Reactance and Impedance of capacitor and inductor: How to Calculate Impedance Triangle

The alternating current (AC) is different from the Directa current (DC) because of its changing behavior. So, the basic elements of a circuit are expected to behave differently in the AC circuit as compared to the DC circuit. The opposition to the alternating current is called impedance rather than resistance. It can be represented with an impedance triangle. We will learn how to calculate impedance triangle? The AC source provides usually provide sinusoidal voltage to the circuit, i.e. mathematically

$v=V_{m}sin \omega t\cdots (1)$

Which cause an AC circuit to flow an AC sinusoidal current in the circuit, i.e. mathematically

$i=I_{m}sin \omega t+\theta $

Where $\omega $ is the angular frequency given by the following relation to the frequency

$\omega =2\pi f$

Before knowing how each element responds to the AC sinusoidal source, let me define the difference in impedance, reactance, and resistance.

Difference Between Impedance, Reactance, and Resistance :

Resistance:

Resistance is the friction experienced by the free electron in every conductor except the superconductor. It is denoted by the capital letter “R” and measured in “Ohms”. The frequency has no effect on the resistance.

Reactance :

Reactance is inertia because of the motion of electron which is present in every current-carrying conductor but most prominent in inductor and capacitor in AC circuit. Reactance has two types; inductive and capacitive reactance. As the name suggests, the inductor provided opposition is called inductance reactance where opposition by the capacitor is called capacitive reactance. Both are denoted by the capital letter “X” with a subscript of “L” for the inductor and “C” for the capacitor. It is also measured in “Ohms”. Reactance changes with a change in frequency.

Impedance :

Any opposition to current an in AC circuit is called impedance, which may be provided by any circuit element like capacitor, inductor, and resistor. The symbol for that is “Z” and the unit for impedance is “Ohm”. This much broader term than resistance and reactance. Where impedance can be obtained from the combination of resistance and reactance vector addition. Impedance can also be affected by a change in frequency.

$Z=\sqrt{R^{2}+(X_{L}sim X_{C})^{2}} $

How to Calculate impedance Triangle

The impedance triangle is the Phasor representation of the impedance of an ac circuit. Resistance is shown on the real axis (x-axis) and reactance is shown on the imaginary axis (y-axis). Their vector addition makes the impedance triangle complete. As the resistance can only be positive, that’s why the triangle always occupies the first and fourth quadrant of the axes.

Using the Pythagoras theorem on the right-angle impedance triangle, we can find the impedance formula which is given below.

$Z=\sqrt{R^{2}+(X_{L}sim X_{C})^{2}} $

Impedance triangle, impedance, reactance and resistance how to calculate impedance triangle

Impedance of Resistor

Suppose the following circuit with an AC source and a resistor. According to the ohm law, the voltage and current are linearly related, i.e.

Resistor in ac circuit

$i=\frac{v}{R}$

By putting the AC voltage from equation (1) into it will give us

$i=\frac{V_{m}sin \omega t}{R}$

$i=\frac{V_{m}}{R} sin \omega t\cdots (2a)$

$i=I_{m}sin \omega t \cdots (2b) $

Comparing the voltage and current equation (1) and (2b), we observe that both current and voltage has the same frequency and phase. See the following graph for better comprehension.

phase relation of current and voltage in AC circui

Putting the equation (2b) into (2a) will give us

$I_{m}sin \omega t=\frac{V_{m}}{R}sin \omega t$

$I_{m}=\frac{V_{m}}{R}\ldots (2c)$

Now comparing the equation (2c) with the following equation

$I=\frac{V}{Z}$

Will give us that

$Z=R\ldots (2d)$

The only opposition to the alternating current is the resistance itself.

Looking at the equation (2d) for the impedance of a resistor, we observe that there is no frequency involvement, so the resistance doesn’t change with a change in frequency.

