Impedance vs Reactance Simple Examples

Impedance vs reactance along with impedance and admittance sound strange for people who do not study electric circuits at an advanced level. Impedance is often used in ac electric circuit analysis as well as resistance in dc electric circuits. The difference between these two is, impedance has magnitude and phase, while resistance only has magnitude. Now we will learn about impedance vs reactance.

Make sure to read the ac circuit first.

Definition of Impedance

From the previous post about sinusoidal and phasor, we knew the voltage-current relations for three passive elements of R, L, and C as

(1)   \begin{equation*}V=RI,\quad V=j\omega LI, \quad V=\frac{I}{j\omega C}\end{equation*}

These equations may be written in terms of the ratio of the phasor voltage to the phasor current as

(2)   \begin{equation*}\frac{V}{I}=R, \quad \frac{V}{I}=j\omega L, \quad \frac{V}{I}=\frac{1}{j\omega C}\end{equation*}

From these three expressions, we obtain Ohm’s law in phasor form for any type of element as

(3)   \begin{equation*}Z=\frac{V}{I} \quad \mbox{or} \quad V=ZI \end{equation*}

where Z is a frequency-dependent quantity known as impedance, measured in Ω.

The impedance Z of a circuit is the ratio of the phasor voltage V to the phasor current I, measured in ohms (Ω)
The impedance shows the opposition that the circuit exhibits to the flow of sinusoidal current.
Although the impedance is the ratio of two phasors, it is not a phasor, because it does not correspond to a sinusoidally varying quantity.

Impedance vs Reactance vs Resistance

Just as the title, we will learn the difference between impedance vs reactance. AC circuits are harder to analyze compared to the DC circuits because the current is flowing in two directions. The DC circuit current will flow from positive polarity to the negative polarity, while the AC circuit current will flow from positive to negative polarity then negative to positive polarity and so on.

Not only the current direction, we need to deal with the voltage and current frequency. We have resistance for DC circuit and impedance for AC circuit. Impedance is the mix between resistance and reactance.

impedance vs reactance

Impedance is the mix of resistance and reactance, for both capacitance and inductance. Impedance consists of complex numbers (real and imaginary parts). The real part is the resistance and the imaginary part is reactance. The impedance result has magnitude and phase.

If the resistance is a friction to the electric current, then the impedance is the friction to the change of current in the circuit. Just like resistance, impedance is also measured in ohms (ohm).

Impedance is harder to analyze because it takes inductance and capacitance into account with various frequencies of the voltage and current. You can say that impedance  depends on the frequency.

We can divide impedance into two important parts:

  • Resistance, R which is the real part (a constant doesn’t depend on frequency), exists because of the resistors in the circuit.
  • Reactance, X which is the imaginary part (a complex number depends on frequency), exists because of the capacitor and/or inductor in the circuit.

impedance vs reactance formula

Both capacitance and inductance will result in phase shift between voltage and current. In order to get impedance from resistance and reactance, we can sum them in a vector way like the figure above.

We will use these important elements while calculating impedance (Z) in the ac circuit:

  • Resistance (R)
  • Capacitance (C)
  • Inductance (L)
  • Frequency (f)

    \begin{align*}Z=\sqrt{R^{2}+X_{T}^{2}}\end{align*}

where:

Z   = magnitude of impedance (Ω)

XT = total reactance (XL – XC)

    \begin{align*}\theta = \arctan (\frac{X_{T}}{R})\end{align*}

θ = phase of impedance (degrees)

Resistor Impedance

How we analyze a resistor in an AC circuit is not different from a DC circuit. Since resistance only consists of real numbers we can still use the basic Ohm’s law. Hence the resistor impedance is:

    \begin{align*}Z_{R}=R\end{align*}

where:
ZR = resistor impedance
R   = resistance

Resistance doesn’t produce phase shift because there is no imaginary part, thus the voltage and current will have the same phase. You can see the graph below:

resistor impedance

Resistance, circuit element which prevents the current from flowing – In order to control the resistance in the circuit, we need the resistors. This element can be found in DC and AC circuits. Resistor will produce heat in exchange preventing the energy in a circuit to a degree.

The resistance can be expressed such as,

    \begin{align*}R=\frac{V}{I}\end{align*}

Reactance, X

Reactance, circuit element which opposes the change in current – Reactance, represented by X, is the element opposed to inductance and capacitance. This measurement value depends on the frequency in the circuit. Reactance is measured in ohm (Ω) just like resistance.

