4.2 Purely Resistive Circuit

4.2 Purely Resistive Circuit

When we apply a sinusoidal voltage source to a resistor, a current appears according to the characteristic or defining equation of the "resistive" element. This current generates heat due to the Joule effect. Slightly different, a circuit that contains a sinusoidal voltage source can also have resistors, coils, capacitors, among others (Fig. 4.3).

Fig. 4.3 Purely resistive circuit. Source Own elaboration

Only by dividing the instantaneous value of the voltage by the value of the resistance (Z=R), we get the instantaneous value of the electric current (Eq. 4.3). If we replace u with the sine function, we get:

$$i=\frac{u}{R}=\frac{U_0\cdot \sin \omega t}{R}=\frac{U_0}{R}\cdot \sin \omega t\text{ A}$$ (4.3)

Obviously, if the instantaneous voltage value equals the maximum value U₀, the result of dividing it by the resistance is the current I₀:

$$I_0=\frac{U_0}{R}$$ (4.4)

As a result, we express the instantaneous value of the current i, as:

$$i=I_0\cdot \sin \omega t$$ (4.5)

We can plot the phasor diagram of voltage (U) and current (I) (Fig. 4.4). We graph these quantities as two vectors rotating at speed ω, counterclockwise. The Y-axis projection of voltage and current waves corresponds to the instantaneous values of current and voltage.

Fig. 4.4 Voltage and current waves in a resistive circuit. Source Own elaboration

As a result, it is possible to conclude that the current oscillates in tandem with the applied voltage. Specifically, the voltage and current fluctuate simultaneously (although with distinct amplitudes), resulting in a phase relationship between them (Fig. 4.5).

Fig. 4.5 Voltage and current Phasors. Source Own elaboration

For practical calculations, it is crucial to use the effective values of voltage and current, known as U and I. The rms values are proportional to the maximum values of current and voltage by √2, and we express it as

$$U_0=\sqrt{2}\cdot U\text{ V}$$ (4.6)

$$I_0=\sqrt{2}\cdot I\text{ A}$$ (4.7)

Substituting Eqs. 4.6 and 4.7 into 4.2 yields:

$$\sqrt{2}\cdot I=\frac{\sqrt{2}\cdot U}{R}$$ (4.8)

This brings us to Ohm's law:

$$I=\frac{U}{R}$$ (4.9)

When we have a circuit where a current with a rms value, I passes through a pure ohmic resistor, a voltage drop occurs. Then, we know it as the "resistance drop or active voltage drop", which is expressed as follows by Ohm's law (Eq. 4.10):

$$U_R=I\cdot R$$ (4.10)