Circuit Theory/Circuit Basics

Circuits

Circuits (also known as "networks") are collections of circuit elements and wires. Wires are designated on a schematic as being straight lines. Nodes are locations on a schematic where 2 or more wires connect, and are usually marked with a dark black dot. Circuit Elements are "everything else" in a sense. Most basic circuit elements have their own symbols so as to be easily recognizable, although some will be drawn as a simple box image, with the specifications of the box written somewhere that is easy to find. We will discuss several types of basic circuit components in this book.

Ideal Wires

For the purposes of this book, we will assume that an ideal wire has zero total resistance, no capacitance, and no inductance. A consequence of these assumptions is that these ideal wires have infinite bandwidth, are immune to interference, and are--in essence--completely uncomplicated. This is not the case in real wires, because all wires have at least some amount of associated resistance. Also, placing multiple real wires together, or bending real wires in certain patterns will produce small amounts of capacitance and inductance, which can play a role in circuit design and analysis. This book will assume that all wires are ideal.

Ideal Nodes

Like Ideal Wires, we assume that connecting nodes have zero resistance, et al. Nodes connect two or more wires together. On a schematic, nodes are frequently denoted with a small filled-in black dot. When 2 wires cross on a schematic, but they do not physically intersect (for instance if one wire lays on top of another wire), there is no node drawn.

In real life, nodes are often connected together, either by wire nuts, or solder, or other connectors. These connectors can have a certain amount of associated resistance, capacitance, or inductance associated with them. This book will not, however, take this interference into account, as it is usually negligible.

Active vs Passive

The elements which are capable of delivering energy are called "Active elements".The elements which will receive the energy and dissipate or store it are called "Passive elements".

Voltage and Current Generators are examples of active elements that can deliver the energy from one point to some other point. These are generally considered independent generators of electric energy. From a system point of view, this is not an accurate depiction since the energy output will be directly related to the energy put into the system or stored in the system previously. Some examples of these generators are alternators, batteries etc...

A previous definition stated; "A dependant source will generate current or voltage but the energy output will depend on some other individual parameter(may be voltage or current) in the same circuit, whereas an independent source will generate regardless of the connections of the circuit."

From a localized perspective, this definition can still be useful. This definition can be used to differentiate a power source ("independant source") from an active power control device, or amplifier ("dependant source"). It is probably more useful to think of "dependant sources" as "energy amplifiers" or "active devices".

The three linear passive elements are the Resistor, the Capacitor and the Inductor. Examples of non-linear passive devices would be diodes, switches and spark gaps. Examples of active devices are Transistors, Triacs, Varistors, Vacuum Tubes, relays, solenoids and piezo electric devices.

Open and Closed Circuits

A closed circuit is one in which a series of device(s) complete a connection between the terminals, and charge is allowed to flow freely.

An open circuit is a section of a circuit for which there is no connection. Current does not flow between the terminals of an open circuit, although a voltage may be applied between the terminals, and a capacitance may exist between them. At steady state, there is no current flow in an open circuit, and most examples will assume that there is no capacitance between nodes of an open circuit, for simplicity.

"Shorting" an element

We will often hear the term "shorting an element" in later chapters of circuit analysis. Shorting a circuit is equivalent to placing an ideal wire across the terminals of the element. Because current will take the path of least resistance, shorting an element redirects all current around the element. Because there is no current, the element also has no voltage across its terminals. This practice must be done with care, because reducing the resistance of a certain portion of a circuit to zero can potentially raise the current to infinity, and the circuit will explode.

Ideal Voltmeters

Voltmeters and Ammeters are devices that are used to measure the voltage across an element, and the current flowing through a wire, respectively.

An ideal voltmeter has an infinite resistance (in reality, several megohms), and acts like an open circuit. A voltmeter is placed across the terminals of a circuit element, to determine the voltage across that element.

Voltmeters should never be placed in-line with other circuit elements.

Ideal Ammeters

An ideal ammeter has zero resistance (practically, a few ohms or less), and acts like a short circuit. Ammeters are placed in-line in a circuit, so that that all the current from one terminal flows through the other terminal. By convention, current into the + terminal is displayed as positive.

Ammeters should never be placed parallel to other circuit elements, without some resistance.

Sources

Sources come in 2 basic flavors: Current sources, and Voltage sources. These sources may be further broken down into independent sources, and dependent sources.

Current Sources

Current sources are sources that output a specified amount of current. The voltage produced by the current source will be dependent on the current output, and the resistance of the load (ohm's law).

Voltage Sources

Voltage sources produce a specified amount of voltage. The amount of current that flows out of the source is dependent on the voltage and the resistance of the load (again, ohm's law). This can be dangerous because if a voltage source is shorted (a resistance-less wire is placed across its terminals), the resulting current output approaches infinity! No voltage source in existance can output infinity current, so the source will usually melt or explode long before it reaches that value. This is an important point to keep in mind, however.

An example of a voltage source is a battery, which is specified as being "9V" or "6V" or something similar. The amount of current that the circuit draws from the battery determines how long the "battery life" is.

Ideal Op Amps

Op amps, (short for operational amplifiers) is an active circuit component. We will not discuss the internals of op amps in this wikibook, but will instead only consider the ideal case. Op amps have 2 input terminals and 1 output terminal.

Ideal op amps are governed by some very simple rules that allow an engineer to solve a circuit without having to know exactly how an op amp does what it does. These rules are enumerated as follows.

We will consider an op amp with 2 inputs (x and y), and an output (z).

1. There is 0 voltage difference between the terminals x and y.
2. There is 0 current flowing on terminals x and y.
3. There is 0 current flowing on terminal z.

For instance, if we know that x has a voltage with respect to ground of +5V, then we know that y also has a voltage with respect to ground of +5V.

Independent Sources

Independent sources produce current/voltage at a particular rate that is dependent only on time. These sources may output a constant current/voltage, or they may output current/voltage that varies with time.

Dependent Sources

Dependent sources are current or voltage sources whose output value is based on time or another value from the circuit. A dependent source may be based on the voltage over a resistor for example, or even the current flowing through a given wire. The following sources are possible:

• Current-controlled current source
• Current-controlled voltage source
• Voltage-controlled current source
• Voltage-controlled voltage source

Dependent sources are useful for modelling transistors or vacuum tubes.

Turning Sources "Off"

Occasionally (specifically in Superposition) it is necessary to turn a source "off". To do this, we follow some general rules:

1. Dependent sources cannot be turned off.
2. Current Sources become an Open Circuit when turned off.
3. Voltage Sources become a Closed Circuit when turned off.

Occasionally it is written that the source is "removed" from the circuit, because often it is physically possible to remove the source component (be it a battery, or a plug, or any other source component) physically from the circuit.

Source Warnings

The following image shows some configurations of current and voltage sources that are not permissable, and will cause a problem in your circuit:

Switches

A switch then is a circuit element that is an open-circuit for all time $t<T_0$, and acts like a closed-circuit for all time $t\ge T_0$.

Unit Step Function

Before talking about switches, we will introduce the Heaviside step function $u(t)$ (also known as the unit step function). The step function is defined piecewise as such:

$u(t)=\{\begin{array}{cc}0,& \text{if }t<0\\ 1,& \text{if }t\ge 0\end{array}$

This function provides a mathematical model for electrical engineers to describe circuit elements that change between boolean states (on/off, high/low, etc).

Transducers

A transducer is a circuit component that transforms electrical energy into another type of energy. Some examples of transducers are actuators and motors.