2.2 Concepts and Definitions
2.2.7 Power
Electric power is electric energy transferred per unit time, P (t) = I(t)V (t). Using Ohm’s law, it can be written also as P (t) = I2(t)R. This
implies that the SI unit for electric power is J/s or watt (W).
2.2.8
Capacitance
When a voltage is applied across two conducting plates separated by an insulated gap, a charge will accumulate on each plate. One plate becomes charged positively (+q) and the other charged equally and negatively (−q). The amount of charge acquired is linearly proportional to the applied volt- age. The constant of proportionality is the capacitance, C. Thus, q = CV . The SI unit of capacitance is coulombs per volt (C/V). The symbol for capacitance, C, should not be confused with the unit of coulomb, C. The SI unit of capacitance is the farad (F), named after the British scientist Michael Faraday (1791-1867).
2.2.9
Inductance
When a wire is wound as coil and current is passed through it by applying a voltage, a magnetic field is generated that surrounds the coil. As the current changes in time, a changing magnetic flux is produced inside the coil, which in turn induces a back electromotive force (emf). This back emf opposes the original current, leading to either an increase or a decrease in the current, depending upon the direction of the original current. The resulting magnetic flux, φ, is linearly proportional to the current. The constant of proportionality is called the electric inductance, denoted by L. The SI unit of inductance is the henry (H), named after the American Joseph Henry (1797-1878). One henry equals one weber per ampere.
Element Unit Symbol I(t) V (t) VI=const
Resistor R V (t)/R RI(t) RI
Capacitor C CdV (t)/dt (1/C)R0tI(τ )dτ It/C
Inductor L (1/L)R0tV (τ )dτ LdI/dt 0
TABLE 2.1
Resistor, capacitor, and inductor current and voltage relations.
Example Problem 2.1
Statement: 0.3 A of current passes through an electrical wire when the voltage difference between its ends is 0.6 V. Determine [a] the wire resistance, R, [b] the total amount of charge that moves through the wire in 2 minutes, qtotal, and [c] the electric
power, P .
Solution: [a] Application of Ohm’s law gives R = 0.6 V/0.3 A = 2 Ω. [b] Integration of Equation 2.1 gives q(t) =Rt2
t1 I(t)dt. Because I(t) is constant, qtotal= (0.3 A)(120 s) = 36 C. [c] The power is the product of current and voltage. So, P = (0.3 A)(0.6 V) = 0.18 W = 0.2 W, with the correct number of significant figures.
2.3
Circuit Elements
At the heart of all electrical circuits are some basic circuit elements. These include the resistor, capacitor, inductor, transistor, ideal voltage source, and ideal current source. The symbols for these elements that are used in circuit diagrams are presented in Figure 2.2. These elements form the basis for more complicated devices such as operational amplifiers, sample-and-hold circuits, and analog-to-digital conversion boards, to name only a few (see [2]).
The resistor, capacitor, and inductor are linear devices because the complex amplitude of their output waveform is linearly proportional to the amplitude of their input waveform. A device is linear if [1] the response to x1(t) + x2(t) is y1(t) + y2(t) and [2] the response to ax1(t) is ay1(t), where
a is any complex constant [4]. Thus, if the input waveform of a circuit com- prised only of linear devices, known as a linear circuit, is a sine wave of a given frequency, its output will be a sine wave of the same frequency. Usually, however, its output amplitude will be different from its input am- plitude and its output waveform will lag the input waveform in time. If the lag is between one-half to one cycle, the output waveform appears to lead the input waveform, although it always lags the input waveform. The re-
sponse behavior of linear systems to various input waveforms is presented in Chapter 4. The current-voltage relations for the resistor, capacitor, and inductor are summarized in Table 2.1.
2.3.1
Resistor
The basic circuit element used more than any others is the resistor. Its current-voltage relation is defined through Ohm’s law,
R = V /I. (2.6)
Thus, the current in a resistor is related linearly to the voltage difference across it, or vice versa. The resistor is made out of a conducting material, such as carbon, carbon-film, or metal-film. Typical resistances range from a few ohms to more than 107 Ω.
