4.1 Circuit Theory

Materials are made up of atoms which are composed of protons, neutrons, and electrons. The atom possess the ability to release electrons and gain electrons. This ability is different for different types of materials. In a closed loop of atoms a flow of electrons around the loop can be established by providing a driving force. The closed loop that contains a flow of electrons is referred to as an electric circuit. Materials that establish this flow easily are called conductors. Examples of such materials are metals, water, etc. Materials that do not allow a flow to be established are called insulators. Examples of such materials are dry wood, rubber air, etc. Conductors can be viewed as being on one end of the conductivity spectrum and insulators at the other end, with all other materials falling in this spectrum somewhere in between. To understand the behavior of circuit a good understanding of three concepts are needed; voltage, current, and resistance.

Voltage

The voltage, also known as the potential difference, is the driving force in an electric circuit. If a potential difference (voltage) does not exist between two points of a circuit, then an electron flow or current will not be created. The unit of voltage is the volt (V). Some commonly used voltage sources are: batteries, where the voltage is created through a chemical reaction, and generators, where the voltage is created through mechanical action. When the voltage of a source is high that implies that there is a strong driving force in a circuit. In a circuit the voltages across a source is considered to increase (+), and the voltages across devices or loads are considered to decrease or drop (-).

Current

The current is the rate of charge flow in a circuit. By definition, the flow of current in a circuit is in the opposite direction to the actual flow of electrons. The unit of current is the Ampere or Amp (A). The presence of a voltage in a circuit will establish a current flow within the circuit. If the voltage or driving force is not present, then a current will not be present in the circuit. There are two types of currents to consider: direct current (DC) and alternating current (AC). In DC, the flow of the current is in one direction only and in AC, the current flow alternates between one direction and the opposite direction.

 Fig. 4.1. Direct current (DC)                    Fig. 4.2. Alternating current (AC)

The electrical power source from a wall outlet is an example of AC and the current flow from batteries is an example of DC. Electrical devices behave differently under the two different types of current flow; therefore, it is important to recognize the difference between the two. Fortunately, it is possible to convert from one type to another. Digital circuit components operate generally on 5V DC, so for a computer to use an AC source it must first convert it to DC and then use it. This conversion process is handled by the power supply unit within the computer.

Resistance

Resistance is the material property that opposes a current flow. This quantity represents a material’s position on the conductor-insulator spectrum. High resistance indicates that the material is a poor conductor or a good insulator. The unit of resistance is the Ohm (W). The electrical resistance could be compared to frictional resistance. When pushing a box on the floor, the type of floor determines the resistance to the motion. For example the resistance on rough concrete is higher than the resistance on ice. Similarly different materials oppose current flow to different degrees.

Ohm’s Law

The relationship that exists between the driving force also called voltage (V), the flow of charge or current (I), and the resistance (R) is called Ohm’s Law. The law expressed in mathematical form is written as:

A simple circuit is shown in Figure 4.3. In this circuit the driving force or voltage is provided by a battery, and resistance to the current flow is provided by the light bulb. In this circuit a fixed current flow will be established as long as the voltage remains constant. In the case of the light bulb, the brightness of the bulb is an indication of the amount of current flow. The brighter the bulb, the higher the current flow. If the battery is replaced with another battery with a higher voltage, then the brightness of the bulb will increase due to the increase in current flow. If the voltage is increased to a level such that the bulb cannot carry the increased current, then the bulb will be damaged. The circuit in figure 4.4 has a battery, bulb and wire to carry a current, but due to the incomplete loop of the circuit a current will not be established. This is called an open circuit. No current flow will be present in an open circuit. The previous circuit was an example of a closed circuit. A closed circuit is needed for a current to be present.

Circuit symbols

Representing a circuit using pictures of the different devices is not convenient. It becomes difficult and impractical when the circuits becomes complex with several connected devices. This problem is solved by using symbols to represent the components in the circuit. Each device is represented with a circuit symbol and the circuit is represented using the various symbols. The advantage of the symbols is that it generalizes the concepts, and eliminates the physical characteristics like the shape and size of the devices form the circuit representation and focuses on the electrical characteristics of the circuit.

The circuit in figure 4.4 can be represented in symbolic form by using the symbols shown in figure 4.5. The symbol on the left indicates a voltage source, and the symbol on the right indicates a light bulb. The break on the top part of the circuit is the symbol for a switch. To make this system of representation useful each circuit device or element needs to have a unique symbol. These symbols will be introduced as each circuit device is discussed.

