3.20 Analog signal processing problems  (Page 2/6)

Bridge circuits

Circuits having the form of [link] are termed bridge circuits .

1. What resistance does the current source see when nothing is connected to the output terminals?
2. What resistor values, if any, will result in a zero voltage for $v_{\mathrm{out}}$ ?
3. Assume $R_1=1\Omega$ , ${\mathrm{R}}_2=2\Omega$ , ${\mathrm{R}}_3=2\Omega$ and ${\mathrm{R}}_4=4\Omega$ . Find the current $i$ when the current source $i_{\mathrm{in}}$ is $\Im ((4+2i)e^{i\times 2\pi \times 20t})$ . Express your answer as a sinusoid.

Cartesian to polar conversion

Convert the following expressions into polar form. Plot their location in the complex plane .

1. $(1+\sqrt{-3})^2$
2. $3+i^4$
3. $\frac{2-i\frac{6}{\sqrt{3}}}{2+i\frac{6}{\sqrt{3}}}$
4. $(4-i^3)(1+i\frac{1}{2})$
5. $3e^{i\pi }+4e^{i\frac{\pi }{2}}$
6. $(\sqrt{3}+i)\times 2\sqrt{2}e^{-(i\frac{\pi }{4})}$
7. $\frac{3}{1+i\times 3\pi }$

The complex plane

The complex variable $z$ is related to the real variable $u$ according to $$z=1+e^{iu}$$

• Sketch the contour of values $z$ takes on in the complex plane.
• What are the maximum and minimum values attainable by $\left|z\right|$ ?
• Sketch the contour the rational function $\frac{z-1}{z+1}$ traces in the complex plane.

Cool curves

In the following expressions, the variable $x$ runs from zero to infinity. What geometric shapes do the following trace in the complex plane?

1. $e^{ix}$
2. $1+e^{ix}$
3. $e^{-x}e^{ix}$
4. $e^{ix}+e^{i(x+\frac{\pi }{4})}$

Trigonometric identities and complex exponentials

Show the following trigonometric identities using complex exponentials. In many cases, they were derived using this approach.

1. $\sin (2u)=2\sin u\cos u$
2. $\cos u^2=\frac{1+\cos (2u)}{2}$
3. $\cos u^2+\sin u^2=1$
4. $\frac{d \sin u}{d u}=\cos u$

Transfer functions

Find the transfer function relating the complex amplitudes of the indicated variable and thesource shown in [link] . Plot the magnitude and phase of the transferfunction.

Circuit a

Circuit b

Circuit c

Circuit d

Using impedances

Find the differential equation relating the indicated variable to the source(s) using impedances for each circuitshown in [link] .

Circuit a

Circuit b

Circuit c

Circuit d

Measurement chaos

The following simple circuit was constructed but the signal measurements were made haphazardly. When the source was $\sin (2\pi f_{0}t)$ , the current $i(t)$ equaled $\frac{\sqrt{2}}{3}\sin (2\pi f_{0}t+\frac{\pi }{4})$ and the voltage $v_2(t)=\frac{1}{3}\sin (2\pi f_{0}t)$ .

1. What is the voltage $v_1(t)$ ?
2. Find the impedances $Z_1$ and $Z_2$ .
3. Construct these impedances from elementary circuit elements.

Transfer functions

In the following circuit , the voltage source equals $v_{\mathrm{in}}(t)=10\sin \left(\frac{t}{2}\right)$ .

1. Find the transfer function between the source and the indicated output voltage.
2. For the given source, find the output voltage.

A simple circuit

You are given this simple circuit .

1. What is the transfer function between the source and the indicated output current?
2. If the output current is measured to be $\cos (2t)$ , what was the source?

Circuit design

1. Find the transfer function between the input and the output voltages for the circuits shown in [link] .
2. At what frequency does the transfer function have a phase shift of zero? What is the circuit's gain atthis frequency?
3. Specifications demand that this circuit have an output impedance (its equivalent impedance) less than 8Ωfor frequencies above 1 kHz, the frequency at which the transfer function is maximum. Find element valuesthat satisfy this criterion.

Equivalent circuits and power

Suppose we have an arbitrary circuit of resistors that we collapse into an equivalent resistor using the series and parallel rules. Is the power dissipated by the equivalent resistor equal to the sum of the powers dissipated by the actual resistors comprising the circuit?Let's start with simple cases and build up to a complete proof.

1. Suppose resistors $R_1$ and $R_2$ are connected in parallel. Show that the power dissipated by $\parallel (R_1, R_1)$ equals the sum of the powers dissipated by the component resistors.
2. Now suppose $R_1$ and $R_2$ are connected in series. Show the sameresult for this combination.
3. Use these two results to prove the general result we seek.