Capacitive reactance is an electrical property that arises in alternating current circuits due to the presence of capacitors. It represents the opposition offered by the capacitor to the passage of alternating current, measured in ohms and represented by the letter Xc. Capacitive reactance can be calculated using the formula Xc = 1 / (2πfC), where f represents the frequency of the alternating current in hertz and C the capacitance of the capacitor in farads. The higher the frequency or capacitance of the capacitor, the greater the capacitive reactance and the greater the opposition to the passage of alternating current.
Calculation of capacitive reactance: step by step to determine impedance in circuits.
Capacitive reactance is a term used in the analysis of electrical circuits involving capacitive components. It represents the resistance to the flow of alternating current offered by a capacitor. To calculate capacitive reactance, follow a few simple steps.
The first step is to identify the capacitance of the component in question, represented by the letter C. Then, the formula for capacitive reactance should be used, given by X C = 1 / (2πfC) , where f is the frequency of the alternating current in hertz.
After calculating the capacitive reactance, it can be used together with the circuit resistance to determine the total impedance, represented by the letter Z. Impedance is the combination of the circuit's resistance and reactance, and can be calculated using the Pythagorean theorem in the form of a right triangle, where the hypotenuse is the total impedance.
By following the correct steps to calculate reactance and combine it with resistance, you can determine the total impedance of the circuit and better understand its behavior in relation to alternating current.
Meaning of capacitive reactance: understand how this property works in electrical circuits.
Capacitive reactance is a property present in electrical circuits that relates to a capacitor's ability to resist the passage of alternating current. It is represented by the symbol Xc and is measured in ohms.
When an alternating current passes through a capacitor, it stores electrical energy in its electric field. Capacitive reactance indicates the capacitor's opposition to the passage of this alternating current, due to energy storage.
To calculate capacitive reactance, the formula Xc = 1 / (2πfC) is used, where Xc is the capacitive reactance in ohms, π is the number pi, f is the frequency of the alternating current in hertz and C is the capacitance of the capacitor in farads.
It is essential to understand how it works to ensure the correct sizing and operation of an electrical circuit.
Inductive reactance formula: What is it and how to calculate its value?
Inductive reactance is a quantity that represents the opposition offered by an inductor to the passage of alternating current. It is responsible for producing a 90-degree phase shift between the applied voltage and the current flowing in the circuit. The formula for inductive reactance is given by X L = 2πfL, where X L is the inductive reactance, f is the frequency of the alternating current, and L is the inductance of the inductor.
To calculate the value of inductive reactance, simply substitute the values of fL into the formula and perform the multiplication. For example, if the frequency of the alternating current is 60 Hz and the inductance of the inductor is 0,5 H, the calculation of the inductive reactance would be X L = 2π * 60 * 0,5 = 188,5 Ω.
Inductive reactance is crucial in electrical circuits with inductors, as it directly influences the behavior of alternating current. Understanding how to calculate inductive reactance is essential to ensure proper circuit sizing and operation.
How to calculate impedance in an electrical circuit effectively.
To effectively calculate the impedance in an electrical circuit, it is necessary to consider capacitive reactance, which is the opposition offered by a capacitor to the passage of alternating current. Capacitive reactance is represented by the symbol Xc and is calculated using the formula:
Xc = 1 / (2 * π * f * C)
Where f is the frequency of the alternating current in hertz and C is the capacitance of the capacitor in farads. After calculating the capacitive reactance, it is possible to determine the total impedance of the electrical circuit, which is the combination of resistance and reactance. The total impedance is represented by the symbol Z and is calculated using the formula:
Z = √(R² + Xc²)
Where R is the resistance of the circuit. With these calculations, it is possible to determine the impedance in an electrical circuit effectively, taking capacitive reactance into account.
What is capacitive reactance and how to calculate it?
Capacitive reactance is a resistance element of the capacitor in the load circuit of the flux regulator that opposes the passage of alternating current.
In a circuit composed of a capacitor and activated by an alternating current source, the capacitive reactance X<sub> C</sub> can be defined as follows:
X C = 1 / ωC
Or also:
X C = 1 / 2πfC
Where C is the capacitor capacity and ω is the angular frequency of the source, related to the frequency f by:
ω = 2πf
Capacitive reactance depends inversely on the frequency; therefore, at high frequencies it tends to be small, while at low frequencies the reactance is large.
The International System unit for measuring capacitive reactance is the ohm (Ω), provided that the capacitance of the capacitor C is denoted by the frequency (abbreviated F) and the frequency is expressed in inverse seconds (s⁻¹ ).
While the charge lasts, an alternating voltage and current are established across the capacitor, whose amplitudes or maximum values, denoted respectively as V<sub> C</sub> and I<sub> C</sub> , are related by capacitive reactance in a manner analogous to Ohm's law:
V C = I C ⋅ X C
In a capacitor, the voltage is 90 degrees behind the current, or 90 degrees ahead of the current, as preferred. Either way, the frequency is the same.
When X<sub> C</sub> is very large, the current tends to be small, and as the value of X<sub> C</sub> approaches infinity, the capacitor behaves like an open circuit and the current is zero.
How to calculate capacitive reactance
An example of how to calculate capacitive reactance: suppose a capacitance of 6 µF is connected to an alternating current of 40 V and a frequency f of 60 Hz.
