6 Feedback

Q 6.1

The figure below shows 2 circuits consisting of an opamp with a feedback network. In this exercise you may assume that the opamps are ideal, with infinite voltage gain.

(a)

Derive a relation for the signal transfer $H(j\omega )$ of the circuit with the capacitor, in figure (a).

(b)

Derive a relation $\frac{v_{out}}{v_{in}}(t)$ of circuit (a) - the one with the capacitor. Note that this is a time domain transfer.

(c)

Derive a relation for the signal transfer $H(j\omega )$ of the circuit with the inductor, in figure (b).

(d)

Derive a relation $\frac{v_{out}}{v_{in}}(t)$ of circuit (b): the one with the inductor. Note that this is a time domain transfer.

Q 6.2

To build a stabilized power supply we use the circuit below to regulate $V_{DD}$ down to $v_{OUT}$. We assume $M_1$ works in saturation and that it has infinite output resistance. The opamp can be assumed to be ideal ($r_i\to \infty$, $r_o=0$, $a_v\to \infty$). Derive an expression for the (generated) voltage $v_{OUT}$ as a function of the various circuit parameters.

Q 6.3

Given is the circuit below; the op-amp can be assumed to be ideal.

(a)

Derive an expression for the input impedance $z_{in}$. Please note that this is the impedance seen into the circuit from the dotted line in the circuit schematic. For this question, assume that the circuit is stable.

(b)

Give a graphical representation of the input impedance as a function of frequency, assuming that $Z$ is a capacitor. If any values are missing, then assume some reasonable value and motivate your choices.

(c)

Derive for which source impedances $R_G$ (values, an interval or …) the circuit operates stable for a resistive $Z$.

Q 6.4

The opamp circuit below uses an ideal opamp ($r_{in}\to \infty ,r_{out}=0\mathrm{\Omega },a_V\to \infty )$.

(a)

Derive an equation for (the large signal) $v_{OUT}(v_{IN})$

(b)

In this circuit schematic, the output signal is fed back to the non-inverting input of the opamp. Show or derive that the system is stable despite feedback to the positive input.

(c)

Derive an equation for the large signal $v_X$ as a function of $v_{IN}$ and draw a graph of it. For the MOS transistor, you can assume that it is operated in saturation with $i_D=\frac{1}{2}k{(v_{GS}-V_T)}^2$.

(d)

It was assumed that the transistor works in saturation, for which the square law relation is valid. Identify the range of $v_{IN}$ where this assumption is satisfied.

(e)

What happens to $v_X$ when $v_{IN}$ approaches $V_{DD}$? Does it asymptotically approach a value? Does it go to 0, $V_{DD}$, $\infty$, -$\infty$, or some other value?

Q 6.5

A non-inverting amplifier has a voltage gain equal to $a_V=5$. It has to amplify a sinusoidal signal with an amplitude of $A_{sig}=5V$ and a frequency $f$. What slew-rate should the opamp have to be able to amplify this signal without slewing-induced distortion?

Q 6.6

In the circuit below, the opamp is ideal ($r_{in}\to \infty ,r_{out}=0,a_V\to \infty )$. For the BJT, you may assume that $i_C=I_{C0}\cdot e^{}$

q$\cdot v_{BE}kT$, with a finite $\alpha fe$.

(a)

Draw a large signal graph of $v_{OUT}$ as a function of $v_{IN}$.

(b)

Describe the range of $v_{OUT}$ for which the specified BJT element equation(s) hold.

(c)

What happens to $v_X$ for values of $v_{IN}$ very close to ground?

Q 6.7

The figure below shows a filter circuit with an opamp.

(a)

For this question, assume that for the opamp $r_{in}\to \infty$. The gain of the opamp is finite and frequency dependent. The opamp is unity gain stable.

Describe what it means that the opamp is unity gain stable.

(b)

With the same assumptions for the opamp, is the circuit stable or not stable?

(c)

Derive an expression for the signal transfer $H(j\omega )$ of the circuit . For this question, please assume an ideal opamp; assume $R_2=R_3$.

(d)

Draw the Bode plot of the transfer (modulus and phase).

Q 6.8

Given is the opamp circuit below. The opamp in this circuit is almost ideal, with $r_{in}\to \infty$, $r_{out}=0$, but the opamp has a finite voltage gain $A$.

Derive an equation for the transfer function $H(j\omega )=\frac{v_{out}(j\omega )}{i_{in}(j\omega )}$ and draw a (half) bode plot of the magnitude, indicating the frequencies of the poles, zeros and the level of the flat parts.

Q 6.9

In the circuit below, the opamp is almost ideal: $r_{in}\to \infty$, $r_{out}=0$), but this opamp does have a finite voltage gain $A$.

(a)

Derive an expression for the output impedance $z_o$ of the circuit (looking into the port identified by $v_{out}$).

(b)

Draw a bode plot (asymptotes only is sufficient) of this output impedance. Indicate the values of the poles/zeros (x-axis) and flat parts (y-axis).

Q 6.10

In the voltage follower below, the opamp is not ideal. It does have a very high input impedance $r_{in}$, its open loop voltage gain is $A(j\omega )=\frac{A_0}{1+j\omega {\tau }_1}$, and its open-loop output resistance $r_{out}$ is non-zero.

(a)

Derive an expression for the output impedance $r_{out,circuit}$ of the circuit above, looking into $v_o$. Manipulate it to standard form and draw a bode plot (magnitude and phase). Indicate the values of poles, zeros (x-axis) and flat parts (y-axis).

Q 6.11

Given is the circuit below; for the op-amp

• the voltage gain is $A_o$ (finite)

• the output resistance is $r_{out}$.

• the input resistance $r_{in}\to \infty$.

The element equation of the BJT is $i_C=I_{C0}\cdot e^{}$

q$\cdot v_{BE}kT$.

(a)

Derive an expression for the transfer of $v_{out}(v_{in})$ of the circuit. You may assume that the input voltage $v_{in}$ is positive. For simplicity, you may use $r_{out}=0\mathrm{\Omega }$ and $A_0\to \infty$ for this question.

(b)

Derive an expression for the (small signal) output resistance $r_{out,circuit}$ of the circuit as a function of the input voltage of the circuit.

Q 6.12

Given is the following second-order Sallen-Key low-pass filter:

(a)

Show that the transfer function of this filter is $H(j\omega )=\frac{1}{1+2j\omega RC+j^2{\omega }^2{R}^2{C}^2}$ for $A\to \infty$.

(b)

Draw the Bode plot (magnitude and phase) of this $H(j\omega )$

Q 6.13

Given is the following second-order filter:

(a)

Show that the transfer function of this filter is $H(j\omega )=j\omega \frac{L}{R}\cdot \frac{(1-j\omega RC)}{1+j\omega \frac{L}{R}}$.

(b)

Draw the Bode plot (magnitude and phase) of this $H(j\omega )$. You may assume that $RC=L∕R={\omega }_0^{-1}$ for simplicity reasons.