# Phase Splitting

Phase splitting means the creation of a sinusoidal signal like a given one, but displaced in phase by a certain amount. A sinusoidal signal repeats a sequence of values as time passes, and the cycle is divided into 360 electrical degrees or 2π radians, like a circle. A rotating phasor is a graphic representation of these changes. The unit phasor exp(jωt) is said to have phase ωt in degrees or radians, where ω is the angular frequency 2πf radians per second. For 60 Hz, the angular frequency is the familar 377 rad/s. One signal is said to lead another if it goes through its phases a constant time before the other does, and this time can be specified as a fixed phase difference φ, which is more useful than a specification in terms of time. The phase lead is expressed as multiplication of the phasor by a complex factor exp(jφ). A signal is said to lag if it goes through its phases at a constant later time than the reference signal. Of course, a lag of 240° is the same as a lead of 120°, since the sinusoidal signals are periodic with period 360°.

Phase leads and lags are familiar from AC circuits. For a capacitor, some current must flow before the voltage can rise, so current leads voltage, which can be remembered by iCe. An inductor, on the other hand, requires a voltage to produce an increase in the current, so voltage leads current, or eLi. The phasor diagrams for series RC and RL circuits should be familiar. Since V = ZI, the inductive impedance jωL implies that V = ωLI will lead by 90°, since j = exp(j90°). Similarly, capacitative impedance 1/jωC implies that V will lag I, since -j = exp(-j90°).

Phase Inversion

A special case of phase splitting is phase inversion, where a copy of a signal in antiphase is required. A phase inverter is required when one wishes to go from single-ended to double-ended operation. We have already seen how this can be done with a differential amplifier, where the signals taken from the two collectors are opposite in phase.

A transformer with a center-tapped secondary also produces phase inversion, since the signals at the ends of the secondary are in antiphase with respect to the center tap. This is a passive phase inverter, but a very practical one that was often used with vacuum tubes.

Two transistors or vacuum tubes can be used in push-pull. In this case, they are fed by opposite phases and biassed to be normally off. The tubes conduct on alternate half-cycles, each supplying half the output. This is called Class B operation, and is much more efficient than Class A operation where a tube or transistor must be biased so that its ouput can swing both ways. In Class B, no output current flows when there is no signal, saving a great waste of power and greatly increasing the efficiency of the amplifier. This has been discussed in more detail in another page.

There are two convenient methods of phase inversion (in addition to the differential amplifier). The first depends on taking the outputs from resistors carrying the same current, which is controlled by the input signal, so that the phases are opposite. There is no voltage gain in this kind of phase inverter, so a preceding voltage amplification stage is usually required. A transistor phase inverter is shown at the left. The circuit is often called a paraphase amplifier. It should be biassed so that there is sufficient “headroom” for the two ouput signals. As the emitter voltage increases, the collector voltage decreases, and the limit is set when they collide, saturating the transistor. In this circuit, I have allowed about 5 V between collector and emitter when resting, which allows a peak-to-peak output of about this amount. The bias divider could probably be increased in value by a factor of 10 if a higher input impedance is required.

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