In more complex circuits, the complete loop is not usually drawn, but instead one branch or side of the loop is indicated using a symbol called the ground as shown below.

The ground symbol indicates that the circuits at that point are connected to a common conductor which will complete the loop for the flow of current in the circuit.

Measuring Voltage, Current, and Resistance

The different electrical quantities; voltage, current and resistance, are measured using different measuring devices. A voltmeter is used to measure voltage, an ammeter is used to measure current, and an ohmmeter is used to measure resistance. Since these three quantities are always of interest in circuit analysis, a device called a multimeter that is capable of performing the function of all three measuring devices is commonly used. A multimeter does not perform all three measurements at the same time, but instead it has all three measuring devices built into one unit for convenience.  Like circuit symbols, different symbols are used for the various measuring devices as shown below.

The different meters do not measure the quantities in the same manner. Each has to be connected differently to get the proper measurement.

To take a voltage measurement the voltmeter is connected across a device while the circuit is operational as shown in figure 4.8. This type of connection is referred to as a parallel connection which will be discussed in greater detail later.

In the case of the ammeter, it has to be connected in series (one behind the other) to measure the current in a particular part of the circuit as shown in figure 4.9.

The ohmmeter is special in that it should be used with a device without any power going through it. The leads of the ohmmeter are simply connected across the device to measure the resistance of the device as shown figure 4.10.

If a meter is not connected in the right manner then it would not be able to make the right measurement. Therefore it is important to be familiar with the connection method for the various meters. In the case of the multimeter, the right connection should be employed for the type of measurement being made.

Kirchhoff’s Laws

Ohm’s Law, which was discussed earlier, can only be used to analyze simple circuits with a single loop. This unfortunately is not sufficient for analyzing complex circuits that are made up of several devices and several paths for current flow. Kirchhoff’s circuit laws are two simple rules for current and voltage values within a complex circuit or circuit with multiple components. These two laws complement Ohm’s law and allow for the analysis of complex circuit designs.

The Current Law:            The currents entering a node must equal the sum of the currents leaving the node.

Kirchhoff’s Current Law (KCL) is centred on the concept of a node which is essentially a connection point of current carrying conductors. Imagine a node in terms of an intersection of two roads. The number of cars entering the intersection should be equal to the number of cars leaving. This same idea is valid for the flow of charge through a node (intersection). If the current law is not valid, then it implies that charge is created, destroyed, or stored at the node, and that does not happen.

Figure 4.11 shows a node that has been created by connecting wires together at a single point. If the current flow in each wire is represented using variables I₁, I₂, I₃, I₄, and I₅, then  according to Kirchhoff’s current law the relationship between the currents must be given by   I₁ + I₄ = I₂ + I₃ + I₅ . In this case I₁ + I₄ represents the current entering the node, and I₂ + I₃ + I₅ represents the currents leaving the node. It is important to note that a node does not have to be a single point of contact, but it could be a distributed over a single common connector as shown in figure 4.12.

In this case the node is a logical node instead of a physical one, meaning it also satisfies the current law. Here also it is seen that the current coming in from I₂ must leave through one of the other connectors which is essentially the concept of the node. Kirchhoff’s law applied to this case will result in the current relationship I₂ + I₆ = I₁ + I₃ + I₄ + I₅.

The Voltage Law:            The sum of the voltages around a loop is equal to zero.

Kirchhoff’s Voltage Law (KVL) is centered on a single loop within a circuit. A complex circuit could be analyzed in terms of multiple loops that are present in the circuit. The voltage law simply states that the voltage around any given loop in a circuit will be equal to zero. Remember the voltage across a source is considered to increase (+), and the voltage across a load or device is considered to decrease (-). Therefore, the sum of all the increases and decreases around a single loop must be equal to each other. Consider the multi loop circuit shown below. In this case the sum of the voltages around each loop will be zero.

For the circuit shown in figure 4.13 there are a total of 3 loops. The three loops are indicated in the diagram using the dotted lines and labeled using Roman numerals. The rectangles in the circuit are used to indicate devices with resistance.  In each of the loops, Kirchhoff’s voltage law will hold. The voltage law applied to the three loops will be as follows.

The validity of the two laws could be verified experimentally by creating a simple circuit, and then measuring the voltages around any given loop.