2.3.2
Capacitor
The current flowing through a capacitor is related to the product of its capacitance and the time rate of change of the voltage difference, where
I = dq dt = C
dV
dt. (2.7)
For example, 1 µA of current flowing through a 1 µF capacitor signifies that the voltage difference across the capacitor is changing at a rate of 1 V/s. If the voltage is not changing in time, there is no current flowing through the capacitor. The capacitor is used in circuits where the voltage varies in time. In a DC circuit, a capacitor acts as an open circuit. Typical capacitances are in the µF to pF range.
2.3.3
Inductor
Faraday’s law of induction states that the change in an inductor’s magnetic flux, φ, with respect to time equals the applied voltage, dφ/dt = V (t). Because φ = LI,
V (t) = LdI
dt. (2.8)
Thus, the voltage across an inductor is related linearly to the product of its inductance and the time rate of change of the current. The inductor is used in circuits in which the current varies in time. The simplest inductor is a wire wound in the form of a coil around a nonconducting core. Most inductors have negligible resistance when measured directly. When used in an AC circuit, the inductor’s back emf controls the current. Larger inductances impede the current flow more. This implies that an inductor in an AC circuit acts like a resistor. In a DC circuit, an inductor acts as a short circuit. Typical inductances are in the mH to µH range.
2.3.4
Transistor
The transistor was developed in 1948 by William Shockley, John Bardeen, and Walter Brattain at Bell Telephone Laboratories. The common transis- tor consists of two types of semiconductor materials, n-type and p-type. The n-type semiconductor material has an excess of free electrons and the p-type material a deficiency. By using only two materials to form a pn junction, one can construct a device that allows current to flow in only one direc- tion. This can be used as a rectifier to change alternating current to direct current. Simple junction transistors are basically three sections of semicon- ductor material sandwiched together, forming either pnp or npn transistors. Each section has its own wire lead. The center section is called the base, one end section the emitter, and the other the collector. In a pnp tran- sistor, current flow is into the emitter. In an npn transistor, current flow is out of the emitter. In both cases, the emitter-base junction is said to be forward-biased or conducting (current flows forward from p to n). The op- posite is true for the collector-base junction. It is always reverse-biased or non-conducting. Thus, for a pnp transistor, the emitter would be connected to the positive terminal of a voltage source and the collector to the nega- tive terminal through a resistor. The base would also be connected to the negative terminal through another resistor. In such a configuration, current would flow into the emitter and out of both the base and the collector. The voltage difference between the emitter and the collector causing this cur- rent flow is termed the base bias voltage. The ratio of the collector-to-base current is the (current) gain of the transistor. Typical gains are up to ap- proximately 200. The characteristic curves of a transistor display collector current versus the base bias voltage for various base currents. Using these curves, the gain of the transistor can be determined for various operating conditions. Thus, transistors can serve many different functions in an elec- trical circuit, such as current amplification, voltage amplification, detection, and switching.
FIGURE 2.4
2.3.5
Voltage Source
An ideal voltage source, shown in Figure 2.4, with Rout = 0, maintains a
fixed voltage difference between its terminals, independent of the resistance of the load connected to it. It has a zero output impedance and can supply infinite current. An actual voltage source has some internal resistance. So the voltage supplied by it is limited and equal to the product of the source’s current and its internal resistance, as dictated by Ohm’s law. A good voltage source has a very low output impedance, typically less than 1 Ω. If the voltage source is a battery, it has a finite lifetime of current supply, as specified by its capacity. Capacity is expressed in units of current times lifetime (which equals its total charge). For example, a 1200 mA hour battery pack is capable of supplying 1200 mA of current for 1 hour or 200 mA for 6 hours. This corresponds to a total charge of 4320 C (0.2 A × 21 600 s).
