Simple Circuit Analysis

Figure 1

Circuit a Circuit b Circuit c (a) (b) (c)

For each circuit shown in Figure 1, the current ii equals cos2⁢π⁢t 2t .

Solving Simple Circuits

Figure 2

Circuit A Circuit B (a) (b)

Equivalent Resistance

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

Figure 3

circuit a circuit b circuit c circuit d (a) (b) (c) (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 the currents or voltages produced in the element by the independent sources. This Principle has important consequences in simplifying the calculation of ciruit variables in multiple source circuits.

Figure 4

Current and Voltage Divider

Use current or voltage divider rules to calculate the indicated circuit variables in Figure 5.

Figure 5

circuit a circuit b circuit c (a) (b) (c)

Thévenin and Mayer-Norton Equivalents

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

Figure 6

circuit a circuit b circuit c (a) (b) (c)

Detective Work

In the depicted circuit, the circuit N 1 N 1 has the v-i relation v 1 =3⁢ i 1 +7 v 1 3 i 1 7 when i s =2 i s 2 .

Figure 7

Bridge Circuits

Circuits having the form of Figure 8 are termed bridge circuits.

Figure 8

Cartesian to Polar Conversion

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

The Complex Plane

The complex variable z z is related to the real variable u u according to z=1+ei⁢u z 1 u

Cool Curves

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

Trigonometric Identities and Complex Exponentials

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

Transfer Functions

Find the transfer function relating the complex amplitudes of the indicated variable and the source shown in Figure 9. Plot the magnitude and phase of the transfer function.

Figure 9

circuit a circuit b circuit c circuit d (a) (b) (c) (d)

Using Impedances

Find the differential equation relating the indicated variable to the source(s) using impedances for each circuit shown in Figure 10.

Figure 10

circuit a circuit b circuit c circuit d (a) (b) (c) (d)

Measurement Chaos

The following simple circuit was constructed but the signal measurements were made haphazardly. When the source was sin2⁢π⁢f0⁢t 2 f0 t , the current i⁢t it equaled 23⁢sin2⁢π⁢f0⁢t+π4 2 3 2 f0 t 4 and the voltage v2⁢t=13⁢sin2⁢π⁢f0⁢t v2t 13 2 f0 t .

Figure 11

Transfer Functions

In the following circuit, the voltage source equals v in ⁢t=10⁢sint2 v in t 10 t 2 .

Figure 12

A Simple Circuit

You are given this simple circuit.

Figure 13

Circuit Design

Figure 14

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.

Power Transmission

The network shown in (Reference) represents a simple power transmission system. The generator produces 60 Hz and is modeled by a simple Thévenin equivalent. The transmission line consists of a long length of copper wire and can be accurately described as a 50Ω resistor.

Figure 15

Simple power transmission system Modified load circuit (a) (b)

Optimal Power Transmission

The following figure shows a general model for power transmission. The power generator is represented by a Thévinin equivalent and the load by a simple impedance. In most applications, the source components are fixed while there is some latitude in choosing the load.

Figure 16

Sharing a Channel

Two transmitter-receiver pairs want to share the same digital communications channel. The transmitter signals will be added together by the channel. Receiver design is greatly simplified if first we remove the unwanted transmission (as much as possible). Each transmitter signal has the form ∀ t ,0≤t≤T: x i ⁢t=A⁢sin2⁢π⁢ f i ⁢t t 0 t T x i t A 2 f i t where the amplitude is either zero or AA and each transmitter uses its own frequency f i f i . Each frequency is harmonically related to the bit interval duration T T, where the transmitter 1 uses the the frequency 1T 1 T . The datarate is 10Mbps.

Circuit Detective Work

In the lab, the open-circuit voltage measured across an unknown circuit's terminals equals sint t . When a 1Ω resistor is place across the terminals, a voltage of 12⁢sint+π4 1 2 t 4 appears.

Mystery Circuit

We want to determine as much as we can about the circuit lurking in the impenetrable box shown in Figure 17. A voltage source vin=2 vin 2 volts has been attached to the left-hand terminals, leaving the right terminals for tests and measurements.

Figure 17

More Circuit Detective Work

The left terminal pair of a two terminal-pair circuit is attached to a testing circuit. The test source v in ⁢t v in t equals sint t (Figure 18).

Figure 18

We make the following measurements.

Linear, Time-Invariant Systems

For a system to be completely characterized by a transfer function, it needs not only be linear, but also to be time-invariant. A system is said to be time-invariant if delaying the input delays the output by the same amount. Mathematically, if S⁢x⁢t=y⁢t S x t y t , meaning y⁢t y t is the output of a system S⁢• S • when x⁢t x t is the input, S⁢• S • is the time-invariant if S⁢x⁢t−τ=y⁢t−τ S x t τ y t τ for all delays τ τ and all inputs x⁢t x t . Note that both linear and nonlinear systems have this property. For example, a system that squares its input is time-invariant.

Long and Sleepless Nights

Sammy went to lab after a long, sleepless night, and constructed the circuit shown in Figure 19. He cannot remember what the circuit, represented by the impedance ZZ, was. Clearly, this forgotten circuit is important as the output is the current passing through it.

Figure 19

A Testing Circuit

The simple circuit here was given on a test.

Figure 20

When the voltage source is 5⁢sint 5 t , the current i⁢t=2⁢cost−arctan2−π4 i t 2 t 2 4 .

Black-Box Circuit

You are given a circuit that has two terminals for attaching circuit elements.

