How Much Lead?

If you know the ratio of capacitive reactance to resistance, or XC/R, in an RC circuit, then you can find the phase angle. Of course, you can find this angle if you know the precise values, too.

Pictorial Method

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Pictorial method of finding phase angle in a circuit containing resistance and capacitive reactance.

You can use a protractor and a ruler to find phase angles for RC circuits, just as you did with RL circuits in the previous topics, as long as the angles aren’t too close to 0° or 90°. First, draw a line somewhat longer than 10 cm, going from left to right on the paper. Then, use the protractor to construct a line going somewhat more than 10 cm vertically downward, starting at the left end of the horizontal line. The horizontal line is the R axis of an RC plane. The line going down is the XC axis. If you know the actual values of XC and R, divide or multiply them by a constant, chosen to make both values fall between −100 and 100. For example, if XC = −3800 Ω and R = 7400 Ω, divide them both by 100, getting −38 and 74. Plot these points on the lines. The XC point goes 38 mm down from the intersection point between your two axes. The R point goes 74 mm to the right of the intersection point. Next, draw a line connecting the two points, as shown in Above figure. This line will be at a slant and will form a triangle along with the two axes. This is a right triangle, with the right angle at the origin of the RC plane. Measure the angle between the slanted line and the R axis. Use the protractor for this. Extend the lines, if necessary, using the ruler, to get a good reading on the protractor. This angle will be between 0 and 90°. Multiply this reading by −1 to get the RC phase angle. That is, if the protractor shows 27°, the RC phase angle is −27°.

The actual vector is found by constructing a rectangle using the origin and your two points, making new perpendicular lines to complete the figure. The vector is the diagonal of this rectangle, running out from the origin (following figure). The phase angle is the angle between the R axis and this vector, multiplied by −1. It will have the same measure as the angle of the slanted line you constructed in top figure.

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Another pictorial method of finding phase angle in a circuit containing resistance and capacitive reactance. This method shows the actual impedance vector.

Trigonometric Method

Using trigonometry, you can determine the RC phase angle more precisely than the pictorial method allows. Given the values of XC and R, the RC phase angle is the arctangent of their ratio. Phase angle in RC circuits is symbolized by the lowercase Greek letter φ, just as it is in RL circuits. Here are the formulas:

φ = tan−1 (XC /R) or φ = arctan (XC /R)

When doing problems of this kind, remember to use the capacitive reactance values for XC, and not the capacitance values. This means that, if you are given the capacitance, you must use the formula for XC in terms of capacitance and frequency and then calculate the phase angle. You should get angles that come out negative or zero. This indicates that they’re RC phase angles rather than RL phase angles (which are always positive or zero).

Problem 1

Suppose the capacitive reactance in an RC circuit is −3800 Ω and the resistance is 7400 Ω. What is the phase angle?
Find the ratio XC /R = −3800/7400. The calculator display should show you something like −0.513513513. Find the arctangent, or tan−1, getting a phase angle of −27.18111109° on the calculator display. Round this off to −27.18°.

Problem

Suppose an RC circuit works at a frequency of 3.50 MHz. It has a resistance of 130 Ω and a capacitance of 150 pF. What is the phase angle?
First, find the capacitive reactance for a capacitor of 150 pF at 3.50 MHz. Convert the capacitance to microfarads, getting C = 0.000150 μF. Remember that microfarads go with megahertz (millionths go with millions to cancel each other out). Then:

XC = −1/(6.28 × 3.50 × 0.000150)
= −1/0.003297 = −303 Ω
Now you can find the ratio XC /R = −303/130 = −2.33. The phase angle is equal to the arctangent of −2.33, or −66.8°.

Problem 2

What is the phase angle in the preceding circuit if the frequency is raised to 7.10 MHz?
You need to find the new value for XC, because it will change as a result of the frequency change. Calculating:
XC = −1/(6.28 × 7.10 × 0.000150)
= −1/0.006688 = −150 Ω
The ratio XC /R in this case is equal to −150/130, or −1.15. The phase angle is the arctangent of −1.15, which turns out to be −49.0°.