Impedance of Resistive Elements

In the previous chapter, we found, for the purely resistive circuit of [Fig. 1], that v and i were in phase, and the magnitude In phasor form, where $V = 0.707 V_m$. Applying Ohm's law and using phasor algebra, we have Since i and v are in phase, the angle associated with i also must be $0^\circ$.
To satisfy this condition, $\theta_R$ must equal $0^\circ$. Substituting $\theta_R = 0^\circ$, we find so that in the time domain, The fact that $\theta_R = 0^\circ$ will now be employed in the following polar format to ensure the proper phase relationship between the voltage and current of a resistor: The boldface roman quantity $Z_R$, having both magnitude and an associated angle, is referred to as the impedance of a resistive element. It is measured in ohms and is a measure of how much the element will "impede" the flow of charge through the network. The above format will prove to be a useful "tool" when the networks become more complex and phase relationships become less obvious. It is important to realize, however, that $Z_R$ is not a phasor, even though the format $R \angle 0$ is very similar to the phasor notation for sinusoidal currents and voltages. The term phasor is reserved for quantities that vary with time, and R and its associated angle of $0^\circ$ are fixed, nonvarying quantities.