To find the capacitive reactance, the definition given at the beginning is used. The angular frequency ω is given by:
ω = 2πf = 2π x 60 Hz = 377 s -1
Then this result is substituted into the definition:
X C = 1 / ωC = 1 / (377 s -1 x 6 x10 -6 F) = 442,1 ohm
Now let's look at the amplitude of the current flowing in the circuit. Since the source provides a voltage with amplitude V<sub> C</sub> = 40 V, we use the ratio of capacitive reactance, current, and voltage to calculate the current amplitude or peak current:
I C = V C / X C = 40 V / 442,1 ohm = 0,09047 A = 90,5 m A.
If the frequency becomes very large, the capacitive reactance becomes small, but if the frequency becomes 0 and we have a direct current, the reactance will tend to be infinite.
Current and voltage in the capacitor
When a capacitor is connected to an alternating current source, as it oscillates and changes its polarity, the capacitor alternately charges and discharges.
For a frequency of 60 Hz, as in the example, the voltage is positive 60 times per second and negative another 60 times per second.
As the voltage increases, it drives current in one direction, but if the capacitor is discharging, reverse current is produced which opposes the first.
If vC ( t) = Vm sin ωt, knowing that capacitance is the ratio between load and voltage, we will have the load:
C = q / V → q (t) = CV = CV m sin ωt
And, having the charge as a function of time, we will have the current, which is the derivative of this:
i C (t) = CV m ω cos ωt
But sine and cosine are related by: cos α = sin (α + π / 2), therefore:
i C (t) = CV m ω sin (ωt + π / 2) = I C sin (ωt + π / 2)
With I C = CV C ω
As you can see, there is a 90º difference in current advancement compared to voltage, as mentioned at the beginning.
In describing this type of circuit, the concept of a phasor is used , which is very similar to a vector and allows any alternating quantity, such as current, voltage, or impedance, to be represented in the complex plane.
The following figure shows, on the right, the voltage and current phasors in the capacitor, which form a 90º angle between them, which is the phase shift between the two.
On the left are the respective graphs, with different amplitudes but the same frequency. Over time, the current increases to the voltage, and when it is maximum, the current is zero, and when the voltage is zero, the current is maximum, but with reversed polarity.
Complex capacitor impedance
In a circuit with resistors, capacitors and inductances, reactance is the imaginary part of the impedance Z, a complex quantity that, in AC circuits, plays a role similar to that of electrical resistance for direct current circuits.
In fact, the impedance of a circuit is defined as the ratio of voltage to current:
Z = V / I
For a capacitor or capacitor, its impedance is given by the quotient:
Z C = v (t) / i (t) = V C sin ωt / I C sin (ωt + π / 2)
One way to express voltage and current as phasors is by indicating the amplitude and phase angle (polar form):
v(t) = V C ∠ 0º
i(t) = I C ∠ 90º
Therefore:
Z C = V C ∠ 0º / I C ∠ 90º = (V C / I C ) ∠ 0º -90º =
= V C / CV C ω ∠ -90º = (1 / ωC) ∠ -90º =
Z C = (- j) X C
In other words, the impedance of the capacitor is its capacitive reactance multiplied by the negative of the imaginary unit.
Impedance of a series RC circuit
The impedance of an alternating current circuit with resistors, capacitors and inductors can also be represented binomially by:
Z = R + jX
In this equation, R represents the resistance, which corresponds to the real part, j is the imaginary unit and X is the reactance, which can be capacitive or inductive or a combination of both, if these elements are present at the same time in the circuit.
If the circuit contains a resistor and a capacitor in series, its impedance is:
Z = Z R + Z C
Since voltage and current are in phase across a resistance, resistive impedance is simply the value of resistance R.
In the case of capacitive impedance, we have already seen that Z<sub> C</sub> = -jX<sub> C </sub>, therefore, the impedance of the RC circuit is:
Z = R – jX C = R – j (1 / ωC)
For example, in the circuit shown below, whose source is of the form:
100 V ⋅ sen (120πt)
Noting that ω = 120π, the impedance is:
Z = 83,0 – j [(1 / (120π ⋅ 6 x 10 -6 )] ohm = 83,0 – 442,1 j ohm.
Capacitive reactance applications
High-pass filters, low-pass filters, bridge-type circuits for measuring capacitances and inductances, and phase-shift circuits are among the main applications of circuits containing capacitive reactances, in combination with electrical inductances and resistances.
For audio equipment, some speakers come with separate woofer -type drivers (larger) for low frequencies and a tweeter or small speaker for high frequencies. This improves performance and audio quality.
They use capacitors that prevent low frequencies from reaching the tweeter, while an inductor is added to the woofer to prevent high-frequency signals, since inductance has a reactance proportional to frequency: X L = 2πfL .
References
- Alexander, C. 2006. Fundamentals of Electrical Circuits. 3rd Edition. McGraw Hill.
- Bauer, W. 2011. Physics for Engineering and Science. Volume 2. McGraw Hill.
- Figueroa, D. 2005. Series: Physics for Science and Engineering. Volume 6. Electromagnetism. Edited by Douglas Figueroa (USB).
- Giancoli, D. 2006. Physics: Principles with Applications. 6th Ed. Prentice Hall.
- Serway, R., Jewett, J. 2008. Physics for Science and Engineering. Volume 1. 7 ma . Ed. Cengage Learning.