3.20 Analog signal processing problems

Simple circuit analysis

Circuit a

Circuit b

Circuit c

For each circuit shown in [link] , the current $i$ equals $\cos (2\pi t)$ .

1. What is the voltage across each element and what is the voltage $v$ in each case?
2. For the last circuit, are there element values that make the voltage $v$ equal zero for all time? If so, what element values work?
3. Again, for the last circuit, if zero voltage were possible, what circuit element could substitute for the capacitor-inductor series combination that would yield the same voltage?

Solving simple circuits

1. Write the set of equations that govern Circuit A's behavior.
2. Solve these equations for $i_1()$ : In other words, express this current in terms ofelement and source values by eliminating non-source voltages and currents.
3. For Circuit B, find the value for $R_{\text{L}}$ that results in a current of 5 A passing through it.
4. What is the power dissipated by the load resistor $R_{\text{L}}$ in this case?

Circuit a

Circuit b

Equivalent resistance

For each of the following circuits , find the equivalent resistance using series and parallel combination rules.

Circuit a

Circuit b

Circuit c

Circuit d

Calculate the conductance seen at the terminals for circuit (c) in terms of each element's conductance.Compare this equivalent conductance formula with the equivalent resistance formula you found for circuit (b).How is the circuit (c) derived from circuit (b)?

Superposition principle

One of the most important consequences of circuit laws is the Superposition Principle : The current or voltage defined for any element equals the sum of thecurrents or voltages produced in the element by the independent sources. This Principle has importantconsequences in simplifying the calculation of ciruit variables in multiple source circuits.

1. For the depicted circuit , find the indicated current using any technique you like (youshould use the simplest).
2. You should have found that the current $i$ is a linear combination of the two source values: $i=C_1()v_{\mathrm{in}}()+C_2()i_{\mathrm{in}}()$ . This result means that we can think of the current asa superposition of two components, each of which is due to a source. We can find each component by settingthe other sources to zero. Thus, to find the voltage source component, you can set the current source tozero (an open circuit) and use the usual tricks. To find the current source component, you would set thevoltage source to zero (a short circuit) and find the resulting current. Calculate the total current $i$ using the Superposition Principle. Is applying the SuperpositionPrinciple easier than the technique you used in part (1)?

Current and voltage divider

Use current or voltage divider rules to calculate the indicated circuit variables in [link] .

Circuit a

Circuit b

Circuit c

Thévenin and mayer-norton equivalents

Find the Thévenin and Mayer-Norton equivalentcircuits for the following circuits .

Circuit a

Circuit b

Circuit c

Detective work

In the depicted circuit , the circuit $N_1()$ has the v-i relation $v_1()=3i_1()+7$ when $i_s()=2$ .

1. Find the Thévenin equivalent circuit for circuit $N_2()$ .
2. With $i_s()=2$ , determine $R$ such that $i_1()=-1$ .