Circuit Analysis For Dummies
Overview
Circuits overloaded from electric circuit analysis?
Many universities require that students pursuing a degree in electrical or computer engineering take an Electric Circuit Analysis course to determine who will "make the cut" and continue in the degree program. Circuit Analysis For Dummies will help these students to better understand electric circuit analysis by presenting the information in an effective and straightforward manner.
Circuit Analysis For Dummies gives you clear-cut information about the topics covered in an electric circuit analysis courses to help further your understanding
of the subject. By covering topics such as resistive circuits, Kirchhoff's laws, equivalent sub-circuits, and energy storage, this book distinguishes itself as the perfect aid for any student taking a circuit analysis course.
- Tracks to a typical electric circuit analysis course
- Serves as an excellent supplement to your circuit analysis text
- Helps you score high on exam day
Whether you're pursuing a degree in electrical or computer engineering or are simply interested in circuit analysis, you can enhance you knowledge of the subject with Circuit Analysis For Dummies.
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About The Author
John M. Santiago Jr., PhD, served in the United States Air Force (USAF) for 26 years. During that time, he held a variety of leadership positions in technical program management, acquisition development, and operation research support. While assigned in Europe, he spearheaded more than 40 international scientific and engineering conferences/workshops.
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circuit analysis for dummies
CHEAT SHEET
When doing circuit analysis, you need to know some essential laws, electrical quantities, relationships, and theorems.Ohm’s law is a key device equation that relates current, voltage, and resistance. Using Kirchhoff’s laws, you can simplify a network of resistors using a single equivalent resistor. You can also do the same type of calculation to obtain the equivalent capacitance and inductance for a network of capacitors or inductors.
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Circuitry
Laplace transforms can be used to predict a circuit's behavior. The Laplace transform takes a time-domain function f(t), and transforms it into the function F(s) in the s-domain. You can view the Laplace transforms F(s) as ratios of polynomials in the s-domain. If you find the real and complex roots (poles) of these polynomials, you can get a general idea of what the waveform f(t) will look like.
When dealing with complicated circuits, such as circuits with many loops and many nodes, you can use a few tricks to simplify the analysis. The following circuit analysis techniques come in handy when you want to find the voltage or current for a specific device. They’re also useful when you have many devices connected in parallel or in series, devices that form loops, or a number of devices connected to a particular node.
You can analyze circuits with dependent sources using node-voltage analysis, source transformation, and the Thévenin technique, among others. For analyzing circuits that have dependent sources, each technique has particular advantages.
Utilize node-voltage analysis to analyze circuits with dependent sources
Using node voltage methods to analyze circuits with dependent sources follows much the same approach as for independent sources.
Circuitry
Use superposition to analyze circuits that have lots of voltage and current sources. Superposition helps you to break down complex linear circuits composed of multiple independent sources into simpler circuits that have just one independent source. The total output, then, is the algebraic sum of individual outputs from each independent source.
Circuitry
Use superposition to analyze circuits that have lots of voltage and current sources. Superposition helps you to break down complex linear circuits composed of multiple independent sources into simpler circuits that have just one independent source. The total output, then, is the algebraic sum of individual outputs from each independent source.
An inverting amplifier takes an input signal and turns it upside down at the op amp output. When the value of the input signal is positive, the output of the inverting amplifier is negative, and vice versa. Here is an inverting op amp. The op amp has a feedback resistor R2 and an input resistor R1 with one end connected to the voltage source.
Use op amp circuits to build mathematical models that predict real-world behavior.The mathematical uses for signal processing include noninverting and inverting amplification. One of the most important signal-processing applications of op amps is to make weak signals louder and bigger.
Analyze a basic noninverting op amp circuit
The following example shows how the feedback affects the input-output behavior of an op amp circuit.
Circuitry
Using the Laplace transform as part of your circuit analysis provides you with a prediction of circuit response. Analyze the poles of the Laplace transform to get a general idea of output behavior. Real poles, for instance, indicate exponential output behavior.Follow these basic steps to analyze a circuit using Laplace techniques:
Develop the differential equation in the time-domain using Kirchhoff’s laws and element equations.