Figure 21

When you attach a voltage source equaling sint t to the terminals, the current through the source equals 4⁢sint+π4−2⁢sin4⁢t 4 t 4 2 4 t . When no source is attached (open-circuited terminals), the voltage across the terminals has the form A⁢sin4⁢t+φ A 4 t φ .

Solving a Mystery Circuit

Sammy must determine as much as he can about a mystery circuit by attaching elements to the terminal and measuring the resulting voltage. When he attaches a 1Ω resistor to the circuit's terminals, he measures the voltage across the terminals to be 3⁢sint 3 t . When he attaches a 1F capacitor across the terminals, the voltage is now 3×2⁢sint−π4 3 2 t 4 .

Find the Load Impedance

The depicted circuit has a transfer function between the output voltage and the source equal to H⁢f=-8⁢π2⁢f28⁢⁢π2⁢f2+4+i⁢6⁢π⁢f H f -8 2 f 2 -8 2 f 2 4 6 f .

Figure 22

Analog “Hum” Rejection

“Hum” refers to corruption from wall socket power that frequently sneaks into circuits. “Hum” gets its name because it sounds like a persistent humming sound. We want to find a circuit that will remove hum from any signal. A Rice engineer suggests using a simple voltage divider circuit consisting of two series impedances.

Figure 23

Figure 24

An Interesting Circuit

Figure 25

A Simple Circuit

You are given the depicted circuit.

Figure 26

An Interesting and Useful Circuit

The depicted circuit has interesting properties, which are exploited in high-performance oscilloscopes.

Figure 27

The portion of the circuit labeled "Oscilloscope" represents the scope's input impedance. R 2 =1⁢MΩ R 2 1 MΩ and C 2 =30⁢pF C 2 30 pF (note the label under the channel 1 input in the lab's oscilloscopes). A probe is a device to attach an oscilloscope to a circuit, and it has the indicated circuit inside it.

A Circuit Problem

You are given the depicted circuit.

Figure 28

Analog Computers

Because the differential equations arising in circuits resemble those that describe mechanical motion, we can use circuit models to describe mechanical systems. An ELEC 241 student wants to understand the suspension system on his car. Without a suspension, the car's body moves in concert with the bumps in the raod. A well-designed suspension system will smooth out bumpy roads, reducing the car's vertical motion. If the bumps are very gradual (think of a hill as a large but very gradual bump), the car's vertical motion should follow that of the road. The student wants to find a simple circuit that will model the car's motion. He is trying to decide between two circuit models (Figure 29).

Figure 29

Here, road and car displacements are represented by the voltages v road ⁢t v road t and v car ⁢t v car t , respectively.

Transfer Functions and Circuits

You are given the depicted network.

Figure 30

Fun in the Lab

You are given an unopenable box that has two terminals sticking out. You assume the box contains a circuit. You measure the voltage sint+π4 t 4 across the terminals when nothing is connected to them and the current 2⁢cost 2 t when you place a wire across the terminals.

Dependent Sources

Find the voltage v out v out in each of the depicted circuits.

Figure 31

circuit a circuit b (a) (b)

Operational Amplifiers

Find the transfer function between the source voltage(s) and the indicated output voltage for the circuits shown in Figure 32.

Figure 32

op-amp a (a) op-amp b (b) op-amp c (c) op-amp d (d)

Op-Amp Circuit

The following circuit is claimed to serve a useful purpose.

Figure 33

Why Op-Amps are Useful

The circuit of a cascade of op-amp circuits illustrate the reason why op-amp realizations of transfer functions are so useful.

Figure 34

Figure 35

Operational Amplifiers

Consider the depicted circuit.

Figure 36

Designing a Bandpass Filter

We want to design a bandpass filter that has transfer the function H⁢f=10⁢i⁢2⁢π⁢f(i⁢f f l +1)⁢(i⁢f f h +1) H f 10 2 f f f l 1 f f h 1 Here, f l f l is the cutoff frequency of the low-frequency edge of the passband and f h f h is the cutoff frequency of the high-frequency edge. We want f l =1⁢kHz f l 1 kHz and f h =10⁢kHz f h 10 kHz .

Pre-emphasis or De-emphasis?

In audio applications, prior to analog-to-digital conversion signals are passed through what is known as a pre-emphasis circuit that leaves the low frequencies alone but provides increasing gain at increasingly higher frequencies beyond some frequency f 0 f 0 . De-emphasis circuits do the opposite and are applied after digital-to-analog conversion. After pre-emphasis, digitization, conversion back to analog and de-emphasis, the signal's spectrum should be what it was.

The op-amp circuit here has been designed for pre-emphasis or de-emphasis (Samantha can't recall which).

Figure 37

Active Filter

Find the transfer function of the depicted active filter.

Figure 38

This is a filter?

You are given a circuit.

Figure 39

Optical Receivers

In your optical telephone, the receiver circuit had the form shown.

Figure 40

This circuit served as a transducer, converting light energy into a voltage v out v out . The photodiode acts as a current source, producing a current proportional to the light intesity falling upon it. As is often the case in this crucial stage, the signals are small and noise can be a problem. Thus, the op-amp stage serves to boost the signal and to filter out-of-band noise.

Figure 41

Reverse Engineering

The depicted circuit has been developed by the TBBG Electronics design group. They are trying to keep its use secret; we, representing RU Electronics, have discovered the schematic and want to figure out the intended application. Assume the diode is ideal.

Figure 42