Chapter 2

Clarifying Basic Circuit Concepts and Diagrams

In This Chapter

Sorting out current-voltage relationships

Mapping out circuits with schematics

Understanding a circuit’s loops and nodes

Before you can begin working with circuits, you need to have a basic understanding of how current and voltage behave in some of the devices most commonly found in circuits. You also need to be able to read basic circuit diagrams, or schematics. This chapter is all about helping you get comfortable with these basics so you can dive confidently into the world of circuit analysis.

Looking at Current-Voltage Relationships

Given that power is a rate of energy transfer, electrical power p(t) is defined as the product of the voltage v(t) and current i(t) as a function of time:

To remember the formula p = iv, I tell students to remember the phrase poison ivy. It may be corny, but it works.

An electrical device absorbs power when p(t) is positive, implying that the current and voltage have the same algebraic sign. The device delivers power when p(t) is negative, implying current and voltage have opposite algebraic signs. See Chapter 1 for details on the passive sign convention and what negative current and negative voltage mean.

Power has units of watts (W), or joules per second. The units of current (coulombs per second) and voltage (joules per coulomb) should cancel out to give you the desired units for power. Here’s the dimensional analysis:

So the power relationship works out as far as units are concerned!

Because power involves current and voltage, understanding the current-voltage (i-v) characteristics of various devices, such as resistors and batteries, is important. Resistors have a very straightforward relationship with voltage and current. In fact, for circuits that contain only resistors and independent power sources, the relationship between current and voltage simply depends on a device’s resistance, which is a constant R. In the following sections, I introduce you to some devices and circuit configurations that provide a certain amount of resistance, no resistance, or infinite resistance.

Absorbing energy with resistors

Resistors are simple electrical devices that appear in almost every circuit. They suck up energy and give it off as heat. An everyday object like a toaster or an incandescent light bulb can be modeled as a resistor.

You may think resistors don’t do much because they waste energy, but they actually have a few important purposes:

Reducing voltage: A resistor can use up some voltage so that not all of the supplied voltage falls on another device. You’re basically dividing the supplied voltage into smaller voltages by adding resistors to a circuit.

Limiting current: If you don’t want current to burn up a device, you can limit current by connecting a resistor to the device.

Timing and filtering: You can use resistors, along with capacitors, to create timing circuits or filters. I discuss timing and filtering in Chapters 12 and 13 and filtering specifically in Chapter 18.

The following sections introduce current-voltage relationships and graph the i-v curves for resistors. They also show you how to calculate the power dissipated as heat.

Applying Ohm’s law to resistors

Ohm’s law says that the current through a linear resistor is proportional to the voltage across the resistor. Mathematically, you have Ohm’s law described as

where v is voltage, i is current, R is resistance, and G is conductance. The resistance R or conductance G is a proportionality constant relating the resistor voltage and its current. For example, if the voltage is doubled, then the current is doubled.

Resistance provides a measure of difficulty in pushing electricity through a circuit. The unit of resistance is ohms (Ω), and the unit of conductance is siemens (S). For fun (if you call algebra fun), you can rearrange Ohm’s law:

When you don’t know the current, use the top equation, and when you don’t know the resistance, use the bottom equation.

Figure 2-1 shows the symbol and i-v characteristic for a linear resistor. The slope of the line gives you conductance G, and the reciprocal of the slope produces the resistance value R.

You can have large current flow for a small applied voltage if the resistance is small enough. Some materials cooled to very low temperatures are superconductors, having near-zero resistance. As soon as current flows in a superconducting circuit, current flows forever unless you disconnect the voltage source.

Illustration by Wiley, Composition Services Graphics

Figure 2-1: The i-v characteristic for resistors.

Calculating the power dissipated by resistors

Because power is p = iv, you can use Ohm’s law, v = iR, to figure the amount of heat a resistor gives off when current flows or voltage is applied across the resistor. Here are two versions of the power-dissipation formula, which you get by plugging in the voltage or current value from Ohm’s law into p = iv:

So by knowing either the voltage or current for a given resistor R, you can find the amount of power dissipated. If you calculate the power dissipated as 0.1 watts, then a ¹⁄₄-watt (0.25-watt) resistor can handle this amount of power. A ¹⁄₈-watt (0.125-watt) resistor should be able to handle that amount as well, but when it comes to power ratings, err on the larger side.

Offering no resistance: Batteries and short circuits

Batteries and short circuits have different i-v characteristics but the same slope (or equivalently, zero resistance), as Figure 2-2 shows. In certain situations, you can remove a battery from a circuit by replacing it with a short circuit, 0 volts (I explain how in Chapter 7). Read on for the details on batteries, short circuits, and their i-v characteristics.

Illustration by Wiley, Composition Services Graphics

Figure 2-2: You get zero resistance and constant voltage from an ideal voltage source or short circuit.

---

---

Batteries: Providing power independently

In circuit analysis, batteries are referred to as independent sources. Specifically, batteries are independent sources of voltage, supplying the circuit with a constant voltage that’s independent of the current. So no matter how much current is drawn from the battery, you still have the same voltage. Figure 2-2 shows the electrical symbol and the i-v characteristic of a battery. Because the slope is infinite, an ideal battery has zero resistance.

You can convert a battery into an independent current source through source transformation, as I explain in Chapter 4. I cover independent current sources later in “Facing infinite resistance: Ideal current sources and open circuits.”