Circuitry
Using the Laplace transform as part of your circuit analysis provides you with a prediction of circuit response. Analyze the poles of the Laplace transform to get a general idea of output behavior. Real poles, for instance, indicate exponential output behavior.Follow these basic steps to analyze a circuit using Laplace techniques:
Develop the differential equation in the time-domain using Kirchhoff’s laws and element equations.
Circuitry
A first-order RL parallel circuit has one resistor (or network of resistors) and a single inductor. First-order circuits can be analyzed using first-order differential equations. By analyzing a first-order circuit, you can understand its timing and delays.Analyzing such a parallel RL circuit, like the one shown here, follows the same process as analyzing an RC series circuit.
Circuitry
A first-order RC series circuit has one resistor (or network of resistors) and one capacitor connected in series. First-order RC circuits can be analyzed using first-order differential equations. By analyzing a first-order circuit, you can understand its timing and delays.
Here is an example of a first-order series RC circuit.
Circuitry
You can extend an inverting amplifier to more than one input to form a summer, or summing amplifier. An inverting amplifier takes an input signal and turns it upside down at the op amp output.
Here is an inverting op amp with two inputs. The two inputs connected at Node A (called a summing point) are connected to an inverting terminal.
Circuitry
There’s a special op amp circuit —a differential amplifier, or subtractor — that is actually a combination of a noninverting amplifier and inverting amplifier. A differential amplifier multiplies the difference between two voltages.
Here is an op amp subtractor.
You use superposition to determine the input and the output relationship.
Using the Laplace transform as part of your circuit analysis provides you with a prediction of circuit response. Analyze the poles of the Laplace transform to get a general idea of output behavior. Real poles, for instance, indicate exponential output behavior.Follow these basic steps to analyze a circuit using Laplace techniques:
Develop the differential equation in the time-domain using Kirchhoff’s laws and element equations.
Circuitry
Second-order RLC circuits have a resistor, inductor, and capacitor connected serially or in parallel. To analyze a second-order parallel circuit, you follow the same process for analyzing an RLC series circuit.
Here is an example RLC parallel circuit. The left diagram shows an input iN with initial inductor current I0 and capacitor voltage V0.
Circuitry
The exponential function is a step function whose amplitude Vk gradually decreases to 0. Exponential functions are important to circuit analysis because they’re solutions to many problems in which a circuit contains resistors, capacitors, and inductors.
The exponential waveform is described by the following equation:
The time constant TC provides a measure of how fast the function will decay or grow.
Circuitry
The impulse function, also known as a Dirac delta function, helps you measure a spike that occurs in one instant of time. Think of the spiked impulse function (Dirac delta function) as one that’s infinitely large in magnitude and infinitely thin in time, having a total area of 1. Impulse forces occur for a short period of time, and the impulse function allows you to measure them.
Circuitry
The unit step (Heavyside) function models the behavior of a switch (off/on). The unit step function can describe sudden changes in current or voltage in a circuit. The unit step function looks like, well, a step. Practical step functions occur daily, like each time you turn mobile devices, stereos, and lights on and off.
Circuitry
Capacitors store energy for later use. The capacitance is the ratio between the amount of charge stored in the capacitor and the applied voltage. Capacitance is measured in farads (F).
Find the equivalent capacitance of parallel capacitors
You can reduce capacitors connected in parallel or connected in series to one single capacitor.
When doing circuit analysis, you need to know some essential laws, electrical quantities, relationships, and theorems.Ohm’s law is a key device equation that relates current, voltage, and resistance. Using Kirchhoff’s laws, you can simplify a network of resistors using a single equivalent resistor. You can also do the same type of calculation to obtain the equivalent capacitance and inductance for a network of capacitors or inductors.