Impedance of Capacitor

In a pure capacitive AC circuit, the only opposition to the current is the capacitive reactance. According to the capacitor current and voltage relation, the current through the capacitor will only flow if there is a change in the capacitor voltage, i.e.

$i_{c}=C \frac{dv_{c}}{dt}$

Capacitor in AC circuit

By putting the AC source voltage from equation (1)

$i_{c}=C \frac{d}{dt}(V_{m}sin \omega t )$

Using some calculus knowledge, we can find the above derivative as follow

$i_{c}=C (\omega V_{m}cos \omega t )$

$i_{c}=(C\omega V_{m})sin \omega t+90^{o}$

$i_{c}=I_{m}sin \omega t+90^{o}\cdots (3a)$

Where

$I_{m}=C\omega V_{m} \cdots (3b)$

Observing the voltage and current equation (1) and (3a), it implies that current in a purely capacitive circuit is 90o leading from the source voltage. To understand what the statement means, take a look at the following graph.

Phase relation of capacitor in AC circuit

The reactance of a capacitive circuit can be shown by the following generic equation

$I=\frac{V}{X}$

Where comparing this generic equation to the specific equation (3b) will show that

$X_{c}=\frac{1}{C\omega }=\frac{1}{2\pi fC}\ldots (3c)$

X$_{c}$ is the capacitance reactance, the only opposition to AC in a capacitive network.

Observe the equation (3c), we notice that capacitive reactance has an inverse relationship with frequency. By increasing the frequency, the reactance will decrease and vice versa. In a special case of a DC circuit, the capacitor will have infinite impedance, which will cause no current to flow through it. That’s why it is stated that the capacitor is almost open to DC. Where for high frequencies the reactance will too low, that’s why capacitors act short for high-frequency AC.

Impedance of Inductor

In a pure inductor AC circuit, the only opposition provided to the current is by the inductor. The current and voltage relation of an inductor is given by the following equation

$v_{L}=L\frac{di_{L}}{dt}$

Where the AC current is given by the equation

$i=I_{m}sin \omega t$

Inductor in AC circuit

In a pure inductor AC circuit, the only opposition provided to the current is inductive reactance. The current and voltage relation is given by the following equation

$v_{L}=L\frac{di_{L}}{dt}$

Where the AC current is given by the following equation

$i=I_{m}sin \omega t$

Putting it into the above equation will give us

$v_{L}=L\frac{d}{dt} (I_{m}sin \omega t)$

Using some calculus knowledge will help us in finding the above derivative as following

$v_{L}=L (\omega I_{m}cos \omega t )$

$v_{L}=(L\omega I_{m})sin (\omega t+90^{o})\cdots (4a)$

$v_{L}=V_{m}sin (\omega t+90^{o})$

Where

$V_{m}=\omega LI_{m}\cdots (4b)$

Comparing the voltage and current through the inductor, the current through the inductor lags the source voltage by 90o. For more comprehension, take a look at the following graph of the current and voltage phase relation of an inductor.

current is leading in inductor

The inductive reactance can be found by comparing the equation (4b) with the following generic equation

$V=IX$

Which will give us

$X_{L}=\omega L=2\pi fL\ldots (4c)$

Inductive reactance X$_{L}$, the only one opponent in a pure inductive network for AC current.

In (4c), the frequency and inductive reactance have a direct relationship with each other, by increasing the frequency, the inductive reactance will increase and vice versa. In the special case of the zero frequency (DC), the reactance will also be zero. Where for high-frequency AC, the reactance is high. So, inductors are short for DC and almost open high-frequency AC.

Conclusion :

  • The AC current is opposed not only opposed by resistance but reactance also.
  • Impedance is the vector combination of resistance and reactance.
  • The impedance triangle is the vector representation of impedance, reactance, and resistance
  • In a purely resistive circuit, voltage and current are always in phase with each other.
  • In a purely inductive circuit, the current lags by 90 from that of voltage.
  • In a purely capacitive circuit, the current leads by 90 from that of voltage.   

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