Example:   For the circuit below determine the missing voltages and currents using KCL and KVL. Solution:   Using KCL at node b the current through section BE can be determined. 0.3A = I +0.1A I = 0.2A Using KVL around loop ABEF the voltage Vbe can be determined. +12V - 6V - Vbe =0 Vbe = - 6V Using KVL again around loop BCDE the voltage Vcd can be determined. Veb + Vcd =0 6V + Vcd =0 Vcd = - 6V The above result can also be arrived at by using KVL around the outer loop ACDF. In which case the analysis would be: +12V - 6V - Vcd =0 Vcd = - 6V

Circuit connections

Circuits can be created by wiring voltage sources and devices in various configurations. There is no limit to the complexity of the circuits that can be created. A CPU in a computer is essentially a complex circuit with millions of devices. To facilitate the analysis of circuits, connections within a circuit could be broken down into two types: series connections and parallel connections. Both types of connections have properties that are unique.  Therefore, a good understanding of each type of connection is essential.

The Series Connection

In a series connection the devices are connected in sequence, one behind the other like a train of circuit elements.  In this case the current has only one path to follow, and the same amount of current flows through each of the circuit elements. If the same amount of current does not flow through each device, then that would mean that electrons will have to collect in one part of the circuit, and this does not happen because an excess of like charges will repel each other. Unlike the current, the voltage across each device can be different with the condition that the sum of the voltages across each device is the same as the voltage of the source. This is due to KVL. For the circuit in figure 4.14 V = V₁ + V₂ + V₃ as shown in figure 4.15.

Using light bulbs, a series circuit can be created as shown above. In a series circuit each light bulb is going to provide resistance to the current flow which will reduce the current in the circuit as increasing number of bulbs are added. Therefore, brightness of the individual bulbs will drop as more bulbs are connected in series. To better understand the process consider a circuit with a source and two resistors in series as shown in figure 4.16.

In this circuit the current must be a constant value given that it is a single path, so let the current be I. Using Ohm’s law the voltage across each resistor can be determined as  and . Using KVL we can say that

Let the equivalent effect of these two resistors working together be called R_eq. Which means that R_eq will have the same effect or produce the same current as the two individual resistors R₁ and R₂. If the effect is going to be the same then the current produced by the equivalent single resistor R_eq must be the same as the current in the circuit that has the two resistors R₁ and R₂. 

If the resistance effect is the same then the current in the circuit of figure 4.17 and the current in the circuit of figure 4.16 must be the same. Based on the circuit of figure 4.17, the relationship between V, I and R_eq can be represented as

Since the current I and the voltage V are the same in both circuits, the two equations could be combined to give:

Therefore,                         

Based on the analysis it can be seen that the equivalent resistance equals to the sum of the individual resistances, when the devices are connected in series. This idea could be generalized as:

If each light bulb had a resistance value of 10 Ω, then the equivalent resistance provided by all three bulbs would be (10 Ω + 10 Ω + 10 Ω = 30 Ω). Knowing this, now the current in the circuit could be determined using Ohm’s law if the voltage of the battery is known. Assuming a 9V battery is being used, the current I = V/R = (9V)/(30Ω) =0.3A.

The parallel connection

In a parallel connection, the devices are connected in a side by side arrangement. In a parallel circuit the voltage across each of the circuit devices remains the same due to KVL. Therefore, if the resistances are different for each of the devices, then the current would be different for each device. An example of a parallel connection with light bulbs is shown below.

Fig. 4.18. A parallel circuit.

In the parallel case, the current is different for each device with the condition that the sum of the currents entering a node, the point where the branches separate or rejoin, is equal to the sum of the currents leaving a node. For the diagram below this can be mathematically stated as I_T = I₁ + I₂ + I₃.

Fig. 4.18. Voltages in a series circuit.

In the case of the light bulbs in figure 4.18 the current passing through each light bulb will be the same provided that the bulbs are identical. Therefore, the intensity of the bulbs will remain constant when additional identical bulbs are added on in parallel.

Consider the case of a circuit with two resistors in parallel, shown in figure 4.20. In this circuit the voltage must be the same across both resistors R₁ and R₂. The resistor voltages must be equal to the voltage of the source V according to KVL.

Fig. 4.20. A series resistor circuit.

Using KCL we can say that the current will divide between the two possible paths. The higher current will be in the path of the lower resistance and the lower current will be present in the path of higher resistance. If the current through each resistor is labeled I₁ and I₂ to correspond with R₁ and R₂, then we can say that the current in the main part of the circuit I must be equal to the sum of the currents in the two branches I₁ and I₂.