Mesh-current analysis (loop-current analysis) can help reduce the number of equations you must solve during circuit analysis. Mesh-current analysis is simply Kircholff’s voltage law adapted for circuits that have many devices connected in multiple loops.
Analyze two-mesh circuits
This section walks you through mesh-current analysis when you have two equations, one for Mesh A and one for Mesh B.
Circuitry
There are many applications for an RLC circuit, including band-pass filters, band-reject filters, and low-/high-pass filters. You can use series and parallel RLC circuits to create band-pass and band-reject filters. An RLC circuit has a resistor, inductor, and capacitor connected in series or in parallel.You can get a transfer function for a band-pass filter with a parallel RLC circuit, like the one shown here.
Circuitry
There are many applications for an RLC circuit, including band-pass filters, band-reject filters, and low-/high-pass filters. You can use series and parallel RLC circuits to create band-pass and band-reject filters. An RLC circuit has a resistor, inductor, and capacitor connected in series or in parallel.
RLC series band-pass filter (BPF)
You can get a band-pass filter with a series RLC circuit by measuring the voltage across the resistor VR(s) driven by a source VS(s).
Circuitry
In circuits, inductors resist instantaneous changes in current and store magnetic energy. Inductors are electromagnetic devices that find heavy use in radiofrequency (RF) circuits. They serve as RF “chokes,” blocking high-frequency signals.
This application of inductor circuits is called filtering. Electronic filters select or block whichever frequencies the user chooses.
Circuitry
If you can use a second-order differential equation to describe the circuit you’re looking at, then you’re dealing with a second-order circuit. Circuits that include an inductor, capacitor, and resistor connected in series or in parallel are second-order circuits. Here are second-order circuits driven by an input source, or forcing function.
Circuitry
Transistors are amplifiers in which a small signal controls a larger signal. One type of transistor is a junction field-effect transistor (JFET). JFET transistors provide a good picture of how transistor circuits work, so it’s a good idea to start with them when studying circuit analysis.
The two primary types of transistors are bipolar transistors and field-effect transistors.
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When analyzing circuits, you can simplify networks consisting of only resistors, capacitors, or inductors by replacing them with one equivalent device. The following equations show equivalent series and parallel connections for resistor-only, capacitor-only, and inductor-only combinations.
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A Thévenin or Norton equivalent circuit is valuable for analyzing the source and load parts of a circuit. Thévenin’s and Norton’s theorems allow you to replace a complicated array of independent sources and resistors, turning the source circuit into a single independent source connected with a single resistor.
Circuitry
A Thévenin or Norton equivalent circuit is valuable for analyzing the source and load parts of a circuit. Thévenin’s and Norton’s theorem allow you to replace a complicated array of independent sources and resistors, turning the source circuit into a single independent source connected with a single resistor.
Using the Thévenin or Norton equivalent of a circuit allows you to avoid having to reanalyze the entire circuit over and over again, just to try out different loads.
Capacitors store energy for later use. The instantaneous power of a capacitor is the product of its instantaneous voltage and instantaneous current. To find the instantaneous power of the capacitor, you need the following power definition, which applies to any device:
The subscript C denotes a capacitance device (surprise!
Circuitry
After finding the zero-input response and the zero-state response of an RL parallel circuit, you can easily find the total response of the circuit. Remember that a first-order RL parallel circuit has one resistor (or network of resistors) and a single inductor.
Here is a sample RL circuit shown with zero-input response and zero-state response.
Circuitry
After finding the zero-input response and the zero-state response of an RC series circuit, you can easily find the total response of the circuit. Remember that a first-order RC series circuit has one resistor (or network of resistors) and one capacitor connected in series.
Here is a sample RC circuit shown with zero-input response and zero-state response.
Circuitry
To find the total response of an RC series circuit, you need to find the zero-input response and the zero-state response and then add them together. A first-order RC series circuit has one resistor (or network of resistors) and one capacitor connected in series.
Here is an RC series circuit broken up into two circuits.
