Frequency

The two variables in any electrical circuit is voltage, V, and current, I. In electronics, all signals are in form of a voltage or a current, physically. Both of these variables can be time varying or constant. Voltages and currents that do not change with respect to time are called d.c. voltages or currents, respectively. The acronym d.c. is derived from direct current.

Voltages and currents that vary with respect to time can, of course, have arbitrary forms. A branch of applied mathematics called Laplace analysis, or its special form Fourier analysis, investigates the properties of such time variation, and shows that all time varying signals can be represented in terms of linear combination (or weighted sums) of sinusoidal waveforms.

A sinusoidal voltage and current can be written as

v(t) =V1cos(cot+9_v), and
i(t) = I1cos(cot+9i).

V1 and I1 are called the amplitude of voltage and current, and have units of Volts (V) and Amperes (A), respectively. "со" is the radial frequency with units of radians per second (rps) and ш = 27if, where "f" is the frequency of the sinusoid with units of Hertz (Hz). "9" is the phase angle of the waveform. These waveforms are periodic, which means that it is a repetition of a fundamental form in every T seconds, where T=1/f seconds (sec).

Quite often, sinusoidal waveforms are referred to by their peak amplitudes or peak-to-peak amplitudes. Peak amplitude of v(t) is V1 Volts peak (or V_p) and peak-to-peak amplitude is 2 V1 Volts peak-to-peak (or Vpp).

Now we can see that a d.c. voltage is in fact a sinusoid with f = 0 Hz. Sinusoidal voltages and currents with non-zero frequency are commonly referred to as a.c. voltages and currents. The acronym a.c. comes from alternating current.

If we know the voltage v(t) across any element and current i(t) through it, we can calculate the power delivered to it as

P(t) = v(t)i(t) = V1I₁cos(cot+9_v) cos(cot+9i)

P(t)= V₁I₁ cos(0 -0i) + V₁I₁ cos(2rot + 0_v+0i).

P(t) is measured in watts (W), i.e. (1V)x(1A)=1 W.

In case of a resistor, both current and voltage have the same phase and hence we can write the power delivered to a resistor as

We shall see that the phase difference between voltage and current in an element or a branch of circuit is a critical matter and must be carefully controlled in many aspects of electronics.

P(t) is called the instantaneous power and is a function of time. We are usually interested in the average power, P_a, which is the constant part of P(t):

P_a = ₁₁ cos(0 -0i),

in general, and

Pa = ₁₁

in case of a resistor.

We note that if the element is such that the phase difference between the voltage across and current through it is 90o, P_a is zero. Inductors and capacitors are such elements.

Radio waves travel at the speed of light, c. The speed of light in air is 3.0 E8 m/sec (through out this book we shall use the scientific notation, i.e. 3.0 E8 for 3x10⁸), to a very good approximation. This speed can be written as

c = X

where X is the wavelength in meters. TRC-10 emits radio waves at approximately 30 MHz (actually between 28 and 29.7 MHz). The wavelength of these waves is approximately 10 meters. The amateur frequency band in which TRC-10 operates is therefore called 10-meter band.