Like the series case, let the equivalent effect of these two resistors working together be called R_eq. Which means that R_eq will produce the same current as the current produced by the two individual resistors R₁ and R₂ working together. If the effect is going to be the same then the current produced by the equivalent single resistor R_eq must be the same as the current in the circuit that has the two resistors R₁ and R₂.

Fig. 4.21. The equivalent resistor circuit.

If the resistance effect is the same, then the current in the circuit of figure 4.21 and the current in the circuit of figure 4.20 must be the same. Based on the above circuit the relationship between V, I and R_eq can be represented as

Since the current I and the voltage V are the same in both circuits the two equations could be combined to give:

Based on the idea of the same voltage across all devices it is determined that for a parallel connection of resistances the net or equivalent effect is given by

Circuit resistance

Due to the fact that all materials create resistance to current flow, it is important to be aware of the effects of resistance on a circuit. Resistance to current flow is provided by devices to a greater extent and also by the connectors to a lesser extent. When a very long wire is used in a circuit, then that wire will provide a significant resistance to the current flow in the circuit. Therefore, to minimize the resistive effects of the conductors in the circuits, good conducting materials need to be used.  Good conductors are essentially metals, but not all metals perform equally. Silver is the best conductor, but it is costly for regular applications. Copper on the other hand is not as good as silver but close to it and significantly cheaper, therefore leading to its widespread use in circuits. Since the resistance of connectors increases with length, it becomes a significant factor when dealing with computer applications that involve long wire lengths; such as connecting wires in networks. Connectors are usually plated with gold to provide a good connection and low resistance.

Electrical Energy and Power

The standard unit of measure for power is in Watts (W). For example, a 100W light bulb consumes 100J of energy every second, and 100J of energy is approximately the energy required to lift 10kg 1m up. Power can also be expressed in terms of the electrical quantities voltage, current, and resistance as R

The electrical energy that is consumed by the circuit is transformed into other forms of energy like thermal energy (heat) and light. In computer circuits the energy released is mainly in the form of heat or thermal energy. This is of concern to the circuit designer because excessive heat release will elevate the temperature of the object and eventually cause it to fail. For example, if a 3V light bulb is connected to a 12V source the heat generated by the bulb will cause the filament of the bulb to melt and fail. Fuses are like a bulb with thin wire that melts when excess current passes through it and creates an open circuit when the fuse is connected in series. Fuses are an important safety device used to limit current flow in a circuit to protect more expensive circuit components and limit the risk of electrical fires due to heat generated. It is important to note that that a fuse must be wired it series for it to act as a safety device.

Overheating of a device is of concern in computers because it will shorten the life of the device. To solve this problem, cooling systems are usually installed in computers where needed. CPU fans, Heat sinks, and cooling fans are all examples of devices that control the thermal environment of the computer.

             

4.1 Practice Questions 1.     Draw a circuit diagram using symbols that has 2 light bulbs that are connected in series to a 9V battery. 2.     Draw a circuit diagram using symbols that has 2 light bulbs that are connected in parallel which are in turn connected in series to another light bulb and to a 9V battery. 3.     Determine the equivalent resistance of the following. a. 2 Ω, 3 Ω, and 5 Ω in series.                    b. 1 Ω, 3 Ω, and 6 Ω in parallel c. 2Ω, 3Ω, in parallel and in series with 2Ω. 4.     A student has a supply of 2 Ω and 5 Ω resistors. Determine the arrangement that is needed to produce the following equivalent resistance.         a. 1 Ω                  b. 2.5 Ω c. 3.5 Ω              d. 4.75 Ω 5.     An electric device has a resistance of 15 Ω. If it is connected to a 12V supply how much power is consumed by the device? 6.    What would happen when a 60 W bulb rated for 120 V is connected to a 240 V supply? Explain your thinking using the relationships discussed. 7.     What is the main function of the power supply within a computer? 8.     Some computers operate on battery power. Explain why AAA, AA, C, and D sized batteries are all 1.5 V, but are of different physical sizes. What would happen if a device that uses AA batteries is connected to AAA batteries? 9.     Using Kirchhoff’s current laws determine the relationships between the currents. 10.   Using KVL and KCL analyze the circuit below. a. Determine the current IT and the voltages Vbe and Vab b. Predict what would happen to IT if the battery voltage is increased.