Circuitry
To find the total response of an RL parallel circuit such as the one shown here, you need to find the zero-input response and the zero-state response and then add them together.After fiddling with the math, you determine that the zero-input response of the sample circuit is this:Now you are ready to calculate the zero-state response for the circuit.
Circuitry
Thévenin’s and Norton’s theorems can be used to analyze complex circuits by focusing on the source and load circuits. One application of Thévenin’s and Norton’s theorems is to calculate the maximum power for a load circuit.
The power p coming from the source circuit to be delivered to the load depends on both the current i flowing through the load circuit and the voltage v across the load circuit at the interface between the two circuits.
Circuitry
Use the concept of impedance to gernalize Ohm’s law in phasor form so you can apply and extend the law to capacitors and inductors. After describing impedance, you use phasor diagrams to show the phase difference between voltage and current. These diagrams show how the phase relationship between the voltage and current differs for resistors, capacitors, and inductors.
Circuit analysis techniques in the s-domain are powerful because you can treat a circuit that has voltage and current signals changing with time as though it were a resistor-only circuit. That means you can analyze the circuit algebraically, without having to mess with integrals and derivatives. Here you learn how to apply voltage and current divider methods in the s-domain.
Circuitry
Here’s your chance to convert light into electricity using simple operational circuits. You can apply a similar approach to develop instruments that measure other physical variables in the environment, such as temperature and pressure.
You use an input transducer to turn a physical variable into an electrical variable.
Circuitry
Filter circuits (such as low-pass filters, high-pass filters, band-pass filters, and band-reject filters) shape the frequency content of signals by allowing only certain frequencies to pass through. You can describe these filters based on simple circuits.
You find the sinusoidal steady-state output of the filter by evaluating the transfer function T(s) at s = jω.
Circuitry
You might need to know how to build band-stop filters to reject line noise. In your stereo or entertainment system, you have unique sounds coming from your favorite music or movies. Special audio effects, different voices, and diverse instruments are synthesized to form a wide range of frequencies. You select, reject, or boost the bass, treble, and midrange frequencies by using an equalizer consisting of many different filters.
Circuitry
When timing is off in your computer, specific events don’t occur in the right order. But if you know the physics and i-v relationships of resistors and capacitors, you can create a circuit that detects pulses; then when a pulse is missing, the circuit can trigger an alarm notifying the user of a timing problem.
Circuitry
If you understand the basic building blocks of op amp circuits, you’re ready to tackle complex processing actions with op amps. Using op amp circuits, you can analyze an instrumentation amplifier, solve mathematical equations, or create systems for signal processing, instrumentation, filtering, process control, or digital-to-analog/analog-to-digital conversion.
Circuitry
By adding a capacitor to an operational-amplifier (op amp) circuit, you can use the op amp circuit to do more-complex mathematical operations, like integration and differentiation. Practically speaking, you use capacitors instead of inductors because inductors are usually bulkier than capacitors.
Create an op amp integrator
Here is an op amp circuit that has a feedback element as a capacitor.
Circuitry
The op amp circuit can solve mathematical equations fast, including calculus problems such as differential equations. To solve a differential equation by finding v(t), for example, you could use various op amp configurations to find the output voltage vo(t) = v(t).
To simplify the problem, assume zero initial conditions: zero initial capacitor voltage for each integrator as shown here.
A phasor is a complex number in polar form that you can apply to circuit analysis. When you plot the amplitude and phase shift of a sinusoid in a complex plane, you form a phase vector, or phasor.
As you might remember from algebra class, a complex number consists of a real part and an imaginary part. For circuit analysis, think of the real part as tying in with resistors that get rid of energy as heat and the imaginary part as relating to stored energy, like the kind found in inductors and capacitors.
Circuitry
Voltages across each device in a circuit can be described using node-voltage analysis (NVA). Node-voltage analysis reduces the number of equations you have to deal with when performing circuit analysis. Key ingredients of NVA include node voltages and reference nodes.
When a voltage source is connected to a node, you end up with fewer unknown node voltage equations because one of the node voltages is given in terms of the known voltage source.
