Circuits that contain both resistors, capacitors, and inductors are common, such as power supply filters, tuners in radios and televisions, and timer circuits.

In the same circuit, the roles of inductive and capacitive reactance are opposite. The total reactance of the circuit is equal to the difference between the inductive and capacitive reactances, as shown in Example 1.

**1. R-L-C series circuit**example 1

A series circuit is shown in Figure 1. Calculate Z, I

_{T}, active power, reactive power, and apparent power, and draw a power triangle.

(1) Calculate impedance

Z=√[R²+(X_{L}-X_{C})²]=√[8²+(12-6)²]=√(8²+6²)Ω=√(64+36)Ω=10Ω

(Note that the reactance of the circuit is equal to the difference between X_{L} and X_{C}.)

(2) Calculate the total current:

I_{T}=E_{T}/Z= 100/10A = 10A

(3) Calculated power:

P_{TRUE} =I²×R=10²×8W=800W

P_{XL}=I²×X_{L}=10²×12W= 1200W

P_{XC}=I²×X_{C}=10I²×6W=600W

P_{VA}=I²_{T}×Z=10²×10W=1000W

and

P_{VA}=√[P²_{TRUE}+(P_{XL}-P_{XC})²]=√[800²+(1200-600)²W=√(800²+600²)W=1000W

(The reactive power of the circuit is the difference between P_{XL} and P_{XC}.)

(4) The power triangle is shown in Figure 2.

**2. R-L-C parallel circuit**Example 2

A parallel R-L-C circuit is shown in Figure 3. Calculate I

_{R}, I

_{XC}, I

_{XL}, I

_{T}, Z, P, P

_{XC}, P

_{XL}, and P

_{VA}, and draw a power triangle.

(1) Calculate I_{R}:

I_{R}=E/R=230/8A=28.75A

(2) Calculate I_{XC}:

I_{XC}=E/X_{C}=230/6A=38.3A

(3) Calculate I_{XL}:

I_{XL}=E/X_{L}=230/12A =19.16A

(4) Calculate I_{T}:

I_{T}=√[I²_{R}+(I_{XL}-I_{XC})²]=√[28.75²+(19.16-38.3)²]A=√(28.75²+(-19.14)²)A=√(826.5+366.3)A=34.5A

(5) Calculate Z:

Z=E_{T}/I_{T}=230/34.5Ω=6.66Ω

(6) Calculate the branch power:

P=I²_{R}×R=28.75²×8W=6612W

P_{XC}=I²_{XC}×X_{C}=38.3²×6W=8801.3W

PXL=I²_{XL}×X_{L}=19.16²×12W=4404.3W

(7) Calculate the apparent power:

P_{VA}=E×I_{T}=230×34.5W=7935W

(8) The power triangle is shown in Figure 4.

Note that the reactive power of this circuit is capacitive (the difference between the calculated result and the power triangle value is due to the rounding error of the calculated number).