Biblio5

584 REVIEW OF FUNDAMENTALS OF ELECTRICITY WITH TELECOMMUNICATION APPLICATIONS

Figure A.19a A simple ac inductive circuit. Here the current (IL) lags the potential difference (voltage VL) by 908. G is the ac emf source.

A simple inductive circuit is shown in Figure A.19a, and the current and voltage phase relationships of this circuit are shown in Figure A.19b. In this circuit, because it is predominantly inductive, the potential difference (voltage VL) across the inductor leads the current by 908.

Figure A.20a is a simple ac capacitive circuit. In this case the current (IC) leads the voltage (VC) by 908. This is illustrated in Figure A.20b.

The magnitudes of current and voltage at some moment in time are usually analyzed using vectors. Vector analysis is beyond the scope of this appendix.

Effective Emf and Current Values. In a practical sense, an arbitrary standard has been adopted so that only the value of current or voltage need be given to define it, its position in time being understood by the convention adopted. One approach is to state the maximum value of voltage or current. However, this approach has some disadvantages. Another and often more useful approach is the average value over a complete half-cycle (i.e., over p radians). For a sine wave this value equals 0.636 of the maximum value.

Figure A.19b Phase relationship between current (IL) and potential difference (voltage VL) in an ac inductive circuit.

Figure A.20a A simple ac capacitive circuit.

One of the most applicable values is based on the heating effect of a given value of alternating current in a resistor that will be exactly the same as the heating value of direct current in the same resistor. This eliminates the disadvantage of thinking that the effects of alternating and direct currents are different. This is known as the effective value and is equal to the square root of the average of the squares of the instantaneous values over 1 cycle (2p radians). This results in 0.707 times the maximum value or:

where E and I without subscripts are effective values. Unless otherwise stated, ac voltages and currents are always given in terms of their effective values.

Figure A.20b The phase relationship between current (IC) and emf (voltage, VC) for an ac capacitive circuit.

588 REVIEW OF FUNDAMENTALS OF ELECTRICITY WITH TELECOMMUNICATION APPLICATIONS

Figure A.22 A simple ac circuit displaying capacitance only.

where f is in Hz and C is in farads.

The more customary capacitance unit is the microfarad (mF). When we use this unit of capacitance, the formula in Eq. (A.33a) becomes:

Example. Figure A.22 illustrates a capacitive reactance circuit with a standard capacitor of 2.16 mF and an emf of 20 V at 1020 Hz. Calculate the current in amperes flowing in the circuit. Use formula (A.33b):

XC c −1 × 106/ 2 × 3.14159 × 1020 × 2.16

c−72.24 Q ,

I c E/ XC

I c −20/ 72.24 c −0.277 A (minus sign means leading current).

Circuits with Combined Inductive and Capacitive Reactance. To calculate the combined or total reactance when an inductance and a capacitance are in series, the following formula is applicable:

A word about signs: If the calculated value of X is positive, inductive reactance predominates, and if negative, capacitive reactance predominates.

Example. Figure A.23 illustrates an example of a circuit with capacitance and inductance in series. The inductance value is 400 mH and the capacitance is 500 nF. The source emf is 20 V. The frequency is 1020 Hz. Calculate the current in the circuit. Assume the resistance is negligible.

Figure A.23 A simple ac circuit with a coil (inductance) and capacitor in series.

Calculate the combined reactance X of the circuit using formula (A.34). Convert units for capacitance to microfarads, and the units of inductance to henrys. That is 0.5 mf and 0.4 H.

X c 2 × 3.14159 × 1020 × 0.4 − 1 × 106/ 2 × 3.14159 × 1020 × 0.5

c2563.537 − 1, 000, 000/ 3204.42

c2563.537 − 312.068

c2251.469 Q .

Apply Ohm’s law variant to calculate current:

I c E/ X c 20/ 2251.469

c 0.00888 A or 8.88 mA.

A.8.3 Calculating Impedance

When we calculate impedance (Z), we must take into account resistance. All circuits are resistive, even though in some cases there is only a minuscule amount of resistance. We first examine the two reactive possibilities; that is, a circuit with inductive reactance and then a circuit with capacitive reactance. For the case with inductive reactance:

Substituting:

For the case with capacitive reactance:

A.10 RESONANCE 591

A.9 RESISTANCE IN ac CIRCUITS

In certain situations, ac resistance varies quite widely from the equivalent dc resistance, given the same circuit. For example, the resistance of a coil wound on an iron core where the magnetizing effect demonstrates hysteresis (“the holding back”) and resulting eddy currents add to the dc resistance. We find that these effects are a function of frequency, the higher the frequency (f), the greater are these effects. Certain comparable losses may occur in the dielectric materials of capacitors, which may have the effect of increasing the apparent resistance of the circuit.

Still more important is the phenomenon called skin effect. As we increase frequency of an ac current being transported by wire means, a magnetic field is set up around the wire, penetrating somewhat into the wire itself. Counter currents are set up in the wire as a result of the magnetic field, and as frequency increases field penetration decreases, and the magnitude of the counter currents increases. The net effect is to force the current in the wire to flow nearer the surface of the wire instead of being evenly distributed across the cross section of the wire. Because the actual current flow is now through a smaller area, the apparent ac resistance is considerably greater than its effective dc resistance. When working in the radio frequency (RF) domain, this resistance may be very much greater than dc resistance. However, when working with power line frequencies (i.e., 60 Hz in North America and 50 Hz in many other parts of the world), skin effect is nearly insignificant.

A.10 RESONANCE [Refs. 1 and 4]

As we are aware, the value of inductive reactance and the value of capacitive reactance depend on frequency. When the frequency (f) is increased, inductive reactance increases, and capacitive reaction decreases. We can say that at some frequency, the negative reactance XC becomes equal and opposite in value to XL. As a result the reactive component becomes zero, and this is where there is resonance. To determine this frequency we set the combined reactive component equal to zero or:

2pfL − 1, 000, 000/ 2pfC c 0.

The resonant frequency, fr, is

To determine the resonant frequency of a series circuit all we have to know is the value of capacitance and inductance of the circuit.

Example. If the inductance of a particular circuit is 20 mH and the capacitance is 50 nf, what is the resonant frequency? Apply formula (A.38):

fr c 1000/ 2 × 3.14159(0.02 × 50 × 10−9)1/ 2 c 5, 032, 991 Hz or 5, 032, 991 MHz.

It should be noted that the units of capacitance have been changed to microfarads and of inductance to henrys. These are the units of magnitude in Eq. (A.38).

592 REVIEW OF FUNDAMENTALS OF ELECTRICITY WITH TELECOMMUNICATION APPLICATIONS

REFERENCES

1. Principles of Electricity Applied to Telephone and Telegraph Work, American Telephone & Telegraph Co., New York, 1961.

2. IEEE Standard Dictionary of Electrical and Electronics Terms, 6th ed., IEEE Std. 100-1996, IEEE, New York, 1996.

3. D. Halliday et al., Fundamentals of Physics—Extended, 4th ed., Wiley, New York, 1993. 4. H. C. Ohanian, Physics, W. W. Norton & Company, New York, 1985.