Circuitry
Laplace transform methods can be employed to study circuits in the s-domain. Laplace techniques convert circuits with voltage and current signals that change with time to the s-domain so you can analyze the circuit's action using only algebraic techniques.
Connection constraints are those physical laws that cause element voltages and currents to behave in certain ways when the devices are interconnected to form a circuit.
Dependent sources are used to model transistors and the operational amplifier IC. A dependent source is a voltage or current source controlled by either a voltage or a current at the input side of the device model. The dependent source drives the output side of the circuit.
Dependent sources are usually associated with components (or devices) requiring power to operate correctly.
Circuitry
With simple RC circuits, you can build first-order RC low-pass (LPF) and high-pass filters (HPF). These simple circuits can give you a foundational understanding of how filters work so you can build more-complex filters.
First-order RC low-pass filter (LPF)
Here's an RC series circuit — a circuit with a resistor and capacitor connected in series.
The op amp circuit is a powerful took in modern circuit applications. You can put together basic op amp circuits to build mathematical models that predict complex, real-world behavior. Commercial op amps first entered the market as integrated circuits in the mid-1960s, and by the early 1970s, they dominated the active device market in analog circuits.
Capacitors store energy for later use. The voltage and current of a capacitor are related. The relationship between a capacitor’s voltage and current define its capacitance and its power. To see how the current and voltage of a capacitor are related, you need to take the derivative of the capacitance equation q(t) = Cv(t), which is
Because dq(t)/dt is the current through the capacitor, you get the following i-v relationship:
This equation tells you that when the voltage doesn’t change across the capacitor, current doesn’t flow; to have current flow, the voltage must change.
Circuitry
To study a range of frequencies, you use Bode plots. Bode plots help you visualize how poles and zeros affect the frequency response of a circuit. You can express the frequency response gain |T(jω)| in terms of decibels. Using decibels compresses the magnitude and the frequency in a logarithmic scale so you don’t need more than 10 feet of paper for your plots.
Circuitry
With transformation, you can modify a complex circuit so that in the transformed circuit, the devices are all connected in series or in parallel. By transforming circuits, you can apply shortcuts such as the current divider technique and the voltage divider technique to analyze circuits.
Each device in a series circuit has the same current, and each device in a parallel circuit has the same voltage.
The sinusoidal functions (sine and cosine) appear everywhere, and they play an important role in circuit analysis. The sinusoidal functions provide a good approximation for describing a circuit’s input and output behavior not only in electrical engineering but in many branches of science and engineering.
The sinusoidal function is periodic, meaning its graph contains a basic shape that repeats over and over indefinitely.
Circuit analysis is a tricky subject, and it’s easy to make certain mistakes, especially when you’re first starting out. You can reduce your odds of making these common mistakes by reviewing the following list.
Failing to label voltage polarities and current directions
When you analyze any circuit, the first step is to properly label the voltage polarities and current direction for each device in the circuit.
Circuitry
At the most basic level, analyzing circuits involves calculating the current and voltage for a particular device. That’s where device and connection equations come in. Device equations describe the relationship between voltage and current for a specific device. One of the most important device equations is Ohm’s law, which relates current (I) and voltage (V) using resistance (R), where R is a constant: V = IR or I = V/R or R = V/I.
Circuitry
Although circuit analysis is typically used to analyze what a circuit is doing, you can also use circuit analysis to design a circuit to perform a particular function. Knowing how to analyze circuits allows you to add the appropriate elements to a circuit during the design phase so that the circuit performs the way you want it to.
Circuitry
Circuit analysis involves designing new circuits as emerging technologies become commonplace. And of course, integrating all the components of these new technologies requires circuit analysis. Here are ten exciting technologies used in current and up-and-coming circuits.
Smartphone touchscreens
The touchscreens found on smartphones use a layer of capacitive material to hold an electrical charge; touching the screen changes the amount of charge at a specific point of contact.
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