1.4 Graphical Representation

From the above analysis (equations 1.2.7 and 1.2.10) we know that the voltage across the resistor (V_R) is in-phase with I through the circuit. The voltage measured across the inductor (V_L) leads the current by T/4 or π/2 radians (90°) and across the capacitor the voltage V_C lags the current by T/4 or -π/2 (-90°). The appropriate vector plots are shown beneath each circuit element in Figure 1.6 below. The reference current, I is constant in magnitude and in-phase across each circuit element, so it's plotted along the x-axis (the real axis).

Figure 1.6. Vector diagrams of current and voltage relations across series RLC circuit elements.

The complex representation of equation 1.2.10 above is

V = RI + i ωLI + (-i/ωC) I

(1.4.1)

This relation can be written in the same form as Ohm's law (equation 1.2.2) yielding

V = Z I

(1.4.2)

where Z is the complex impedance. Z is the complex sum of the impedances of the separate circuit elements. The impedance of the resistor is R; the magnitudes of the impedances of L and C are the inductive reactance (ωL) and capacitive reactance (1/ωC), respectively.

The vector plot of the various voltages is presented in the Figure 1.7.

Figure 1.7. Combined vector plot of voltage relations across series RLC circuit elements.

Various relations for this figure are: V leads V_R by φ. V_R leads V_C by π/2. V_C lags V_L by π.

The vector plot of the impedances is shown in Figure 1.8.

Figure 1.8. Combined vector plot of impedance relations for RLC circuit elements.

 

Quantities plotted on the real axis are called real or in-phase components. Quantities plotted on the imaginary axis are called imaginary, quadrature, or out-of-phase components. The phases in voltage and current are all temporal or time phases. For example, V leads V_R by (φ/2π)T seconds (s), and V_R leads V_C by ()T = (T/4) s.