**First, the characteristics of pure resistance AC circuit**

An AC circuit without capacitance and inductance and only resistance is called a purely resistive AC circuit, as shown in Figure (a). The waveforms of the current in the circuit and the voltage across the circuit were observed through experiments and an oscilloscope, as shown in Figure (b). Thus, the vector diagrams of current and voltage can be obtained as shown in Figure (c).

Pure Resistive Circuit

1. The relationship between current and voltage in a purely resistive AC circuit

In a purely resistive AC circuit, the current and the voltage have the same frequency, the same initial phase, and the same phase. Instantaneous, rms, and maximum values follow Ohm’s law.

Assuming that the voltage of the circuit is u=Umsin(ωt+φ0)V, and the resistance is R, then

Pure Resistance Ohm’s Law

2. The power of the resistive element

In the AC circuit, the product of the instantaneous value of the voltage and the instantaneous value of the current is called the instantaneous power, which is represented by the symbol p. In a pure resistance circuit, set u=Umsinωt, i=Imsinωt, then

pure resistance power

There are only resistive elements in a purely resistive circuit, so the resistive elements are called dissipative elements in the circuit. The waveform of the power in a purely resistive circuit is shown in the figure.

Pure Resistive Power Waveform

From the mathematical expression and waveform diagram, it can be seen that the law of instantaneous power changes with time is: when the current and voltage are zero, the instantaneous power is also zero; when the current and voltage are at the maximum value, the instantaneous power is also the maximum value. The current and voltage change once, and the instantaneous power changes twice.

In engineering and practice, as long as the average power consumed by the resistance is known, the electric energy consumed by the resistance can be calculated within a certain period of time. Since the average power reflects the power consumption of the resistor, the average power is also called the active power. Therefore, it is more meaningful to use the average power than the instantaneous power.

The average value of the instantaneous power in a period, called the average power, is represented by the symbol P.

Pure resistive active power

**Second, the characteristics of pure inductance AC circuit**

1. Inductive reactance

The resistance of the coil to the alternating current is called the inductive reactance, which is represented by XL. The magnitude of the inductive reactance is proportional to the frequency at which the alternating current changes, and is proportional to the inductance of the coil.

When the angular frequency of the alternating current is ω (or the frequency is f or the period is T) and the inductance of the coil is L, the inductive reactance is:

Inductive reactance

2. The relationship between current and voltage of pure inductive circuit

When the voltage of the pure inductance circuit is u=Umsinωt, the waveform diagram and phasor diagram of the circuit voltage and current are shown in the figure.

Pure Inductor Voltage and Current Relationship

Pure Inductor Voltage and Current Relationship

3. The power of pure inductive circuit

The waveforms of voltage, current, and power for a purely inductive circuit are shown in the figure.

Pure Inductive Power Waveform

The power dissipated by pure inductance is reactive power. The formula is as follows:

pure inductive power

**Third, the characteristics of pure capacitive AC circuit**

1. Capacitance

The blocking effect of the capacitor on the alternating current is called capacitive reactance, which is represented by the symbol XC. The higher the frequency of the AC circuit, the smaller the capacitive reactance; the larger the capacitance of the capacitor, the smaller the capacitive reactance. In an AC circuit, the capacitive reactance can be calculated using the following formula:

capacitive reactance

2. The relationship between current and voltage of pure capacitive circuit

When the voltage of the pure capacitive circuit is u=Umsinωt, the waveforms and vector diagrams of the circuit voltage and current are shown in the figure.

Pure Capacitor Voltage vs Current

Pure Capacitor Voltage vs Current

3. The power of pure capacitive circuit

The waveforms of voltage, current, and power for a purely capacitive circuit are shown in the figure.

Pure capacitive power waveform

It can be found that it is the opposite of the power waveform of pure inductance.

Pure capacitive power

4. Summary

The most important content of this chapter is the capacitive reactance and the calculation formula of inductive reactance, which will be of great help in future circuit design.