2.3.6
Current Source
An ideal current source, depicted in Figure 2.4, with Rout = ∞ maintains
a fixed current between its terminals, independent of the resistance of the load connected to it. It has an infinite output impedance and can supply infinite voltage. An actual current source has an internal resistance less than infinite. So the current supplied by it is limited and equal to the ratio of the source’s voltage difference to its internal resistance. A good current source has a very high output impedance, typically greater than 1 MΩ. Actual voltage and current sources differ from their ideal counterparts only in that the actual impedances are neither zero nor infinite, but finite.
2.4
RLC
Combinations
Linear circuits typically involve resistors, capacitors, and inductors con- nected in various series and parallel combinations. Using the current- voltage relations of the circuit elements and examining the potential dif- ference between two points on a circuit, some simple rules for various com- binations of resistors, capacitors, and inductors can be developed.
First, examine Figure 2.5 in which the series combinations of two resis- tors, two capacitors, and two inductors are shown. The potential difference across an i-th resistor is IRi, across an i-th capacitor is q/Ci, and across
an i-th inductor is LidI/dt. Likewise, the total potential difference, VT, for
the series resistors’ combination is VT = IRT, for the series capacitors’
combination is VT = q/CT, and for the series inductors’ combination is
VT = LTdI/dt. Because the potential differences across resistors, capaci-
FIGURE 2.5
Series R, C, and L circuit configurations.
series combination, VT = IR1+ IR2= IRT, which yields
RT = R1+ R2. (2.9)
For the capacitors’ series combination, VT = q/C1+ q/C2 = q/CT, which
implies
1/CT = 1/C1+ 1/C2. (2.10)
For the inductors’ series combination, VT = L1dI/dt + L2dI/dt = LTdI/dt,
which gives
LT = L1+ L2. (2.11)
Thus, when in series, resistances and inductances add, and the reciprocals of capacitances add.
Next, view Figure 2.6 in which the parallel combinations of two resistors, two capacitors, and two inductors are displayed. The same expressions for the i-th and total potential differences hold as before. Hence, for the resistors’ parallel combination, IT = I1+ I2, which leads to
1/RT = 1/R1+ 1/R2. (2.12)
For the capacitors’ parallel combination, qT = q1+ q2, which leads to
CT = C1+ C2. (2.13)
For the inductors’ parallel combination, IT = I1+ I2, which gives
FIGURE 2.6
Parallel R, C, and L circuit configurations.
Thus, when in parallel, capacitances add, and the reciprocals of resistances and inductances add.
Example Problem 2.2
Statement: Determine the total equivalent resistance, RT, and total equivalent
capacitance, CT, for the respective resistance and capacitance circuits shown in Figure
2.7.
Solution: For the two resistors in parallel, the equivalent resistance, Ra, is
1 Ra =1 4+ 1 4= 2 Ω.
The two other resistors are in series with Ra, so RT= Ra+ 2 + 6 = 10 Ω.
For the two capacitors in parallel, the equivalent capacitance, Cb, is Cb= 3 + 3 =
6 µF . This is in series with the two other capacitors, which implies that 1 CT =1 2+ 1 3+ 1 Cb = 1 2+ 1 3+ 1 6= 1. So, CT=1 µF.
To aid further with circuit analysis, two laws developed by G. R. Kirch- hoff (1824 - 1887) can be used. Kirchhoff ’s current (or first) law, which is conservation of charge, states that at any junction (node) in a circuit, the current flowing into the junction must equal the current flowing out of it, which implies that
X node Iin= X node Iout. (2.15)
A node in a circuit is a point where two or more circuit elements meet. Kirchhoff ’s voltage (or second) law, which is conservation of energy, says that around any loop in a circuit, the sum of the potential differences equals zero, which gives
FIGURE 2.7
Resistor and capacitor circuits.
X
i,cl.loop
Vi= 0. (2.16)
A loop is a closed path that goes from one node in a circuit back to it- self without passing through any intermediate node more than once. Any consistent sign convention will work when applying Kirchhoff’s laws to a circuit. Armed with this information, some important DC circuits can be examined now.