# What is an R-L circuit?

The R-L circuit contains resistors and inductors, as shown in Figure 1. If the resistance is very small or equal to zero, the voltage and current waveforms are shown in Figure 2. The power supply voltage has reached the maximum value at the moment when the current has just entered the positive half cycle. The current takes time to build up the magnetic field, and the power supply voltage on the coil has already begun to drop. In a purely inductive circuit, the power supply voltage leads the current by 90°. Note that when the current reaches its maximum value and the magnetic field is built up, the voltage is zero. When the voltage turns negative, the current decreases and the magnetic field gradually collapses. When the resistance in the R-L circuit increases, the phase difference between voltage and current is less than 90°.

The resistance of an inductor to current is called inductive reactance XL, and the unit of measurement is ohms. The calculation formula of inductance is XL=2πfL, where f is frequency; L is inductance; it is equal to 3.14. When calculating impedance, resistance and inductive reactance should be combined, but like capacitive reactance, these two types of values ​​cannot be added directly. The impedance of the R-L series circuit is Z=√(R²+X²L). If R=20Ω and XL=40Ω, then Z=√(20²+40²)Ω. Please note that in this formula, XL will increase with increasing frequency and decrease with decreasing frequency.

①Calculation of R-L series circuit
When applying Ohm’s law, Z can be substituted for R. So I=E/Z, E=I×Z, Z=E/I.

example 1
Using the values ​​in Figure 3, calculate Z, I, VR, and VXL, calculate the sum of the voltage drops of each component, and confirm that it is equal to the power supply voltage.
Impedance is
Z=√(R²+X²L)=√(50²+30²)Ω=58.31Ω
Apply Ohm’s Law
IT=ET/Z=120/58.31A=2.05A
VR=I×R=2.05×50V=103V
VxL=I×XL=2.05×30V=61.5V
The sum of the pressure drops is as follows
E=√(V²R+V²xL)=√(103²+31.5²)V=120V

Example 2
For the circuit shown in Figure 4, when L=79.6mH, calculate the frequency of the voltage source.
The finishing formula XL=2πfL, we get
F=XL/2πL=30/(2×3.14×0.0796)Hz=60)Hz
If the frequency is increased to 120 Hz, calculate XL.
XL=2πfL=2×3.14×120×0.0796Ω=60Ω

As the frequency increases, the inductive reactance increases, so the current in the circuit will decrease; if the frequency decreases, the current will increase. In some applications, such as electric motors on airplanes, the current will decrease when the frequency of the power supply is increased, and the amount of heat generated will also decrease. These will help reduce the weight of the motor.

②Calculation of R-L parallel circuit
The formula for calculating the total impedance in the R-L parallel circuit is
Z=R×XL/√(R²+X²L)

Example 3
For the parallel circuit shown in Figure 4, calculate XL, Z, VR, VC, IR, IxL, and IT.
XL=2πfL=2×3.14×2000×0.0048Ω=60Ω
Impedance is
Z=R×XL/√(R²+X²L)=30×60/√(30²+60²)Ω=1800/67.1Ω=26.8Ω
The voltage on each branch is equal:
VR=VX=E=240V
Use Ohm’s law to calculate IR and IxL:
IR=VR/R=240/30A=8A
IxL=VxL/XL=240/60A=4A
IT is
IT=ET/Z=240/26.8A=8.95A
or
IT=√(I²R+I²xL)=√(8²+4²)A=8.9A
(Either way, the slight difference in the answer comes from the rounding error)

Example 4
In Figure 4, if the frequency becomes 1kHz, calculate XL, Z, VR, VC, IR, IxL, and IT.
XL=2πfL=2×3.14×1000×0.0048Ω=30Ω
Impedance is
Z=R×XL/√(R²+X²L)=30×30/√(30²+30²)Ω=21.2Ω
The voltage on each branch is equal:
VR=VC=E=240V
Calculate the current using Ohm’s law:
IR=VR/R=240/30A=8A
IL=VL/XL=240/30A=8A
IT can be calculated in two ways:
IT=√(I²R+I²L)=√(8²+8²)A=11.3A
or
IT=ET/Z=240/21.2A=11.3A