TWO SINE WAVE GENERATORS IN LINEAR CIRCUITS

A circuit composed of two sine-wave generators, G1 and G2, and two linear impedances, Z1 and Z2, is shown in figure 1-13. The voltage applied to Z1 and Z2 will be the vector sum of the generator voltages. The sum of the individual instantaneous voltages across each impedance will equal the applied voltages.

Figure 1-13. - Two sine-wave generators with linear impedances.

If the two generator outputs are of the same frequency, then the waveform across Z1 and Z2 will be a sine wave, as shown in figure 1-14, views (A) and (B). No new frequencies will be created. Relative amplitude and phase will be determined by the relative values and types of the impedances.

Figure 1-14A. - Waveforms across two nonlinear impedances.

Figure 1-14B. - Waveforms across two nonlinear impedances.

If the two sine wave generators are of different frequencies, then the sum of the instantaneous values will appear as a complex wave across the impedances, as shown in figure 1-15, views (A) and (B). To determine the wave shape across each individual impedance, assume only one generator is connected at a time and compute the sine-wave voltage developed across each impedance for that generator input. Then, combine the instantaneous voltages (caused by each generator input) to obtain the complex waveform across each impedance. The nature of the impedance (resistive or reactive) will determine the shape of the complex waveform. Because the complex waveform is the sum of two individual sine waves, the composite waveform contains only the two original frequencies.

Figure 1-15A. - Sine-wave generators with different frequencies and linear impedances.

Figure 1-15B. - Sine-wave generators with different frequencies and linear impedances.

Linear impedances may alter complex waveforms, but they do not produce new frequencies. The output of one generator does not influence the output of the other generator.

TWO SINE WAVE GENERATORS AND A COMBINATION OF LINEAR AND NONLINEAR

IMPEDANCES

Figure 1-16 illustrates a circuit that contains two sine-wave generators (G1 and G2), linear impedance Z1, and nonlinear impedance Z2, in series. When a single sine-wave voltage is applied to a combined linear and nonlinear impedance circuit, the voltages developed across the impedances are complex waveforms.

Figure 1-16. - Sine-wave generators with a combination of impedances.

When two sine wave voltages are applied to a circuit, as in figure 1-16, nonlinear impedance Z2 reshapes the two sine-wave inputs and their harmonics, resulting in a very complex waveform.

Assume that nonlinear impedance Z2 will allow current to flow only when the sum of the two sine-wave generators (G1 and G2) has the polarity indicated. The waveforms present across the linear impedance will appear as a varying waveform. This will be a complex waveform consisting of:

• a dc level
• the two fundamental sine wave frequencies
• the harmonics of the two fundamental frequencies
• the sum of the fundamental frequencies
• the difference between frequencies

The sum and difference frequencies occur because the phase angles of the two fundamentals are constantly changing. If generator G1 produces a 10-hertz voltage and generator G2 produces an 11-hertz voltage, the waveforms produced because of the nonlinear impedance will be as shown in the following list:

• a 10-hertz voltage
• an 11-hertz voltage
• harmonics of 10 hertz and 11 hertz (the higher the harmonic, the lower its strength)
• the sum of 10 hertz and 11 hertz (21 hertz)
• the difference between 10 hertz and 11 hertz (1 hertz)

Figure 1-17 illustrates the relationship between the two frequencies (10 and 11 hertz). Since the waveforms are not of the same frequency, the 10 hertz of view (B) and the 11 hertz of view (A) will be in phase at some points and out of phase at other points. You can see this by closely observing the two waveforms at different instants of time. The result of the differences in phase of the two sine waves is shown in view (C). View (D) shows the waveform that results from the nonlinearity in the circuit.

Figure 1-17A. - Frequency relationships.

Figure 1-17B. - Frequency relationships.

Figure 1-17C. - Frequency relationships.

Figure 1-17D. - Frequency relationships.

The most important point to remember is that when varying voltages are applied to a circuit which contains a nonlinear impedance, the resultant waveform contains frequencies which are not present at the input source.

The process of combining two or more frequencies in a nonlinear impedance results in the production of new frequencies. This process is referred to as heterodyning.