2. Equivalent Circuits

Equivalent circuits summarize the manner in which a signal source interacts with the arrangement to which its output is attached, referred to as the load.

In the accompanying diagram (a) and (b) are alternative equivalent circuits for the simplest situation of a DC source, such as a dry cell. The source, which may be quite complex is contained within the area delineated by the dashed lines and interaction with the external circuitry takes place at the output terminals indicated by open circles. Again no matter how complex the circuitry attached to the source, for the DC case it may always be represented by a resistance referred to as the load and designated R_L. Equivalent circuit (a), referred to as Thevenin's equivalent circuit represents the signal source by an ideal voltage source in series with a resistance referred to as the source resistance. Clearly the voltage across the output terminals resulting in the presence of an external load is given by

In the case R_L>>R_s the output voltage equals the source voltage independent of the actual magnitude of the load. In this region of operation the source is referred to as a constant voltage source.

Norton's equivalent circuit shown in (b) represents the signal source by an ideal current source with the source resistance in parallel. The output current through the load is given by

and now constant current operation is reached for the situation R_L<<R_s. It is easy to see by considering circuit (a) in the constant current regime that the source current is the ratio of the source voltage to the source resistance.

These circuits may be generalized to time varying signal sources by replacing resistances by impedances.

2.1 Harmonic signals

The fundamental passive circuit elements are the resistor, capacitor and inductor, each of which involves a relationship between the voltage across the element v(t) and the current through the element i(t). These relations are generally differential equations, viz:

A simpler situation arises in the case of AC or time harmonic variations. These may be defined as the class of all variations associated with a single frequency, of general form

where w=2f is the angular frequency. Each member of the class is defined by the pair of variables {A,q}, the amplitude and the phase. An alternative representation is that of the phasor, essentially a two-dimensional vector, which arises from the trigonometric expansion

The individual cosine and sine time variations may be considered as orthogonal, defining an x- and y-axis respectively. Thus every member in the class may now be represented by the pair of numbers (Acosq,Asinq) defining a point in the x-y plane and an associated vector. Thus the quantity Acosq represents the component of the signal oscillating in phase with a pure cosine signal.

Similarly the component Asinq is oscillating in phase with a pure sine variation and 90^o out of phase with the cosine signal. Eqn (2.4) can be arranged as

Thus the phase angle q corresponds to a time delay t_d=q/w. Note that a phase delay of 90^o corresponds to a time delay of one quarter of a period. The choice of the cosine as the standard form in Eqn(2.4) and the x- and y- axes is arbitrary and any consistent convention is applicable. In comparing any two signals, one can always be represented by a phasor along the x-axis. This corresponds to choosing the time origin. The second is then represented by a vector making an angle corresponding to the phase difference between the two signals.

It is well known that there is also an intimate relationship between two-dimensional vectors and complex numbers which leads to the most fruitful representation of harmonic signals. In this approach the signal described by Eqn(2.4)may be considered as the real part of

From the second form in the above equation it is apparent that any member of the class is uniquely described by a single complex amplitude. Note that if the signal were to be graphically represented in this convention by an Argand diagram the y-component would be negative. In signal analysis it is often not necessary to extract the real part of the complex signal, although it is understood that the physical quantity must indeed be real. For example the mean square value of the signal is one half the product of z(t) with z^*(t), the complex conjugate of z.

For harmonic signals the behaviour of the fundamental circuit elements follows Eqn(2.3) by setting

Substituting gives

Using the genreral form V=ZI where Z is the impedance, it follows that the impedance values for the fundamental circuit elements may be written

A combination of circuit elements will behave as a single element of an effective impedance calculated by combining the individual impedances of the components in the combination. For series combinations the combination impedance is the sum of individual impedances. For parallel combinations the inverse of the combination impedance is the sum of the inverse of the individual impedances. The resulting effective impedance is then a general complex number with amplitude and phase. The amplitude is a scaling factor between maximum voltage and current while the phase represents the phase difference between the voltage and current time variations.