j$ all connected through $R_j$ ( $j=1,\cdots,n$) to the non-inverting input of the op-amp. Give the general expression of the output $v_{out}$ as a function of all $m+n$ inputs in terms of all resistors $m+n+1$ resistors.

4. Sallen-Key filter (HP/LP)

   Derive the voltage gain ( $V_{out}/V_{in}$) of the general Sallen-Key fitler (HP or LP) in terms of the impedances $Z_1$ through $Z_4$:

   

5. Sallen-Key filter (BP)

   

   Derive the voltage gain ( $V_{out}/V_{in}$) of the general Sallen-Key fitler (HP or LP) in terms of the impedances $Z_1$ through $Z_4$ and $R_f$, $R_a$, $R_b$.

   (Hint: For simplicity, assume $R_a/(R_a+R_b)=k$.

6. The Howland current source:

   The circuit generates a constant current through the load, independent of the load resistance $R_L$ of the load. Give the expression of this current $I_L$ through $R_L$ as a function of the inputs $V^+$ and $V^-$ and the resistances in the circuit, and show it is independent of $R_L$. Assume $R_2/R_1=R_4/R_3$.

   

   Hint: recall the relationship between the current through and voltage across a diode and use the virtual ground assumption.

7. Exponential and logarithmic amplifiers:

   

   The current $I_D$ through and voltage $V_D$ across a diode are related by the following:

   $$I_D=I_0 \left( e^{V_D/\eta V_T}-1 \right)$$
   where $\eta$ and $V_T$ are some parameters. The direction of the current $I_D$ through the diode is indicated by the arrow of the symbol, and the polarity of the voltage $V_D$ is plus on the arrow side and minus on the other side. When $V_D/\eta V_T$ is large enough, $e^{V_D/\eta V_T}\gg 1$.

   Show that the output voltages $v_{out}$ of the exponential amplifier (left) and logarithmic amplifier (right) are approximately an exponential and logarithmic function of the input voltage $v_{in}$, respectively:

   $$v_{out}\approx C \;\exp(v_{in}/a),\;\;\;\;\;\; v_{out}\approx D\; \ln (v_{in}/b)$$
   Derive these relationships and determine the coefficients $C,\;D$ and $a, b$.