Short circuits: No voltage, no power

Figure 2-2 shows that, like a battery, a short circuit has an infinite slope (and therefore infinite resistance) in its i-v characteristic. And just like a battery, the voltage is also constant: In a short circuit, there’s zero voltage across a wire, no matter how much current flows through it. Because there’s no voltage across a short circuit, there’s zero absorbed power (p = 0 watts).

When you connect two points in a circuit that have different voltages, you get a short circuit. When this happens, you bypass the other parts of a circuit (called the load) and establish a path of low resistance, causing most of the current to flow around or away from some other parts in the circuit. Accidental short circuits, especially between the high and low voltages of a power supply, can cause strong current to flow, possibly damaging or overheating the power supply and the circuit if the circuit isn’t protected by a fuse.

Facing infinite resistance: Ideal current sources and open circuits

Figure 2-3 shows that an ideal current source and an open circuit both have zero slope in their i-v characteristics, meaning that they have infinite resistance. And in both cases, the current is constant.

The infinite resistance makes sense because if current entered the ideal current source, the current would no longer be constant. In circuit analysis, you can remove a current source by replacing it with an infinite resistor or open circuit. (You can read more about this change in Chapter 7.)

Illustration by Wiley, Composition Services Graphics

Figure 2-3: You get infinite resistance and constant current from an ideal current source or open circuit.

An open circuit occurs when there’s no current flow for any applied voltage, like when you blow a fuse. Because there’s no current flow, there’s no power absorbed (p = 0 watts) in an open-circuit device.

All or nothing: Combining open and short circuits with ideal switches

Think of ideal switches as a combination of an open circuit and a short circuit. When a switch is on, you have a short circuit, providing current flow in the circuit. When a switch is off, you have an open circuit, leaving zero current flow. Figure 2-4 illustrates an ideal switch’s i-v characteristic along with its symbol, which shows the switch in the off state.

Because the switch has zero voltage in its on state and zero current in its off state, no power (p = 0 watts) is dissipated in the switch.

Illustration by Wiley, Composition Services Graphics

Figure 2-4: An ideal switch has infinite or zero resistance, depending on whether the switch is on or off.

Mapping It All Out with Schematics

Schematics, which are drawings that symbolize a circuit, help you see the connections between electronic components. They also help you troubleshoot your circuit design during construction. You usually arrange electronic schematics from top to bottom and left to right, following the path to place the components.

Schematics use symbols to represent the different components of circuits. Here are some basic symbols to help you get started:

Wires: Simple conductors, or wires, appear as plain lines in schematics. When two wires cross each other, you know the following:

• If a dot appears at their intersection, the wires are connected (see the top-left diagram in Figure 2-5).

• If the dot is absent or you see a curved bridge over one of the wires, the wires are unconnected (see the top-right diagram in Figure 2-5).

Wires that cross over are found in more-complicated circuits, which appear much later in the book.

Lines don’t necessarily depict actual wires, like the rat’s nest of wires you’d see inside an old radio; the lines simply represent a pathway of conductors. Today you have the more common metallic pathway called traces on a board. If you’ve ever opened up a desktop computer, you’ve seen traces on a big motherboard and wires connecting various devices like power supplies, sound cards, and hard drives.

Gates: Control lines at the gate terminals of switches are represented by dashed lines (see the bottom-left diagram in Figure 2-5). By applying a voltage to the gate terminal, you can control the on and off states of the switch.

Power supplies: Power supplies in schematics incorporate the device symbols I show you in Figures 2-2 and 2-3. You see power supply connections at the bottom right of Figure 2-5. The left diagram shows a way to reduce the clutter found in schematics by not drawing the symbol for the power supply. The schematic on the right shows the ground symbol, which marks a reference point of 0 volts.

Illustration by Wiley, Composition Services Graphics

Figure 2-5: Connection circuit symbols.

Additionally, circuit schematics often depict circular arrangements of electronic devices and junction points. The circular arrangements of electrical devices are called loops, and the junction or connection points are called nodes. I discuss these features next.

Going in circles with loops

When looking at a circuit schematic such as the one in Figure 2-6, you often see a collection of resistors and a battery connected together in some configuration. The loops form circular connections of devices. By definition, a loop occurs when you trace a closed path through the circuit in an orderly way, passing through each device only once.

This method of generating a closed path allows you to get consistent results when analyzing circuits. To form a loop or closed path, you must start at one point in the circuit and end up at the same place, much like going around the block in your neighborhood.

Illustration by Wiley, Composition Services Graphics

Figure 2-6: Schematic with a closed path loop.

As more devices are connected to the circuit, there’s an increased likelihood that more loops will occur. Figure 2-7 shows a circuit with two inner loops (Loops 1 and 2) and one big outer loop (Loop 3).

Illustration by Wiley, Composition Services Graphics

Figure 2-7: Schematic with three loops.

Getting straight to the point with nodes

A node is simply a junction or point where two or more devices are connected. Be sure to add the following important points about nodes to your memory bank:

A node isn’t confined to a point; it includes the wire between devices.

Wires connected to a node have zero resistance.

Figure 2-8, which depicts three nodes (or junctions), emphasizes the preceding points about nodes. The connected devices, which can be either resistors or independent sources (like batteries), are represented as boxes. The dashed lines outline the node points.

Look at Node A, which consists of points 1, 2, and 3. These points are really the same node or point connected by a zero-resistance wire. Similarly, four devices are connected at Node C.

Illustration by Wiley, Composition Services Graphics

Figure 2-8: Circuit schematic with three nodes and five devices.