In order to control the reactance in the circuit, we need the inductor and capacitor. This element is exclusive only for AC circuits where input frequency matters. When the circuit has reactance, it changes the phase shift between voltage and current.

Reactance occurs when an inductor and/or capacitor exists in the circuit. Hence, we divide reactance into two parts for each of them:

  • Inductive reactance, XL
  • Capacitive reactance, XC

Inductive Reactance Formula, XL

Inductive reactance, XL will be low if the frequency is low and vice versa, will be high if the frequency is high. When it comes to DC circuits which have zero frequency (steady-state circuit), the XL value will be zero ohm. It means that DC current passes through an inductor fully and blocks high frequency AC current.

Inductive reactance is a reactance for an existing inductor in the circuit. If the inductive reactance exists in the circuit, the energy will be stored in magnetic field form. The current waveform lags the voltage waveform by 90o when inductive reactance exists in the circuit. This element is caused by a component made from the conductor wire built in a roll (wound circularly), like a coil. The simplest example is transformers.

    \begin{align*}X_{L}=2 \pi fL\end{align*}

where:

XL = inductive reactance, measured in ohm (Ω)
f    = frequency, measured in Hertz (Hz)
L   = inductance, measured in henrys (H)

Inductor Impedance

With the same concept, an inductor provides inductance in the circuit. This component is able to store electrical energy in the magnetic field form. Unlike capacitor, an inductor makes the current lags to voltage by 90 degrees. You can observe the graph below:

inductor impedance

In other words, the voltage leads the current by 90 degrees. We can use the following equation:

    \begin{align*}Z_{L}=j \omega L\end{align*}

where:
ZL = inductor impedance
ω   = 2πf = angular frequency
f    = signal frequency
L   = inductance

Capacitive Reactance Formula, XC

Capacitive reactance, XC value will be high if the frequency is low and vice versa, will be small if the frequency is high. When it comes to DC circuits which have zero frequency (steady-state circuit), the XC value is infinite ohm. It means that DC current is unable to pass through a capacitor unlike AC current.

Capacitive reactance is a reactance for an existing capacitor in the circuit. If the capacitance reactance exists in the circuit, the energy will be stored in electrical field form. The current waveform leads the voltage waveform by 90o when capacitance reactance exists in the circuit. This element is caused by a component made from a pair of conducting plates built in parallel with a small gap between them. The gap is filled with a dielectric material.

    \begin{align*}X_{C}=\frac{1}{2 \pi fC}\end{align*}

where:

XC = capacitive reactance, measured in ohm (Ω)
f     = frequency, measured in Hertz (Hz)
C   = capacitance, measured in farads (F)

Capacitor Impedance

Capacitor, a component which generates capacitance in the circuit. This component is used to store electrical energy in electric field form temporarily. In an AC circuit, this component is often used to make the voltage lagged 90 degrees to the current. You can see the graph below:

capacitor impedance

Like we observe above, the voltage lags behind the current when there are capacitors in the circuit. In other way, we can say that the current leads the voltage for this component by 90 degrees. To make things easier, we can use the equation below:

    \begin{align*}Z_{C}=-j \frac{1}{\omega C}\end{align*}

where:
ZC = capacitor impedance
ω   = 2πf = angular frequency
f    = signal frequency
C   = capacitance

Impedance Formula

The impedance of resistors, inductors, and capacitors can be readily obtained from Equation.(2). Table.1 gives a summary of their impedances.

Table.(1)

    \begin{align*}&&\mbox{Element} \quad && \mbox{Impedance} \quad && \mbox{Admittance} \\&&R \quad && Z=R \quad && Y= \frac{1}{R} \\&&L \quad && Z=j \omega L \quad && Y= \frac{1}{j \omega L} \\&&C \quad && Z=\frac{1}{j \omega C} \quad && Y= j \omega C\end{align*}

Looking from the table, we notice that ZL = jωL and ZC = -j/ωC. Consider two extreme cases of angular frequency.
When ω = 0 (i.e., for dc sources), ZL = 0 and ZC → ∞, confirming what we already know – that the inductor acts like a short circuit, while the capacitor acts like an open circuit.
When ω→ ∞ (i.e., for high frequencies), ZL → ∞ and ZC = 0, showing that the inductor is an open circuit, while the capacitor is a short circuit as can be seen in Figure.(1).

equivalent impedance circuit
Figure 1. Equivalent circuits at dc and high frequencies: (a) inductor, (b) capacitor

