Demystifying Impedance: A Comprehensive Guide

• Introduction

• What Is Phasor?

• What Is Impedance?

• Impedance of Resistor

• Impedance of Inductor

• Impedance of Capacitor

• Application of Impedance

Introduction

In this blog, we will go over a fundamental concept in electrical engineering, specifically a core component in modeling alternating current (AC) circuits. Impedance refers to a component's natural opposition to the flow of alternating current in an electrical circuit.

What Is Phasor?

Before discussing what impedance is, I would like to address what a phasor is. Understanding phasor will help break down the complex concept of impedance.

A phasor is a type of complex number that contains information about both the amplitude and phase angle of a sinusoidal function. A phase angle is the angular difference between two sinusoidal signals of the same frequency. You can refer to Figure 1 for an illustration of the phase angle between two sinusoidal waves.

The mathematical concept is based on Euler's identity, which shows how the exponential and trigonometric functions are related (Nilsson & Riedel, 2015, 310):

e±^jθ = cosθ ± jsinθ = R(e^jθ) + I(e^jθ) (1)

From Equation 1, we can see cosine represented as the real part and sine represented as the imaginary part of the exponential function. The R denotes "the real part" and I denotes "the imaginary part".

In this example, the cosine function is represented as a voltage function to analyze the phasor representation:

V = V₁e^j(ωt+φ) = V₁cos(ωt + φ) + jV₁sin(ωt + φ) (2)

Equation 2 represents the voltage function in the frequency domain, where ω is the angular frequency,

ω = 2πf (rad/s) (3)

From Equation 1, there is another abbreviation form of the exponential function ejθ or phasor form.

e^jθ = 1 < θ (4)

What Is Impedance?

Impedance is a complex quantity that combines both resistance and reactance. Resistance is classified as the opposition of flow in direct current (DC), while reactance is the opposition of flow in AC; this is due to capacitance and inductance. Impedance is also measured in ohms (Ω), where the real part of the complex function is resistance, and the imaginary part denotes reactance. The mathematical representation of impedance is given by the following equation:

Z = R + jX(5)

where Z, R, and X are impedance, resistance, and reactance, respectively (Nilsson & Riedel, 2015, 318).

In AC circuits, impedance plays a critical role in determining the behavior of the circuit, as it affects the voltage drop and the current flow. Impedance of an AC circuit is calculated using Ohm's law, which relates current to voltage and to impedance. The equation for Ohm's law is

V = IZ (6)

where V is the voltage, I is the current, and Z is the impedance.

From here, we will evaluate the impedance of the resistor, inductor, and capacitor.

Impedance of Resistor

The equation of current in terms of cosine is given as i = I₁cos(ωt + θ) and when combined with Ohm's Law (Nilsson & Riedel, 2015, 315), the resulting voltage equation of the resistor is

v = RI₁cos(ωt + θ) (7)

where I₁ is the maximum amplitude, and φ is the phase angle (degree) of the current.

Voltage can then be rewritten as

v = RI₁e^jθ = RI₁ < θ (8)

From Equation 1, I₁ < θ is the phasor form so we can write the voltage equation:

V = RI (9)

Impedance of Inductor

Given the equation of current in terms of cosine, i = I₁ cos(ωt + θ) and the expression of inductor voltage, L di/dt (Nilsson & Riedel, 2015, 316),

v = L di/dt = -ωLI₁ sin(ωt + θ) (10)

we are able to rewrite Equation 10 as a cosine function:

v = -ωLI₁ cos(ωt + θ - 90^°) (11)

The inductor voltage can be illustrated in the phasor form:

V = -ωLI₁ e^j(θ-90°)

= -ωLI₁ e^jθ e^-j90° (12)

From Equation 1, we can simplify e^-j90°

e^-j90° = cos(90^°)-jsin(90^°) = -j (13)

This allows Equation 12 to be further simplified to

V = jωLI₁ e^jθ

jωLI_ (14)

Therefore, the impedance of inductor can be represented as

Z_L=jωL (15)

Impedance of Capacitor

The equation of voltage in terms of cosine is v = V₁ cos(ωt + θ), and the expression of capacitor current is C dv/dt (Nilsson & Riedel, 2015, 317).

i = C dv/dt = -ωCV₁ sin(ωt + θ) (16)

We can rewrite Equation 16 as a cosine function:

i = -ωCV₁ cos(ωt + θ - 90^°) (17)

The capacitor current can be illustrated in the phasor form:

I = -ωCV₁ e^j(θ-90°)

= -ωCV₁ ejθ e^-j90° (18)

Using Equation 13 and combining with Equation 18, we then are able to simplify the current further to

I = jωCV (19)

The voltage of capacitor can be expressed as

V = 1/jωC I (20)

Therefore, the impedance of capacitor can be represented as

Z_C = 1/jωC (21)

Application of Impedance

Impedance has numerous real-world applications in electrical engineering, including filter design, power electronics, antenna design, and audio systems. In filter design, impedance is used to control the flow of current, allowing only specific frequencies to pass through a given system. In power electronics, impedance is used to regulate the flow of current, ensuring efficient operation of the circuit. Antenna design requires impedance matching to ensure maximum power transfer of a signal or signals. Finally, in audio systems, impedance is used to match amplifiers to speakers for optimal sound quality.

1. Impedance in Transmission Line

Impedance matching is a critical factor in the design of transmission lines used to transmit electrical signals over long distances. The impedance of the transmission line needs to match the impedance of the load to ensure maximum power transfer, otherwise, any mismatch in impedance will result in a partial reflection of the energy and signal loss.

The equation for the impedance of a transmission line is

Z₀= √ L/C (22)

where Z₀ is the characteristic impedance of the transmission line, L is the inductance per unit length, and C is the capacitance per unit length.

2. Impedance Measurement and Testing

Impedance measurement and testing are essential in many fields of electrical engineering, including telecommunications, audio systems, and power electronics. Impedance meters and testers are used to measure impedance, and the techniques used depend on the application.

Conclusion

Impedance is an essential function in modeling AC circuits; it can be easily derived as the inherent resistance of electrical components. Understanding impedance allows for real-world applications and problem-solving of systems in the frequency domain. These applications can range from small signal designs of antennas in cell phones to long-distance power transfer of transmission lines.

Interested in becoming an electrical engineer? School of PE offers comprehensive FE and PE exam review courses to help you pass your engineering exams. Register for a course today!

References

Riedel, S. A., & Nilsson, J. W. (2015). Electric Circuits. Pearson

About the Author: Khoa Tran

Khoa Tran is an electrical engineer working at the Los Angeles Department of Water and Power and is currently pursuing his master's in electrical Power from the University of Southern California. He is fluent in both Vietnamese and English and is interested in outdoor activities and exploring new things.