3.20 Analog signal processing problems

Circuit detective work

In the lab, the open-circuit voltage measured across an unknown circuit's terminals equals $\sin t$ . When a 1Ω resistor is place across the terminals, avoltage of $\frac{1}{\sqrt{2}}\sin (t+\frac{\pi }{4})$ appears.

1. What is the Thévenin equivalent circuit?
2. What voltage will appear if we place a 1F capacitor across the terminals?

Mystery circuit

We want to determine as much as we can about the circuit lurking in the impenetrable box shown in [link] . A voltage source $v_{\mathrm{in}}=2$ V has been attached to the left-hand terminals, leaving the right terminals for tests and measurements.

1. Sammy measures $v=10$ V when a 1 Ω resistor is attached to the terminals. Samantha says he is wrong.Who is correct and why?
2. When nothing is attached to the right-hand terminals, a voltage of $v=1$ V is measured. What circuit could produce this output?
3. When a current source is attached so that $i=2$ amp, the voltage $v$ is now 3 V. What resistor circuit would be consistent with this and the previous part?

More circuit detective work

The left terminal pair of a two terminal-pair circuit is attached to a testingcircuit. The test source $v_{\mathrm{in}}(t)$ equals $\sin t$ ( [link] ).

We make the following measurements.

• With nothing attached to the terminals on the right, the voltage $v(t)$ equals $\frac{1}{\sqrt{2}}\cos (t+\frac{\pi }{4})$ .
• When a wire is placed across the terminals on the right, the current $i(t)$ was $-\sin t$ .

1. What is the impedance “seen” from the terminals on the right?
2. Find the voltage $v(t)$ if a current source is attached to the terminals on the right so that $i(t)=\sin t$ .

Linear, time-invariant systems

For a system to be completely characterized by atransfer function, it needs not only be linear, but also to be time-invariant. A system is said to betime-invariant if delaying the input delays the output by the same amount. Mathematically, if $S(x(t))=y(t)$ , meaning $y(t)$ is the output of a system $S(•)$ when $x(t)$ is the input, $S(•)$ is the time-invariant if $S(x(t-\tau ))=y(t-\tau )$ for all delays $\tau$ and all inputs $x(t)$ . Note that both linear and nonlinear systems have thisproperty. For example, a system that squares its input is time-invariant.

1. Show that if a circuit has fixed circuit elements (their values don't change over time), itsinput-output relationship is time-invariant. Hint : Consider the differential equation that describes a circuit's input-outputrelationship. What is its general form? Examine the derivative(s) of delayed signals.
2. Show that impedances cannot characterize time-varying circuit elements (R, L, and C).Consequently, show that linear, time-varying systems do not have a transfer function.
3. Determine the linearity and time-invariance of the following. Find the transfer function of the linear,time-invariant (LTI) one(s).
   1. diode
   2. $y(t)=x(t)\sin (2\pi f_{0}t)$
   3. $y(t)=x(t-{\tau }_0)$
   4. $y(t)=x(t)+N(t)$

Long and sleepless nights

Sammy went to lab after a long, sleepless night, and constructed the circuit shown in [link] . He cannot remember what the circuit, represented by theimpedance $Z$ , was. Clearly, this forgotten circuit is important as the output is thecurrent passing through it.

1. What is the Thévenin equivalent circuit seen by the impedance?
2. In searching his notes, Sammy finds that the circuitis to realize the transfer function $$H(f)=\frac{1}{i\times 10\pi f+2}$$ Find the impedance $Z$ as well as values for the other circuit elements.