1.6 Circuit Elements

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1.6 Circuit Elements

As we discussed in Section 1.1, an element is the basic building block of a circuit. An electric circuit is simply an interconnection of the elements. Circuit analysis is the process of determining voltages across (or the currents through) the elements of the circuit.

There are tw o types of elements found in electric circuits: passive elements and active elements. An active element is capable of generating energy while a passive element is not. Examples of passive elements are resistors, capacitors, and inductors. Typical active elements include generators, batteries, and operational amplifiers. Our aim in this section is to g ain familiarity with some important acti ve elements.

The most important acti ve elements are v oltage or current sources that generally deliver power to the circuit connected to them. There are two kinds of sources: independent and dependent sources.

An ideal independent source is an active element that provides a speci fied voltage or current that is completely independent of other circuit elements.

Figure 1.11

Symbols for independent voltage sources: (a) used for constant or time-varying voltage, (b) used for constant voltage (dc).

In other words, an ideal independent voltage source delivers to the circuit whatever current is necessary to maintain its terminal v oltage. Physical sources such as batteries and generators may be regarded as approximations to ideal v oltage sources. Figure 1.11 sho ws the symbols for inde pendent voltage sources. Notice that both symbols in Fig. 1.11(a) and (b) can be used to represent a dc v oltage source, b ut only the symbol in Fig. 1.11(a) can be used for a time-v arying voltage source. Similarly, an ideal independent current source is an active element that provides a specified current completely independent of the voltage across the source. That is, the current source deli vers to the circuit whate ver

voltage is necessary to maintain the designated current. The symbol for an independent current source is displayed in Fig. 1.12, where the arrow indicates the direction of current i.

An ideal dependent (or controlled) source is an active element in which the source quantity is controlled by another voltage or current.

Dependent sources are usually designated by diamond-shaped sym bols, as shown in Fig. 1.13. Since the control of the dependent source is achieved by a voltage or current of some other element in the circuit, and the source can be voltage or current, it follows that there are four possible types of dependent sources, namely:

• 1. A voltage-controlled voltage source (VCVS).
• 2. A current-controlled voltage source (CCVS).
• 3. A voltage-controlled current source (VCCS).
• 4. A current-controlled current source (CCCS).

Dependent sources are useful in modeling elements such as transistors, operational amplifiers, and integrated circuits. An example of a currentcontrolled voltage source is sho wn on the right-hand side of Fig. 1.14, where the v oltage 10i of the v oltage source depends on the current i through element C. Students might be surprised that the value of the dependent voltage source is 10 i V (and not 10 i A) because it is a v oltage source. The key idea to keep in mind is that a voltage source comes with polarities (+ −) in its symbol, while a current source comes with an arrow, irrespective of what it depends on.

It should be noted that an ideal v oltage source (dependent or in dependent) will produce any current required to ensure that the termi nal voltage is as stated, whereas an ideal current source will produce the necessary voltage to ensure the stated current flow. Thus, an ideal source could in theory supply an infinite amount of energy. It should also be noted that not only do sources supply po wer to a circuit, the y can absorb power from a circuit too. For a voltage source, we know the voltage but not the current supplied or drawn by it. By the same token, we know the current supplied by a current source b ut not the v oltage across it.

i Figure 1.12

Symbols for: (a) dependent voltage source, (b) dependent current source.

Figure 1.14 The source on the right-hand side is a current-controlled voltage source.

Calculate the power supplied or absorbed by each element in Fig. 1.15. Example 1.7

Solution:

We apply the sign convention for power shown in Figs. 1.8 and 1.9. For p1, the 5-A current is out of the positive terminal (or into the negative terminal); hence,

p¹ = 20(−5) = −100 W Supplied power

For p2 and p3, the current flows into the positive terminal of the element in each case.

p² = 12(5) = 60 W Absorbed power p³ = 8(6) = 48 W Absorbed power

For Example 1.7.

For p4, we should note that the voltage is 8 V (positive at the top), the same as the voltage for p3 since both the passive element and the dependent source are connected to the same terminals. (Remember that v oltage is always measured across an element in a circuit.) Since the current flows out of the positive terminal,

$$p_4 = 8(-0.2I) = 8(-0.2 \times 5) = -8$$

W Supplement power

We should observ e that the 20-V independent v oltage source and 0.2I dependent current source are supplying power to the rest of the network, while the two passive elements are absorbing power. Also,

p¹ + p² + p³ + p⁴ = −100 + 60 + 48 − 8 = 0

In agreement with Eq. (1.8), the total po wer supplied equals the total power absorbed.

Practice Problem 1.7

Figure 1.16 For Practice Prob. 1.7.

Compute the power absorbed or supplied by each component of the circuit in Fig. 1.16.

Answer: p¹ = −45 W, p² = 18 W, p³ = 12 W, p⁴ = 15 W.