Exercise 7.2 Input impedances

a)

An answer:
From the description above: $A_v(\mathit{j\omega })=\frac{A_{\mathit{DC}}}{1+\mathit{j\omega }∕{\omega }_0}$ with $A_{\mathit{DC}}=10^5$, ${\omega }_0=2\pi f_{\mathit{UG}}∕A_{\mathit{DC}}$

Indicate if the following statements are true or false. Motivate your answers with a proper explanation or calculation.

b)

An answer:
False. $$\begin{array}{rcll}& z_{\mathit{in}}& \frac{v_{\mathit{in}}}{i_{\mathit{in}}}& \\ & & i_{\mathit{in}}=\frac{v_{-}-v_{\mathit{OUT}}}{R_1}& \\ & & v_{\mathit{OUT}}=-A_v(\mathit{j\omega })\cdot v_{-}& \\ & z_{\mathit{in}}& =\frac{R_1}{1+A_v(\mathit{j\omega })}& \\ & & =\frac{R_1}{1+\frac{A_{\mathit{DC}}}{1+\mathit{j\omega }∕{\omega }_0}}& \end{array}$$

For (radian) frequencies lower than ${\omega }_0$, $A_v$ is almost real and almost $10^5$ and hence there $z_{\mathit{in}}\approx \frac{R_1}{A_{\mathit{DC}}}$ which shows that the statement is false.

c)

An answer:
False. $$\begin{array}{rcll}& z_{\mathit{in}}& =\frac{R_1}{1+A_v(\mathit{j\omega })}& \\ & & =\frac{R_1}{1+\frac{A_{\mathit{DC}}}{1+\mathit{j\omega }∕{\omega }_0}}& \end{array}$$

For all frequencies, the numerator of the equation for $z_{\mathit{in}}(\mathit{j\omega })$ has a real part larger than or equal to 1, and a non-zero imaginary part. The norm of the denominator is hence larger than 1; then $|z_{\mathit{in}}|<R_1$ for all frequencies.

d)

An answer:
True. The opamp has 1 pole. Now an extra pole is added in the feedback path. This results in a phase shift that may be close to $-180°$ at (magnitude of the) loopgain values equal to 1. This then corresponds to a low phase margin.

For nicely behaved amplifiers, the phase margin is usually assumed to be $60°$ or larger.

e)

An answer:
False. The loopgain has a second order low pass transfer function. The gain at a phase shift equal to $-180°$ is zero. The gain margin is therefore large, $\infty$.

f)

An answer:
False.

$$\begin{array}{rcll}& z_{\mathit{in}}& =\frac{R_1}{1+\frac{A_{\mathit{DC}}}{1+\mathit{j\omega }∕{\omega }_0}}& \end{array}$$

This equation may be rewritten in a standard form, but you may also just plug in some extreme values:

$$\begin{array}{rcll}& z_{\mathit{in}}& =\frac{R_1}{1+A_{\mathit{DC}}}\mathrm{\text{(for }}\omega =0)& \\ & z_{\mathit{in}}& =R_1\mathrm{\text{(for }}\omega \to \infty )& \end{array}$$

At lower frequencies, $A$ is higher, resulting in a lower $z_{\mathit{in}}$. There is a frequency range where the $z_{\mathit{in}}$ shows a (first order) frequency dependency, roughly between ${\omega }_0$ and the unity gain (radian) frequency.

g)

An answer:
False.