As a complex quantity, the impedance may be written in rectangular form as

(4)   \begin{equation*}Z=R+jX\end{equation*}

where R = Re Z is the resistance and X = Im Z is the reactance. The reactance X may be negative or positive.
We may say that the impedance is inductive when X is positive or capacitive when X is negative.
Hence, impedance Z = R + jX is said to be inductive or lagging since current lags voltage, while impedance Z = R – jX is capacitive or leading since current leads voltage.
All three impedance, resistance, and reactance are measured in ohms. The impedance may also be expressed in polar form as

(5)   \begin{equation*}Z=|Z|\angle \theta\end{equation*}

Comparing Equations.(4) and (5), we conclude that

(6)   \begin{equation*}Z=R+jX=|Z|\angle \theta\end{equation*}

Where

(7)   \begin{equation*}|Z|=\sqrt{R^{2}+X^{2}}, \quad \theta=\tan^{-1}\frac{X}{R}\end{equation*}

And

(8)   \begin{equation*}R=|Z|\cos \theta, \quad X=|Z|\sin \theta\end{equation*}

Equivalent Series Impedances

If we have two impedances connected in series we can calculate the equivalent series impedance using

    \begin{align*}Z_{e}=Z_{1}+Z_{2}\end{align*}

Because Z is complex number then we use the simple equations below:

    \begin{align*}Z_{1}&=R_{1}+jX_{1}\\Z_{2}&=R_{2}+jX_{2}\\Z_{e}&=\frac{R_{1}+R_{2}}{j(X_{1}+X_{2})}\end{align*}

Example:
If we have a resistor 10 ohm in series with a capacitor 1mF while the frequency is 100Hz, the equivalent series impedance is:

    \begin{align*}Z_{e}=(10-j\cdot 1.59) \Omega\end{align*}

The effective impedance or magnitude impedance is:

    \begin{align*}|Z_{e}|^{2}=R^{2}+X^{2}\end{align*}

Thus,

    \begin{align*}|Z_{e}|=10.12 \Omega\end{align*}

Equivalent Parallel Impedances

If we have two impedances connected in parallel we can calculte the equivalent parallel impedance using the same method as series impedances but we use the admittances instead. Admittance is measured by siemens and an element to measure how easier the current to flow and the inverse of impedance.

    \begin{align*}Y=\frac{1}{Z}\end{align*}

Equivalent admittance in parallel is the same with the equivalent impedance in series.

    \begin{align*}Y_{e}=Y_{1}+Y_{2}\end{align*}

Using previous value in series connection, the equivalent parallel admittance is

    \begin{align*}Y_{1}&=\frac{1}{R_{1}}=0.1S \\Y_{2}&=\frac{1}{Z_{2}}=\frac{1}{-j\cdot 1.59 \Omega}\\&=j\cdot 0.63S\\Y_{e}&=(0.1+j\cdot 0.63)S\\Z_{e}&=\frac{1}{Y_{e}}=(0.24-j\cdot j1.55) \Omega \\\end{align*}

And the impedance magnitude is

    \begin{align*}|Z_{e}|=1.56 \Omega\end{align*}

We will move on to the:

And its applications:

Impedance Example

For better understanding let us review the example below :

impedance circuit example
Figure 2


Find v(t) and i(t) in the circuit in Figure.(2)

Solution :
From the voltage source 10 cos 4t, ω = 4,

    \begin{align*}V_{S}=10 \angle 0^{o} V\end{align*}

The impedance is

    \begin{align*}Z&=5+\frac{1}{j \omega C}\\&=5+\frac{1}{j4\times 0.1}\\&=5-j2.5 \Omega\end{align*}

Thus the current

(1.1)   \begin{align*}I&=\frac{V_{S}}{Z}=\frac{10 \angle 0^{o}}{5-j2.5}\\&=\frac{10(5+j2.5)}{5^{2}+2.5^{2}}\\&=1.6+j0.8=1.789 \angle 26.57^{o} A\end{align*}

The voltage across the capacitor is

(1.2)   \begin{align*}V&=IZ_{C}=\frac{1}{j \omega C}\\&=\frac{1.789 \angle 26.57^{o}}{j4\times 0.1}\\&=\frac{1.789 \angle 26.57^{o}}{0.4\angle 90^{o}}\\&=4.47\angle -63.43^{o}V\end{align*}

Converting I and V in Equations.(1.1) and (1.2) to the time domain results

    \begin{align*}i(t)&=1.789 \cos (4t+26.57^{o})A\\v(t)&=4.47 \cos (4t-63.43^{o})V\end{align*}

Notice that i(t) leads v(t) by 90o as expected.

 

 

 

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