What is Defined Impedance?

Defined impedance is a key concept in electrical engineering that refers to the total opposition a circuit presents to alternating current (AC) at a given frequency. It is a complex quantity that combines resistance and reactance, and is measured in ohms (Ω).

Impedance is defined as the ratio of voltage to current in an AC circuit. In a direct current (DC) circuit, this ratio is simply the resistance. However, in an AC circuit, capacitive and inductive elements introduce an additional opposition to current flow known as reactance. The combination of resistance (R) and reactance (X) is impedance (Z).

Mathematically, impedance can be expressed as a complex number:

Z = R + jX

where:
– Z is impedance in ohms
– R is resistance in ohms
– X is reactance in ohms
– j is the imaginary unit (square root of -1)

The real part of impedance is resistance, while the imaginary part is reactance. Reactance can be either inductive (positive) or capacitive (negative).

Types of Reactance

There are two types of reactance:

1. Inductive Reactance (XL): Opposes changes in current. Increases with frequency. Given by:

XL = 2πfL

where f is frequency in hertz and L is inductance in henries.

1. Capacitive Reactance (XC): Opposes changes in voltage. Decreases with frequency. Given by:

XC = 1 / (2πfC)

where C is capacitance in farads.

The net reactance in a circuit is the difference between inductive and capacitive reactance:

X = XL – XC

If XL > XC, the circuit is said to be inductive. If XC > XL, the circuit is capacitive. If XL = XC, the circuit is resonant and impedance is purely resistive (Z = R).

Importance of Defined Impedance

Defined impedance is a critical parameter in the design of electrical and electronic systems. It affects signal integrity, power transfer, and electromagnetic compatibility. Some key applications include:

Transmission Lines

Transmission lines are used to carry high-frequency signals between circuits. To minimize reflections and ensure maximum power transfer, the impedance of the source, line, and load must be matched. Common values of Characteristic Impedance for transmission lines are 50 Ω, 75 Ω, and 110 Ω.

Filters

Filters are used to selectively attenuate or pass signals based on frequency. The impedance of filter components (resistors, capacitors, inductors) determines the cutoff frequency and attenuation characteristics. Impedance matching is important to avoid loading the source or reflecting power back to it.

Antennas

Antennas are used to transmit and receive electromagnetic waves. The impedance of an antenna affects its radiation efficiency, bandwidth, and directivity. Most antennas are designed for either 50 Ω or 75 Ω impedance to match standard transmission lines.

Audio Systems

In audio systems, impedance matching between sources (e.g. microphones, instruments), amplifiers, and loudspeakers is essential for optimal sound quality and power efficiency. Typical values of loudspeaker impedance are 4 Ω, 8 Ω, and 16 Ω.

Measuring Impedance

Impedance can be measured using various instruments and techniques, depending on the frequency range and required accuracy. Some common methods are:

Impedance Bridge

An impedance bridge is a null-balance circuit that compares an unknown impedance to known reference impedances. It is accurate but limited to low frequencies (< 1 MHz). The most common type is the Wheatstone bridge, which measures resistance and capacitance.

Vector Network Analyzer (VNA)

A VNA is a sophisticated instrument that measures the amplitude and phase of reflected and transmitted signals in a circuit. It can accurately characterize impedance, admittance, s-parameters, and other network properties over a wide frequency range (up to 100 GHz). VNAs are widely used in RF and microwave engineering.

Time-Domain Reflectometry (TDR)

TDR is a technique that measures impedance by sending a fast rise-time pulse into a circuit and observing the reflections caused by impedance discontinuities. It provides a graphical display of impedance vs. distance, making it useful for locating faults in transmission lines and printed circuit boards.

Impedance Matching

Impedance matching is the practice of designing the input impedance of an electrical load to maximize power transfer and minimize reflections from the load. There are several techniques for impedance matching, including:

Transformer Matching

A transformer can be used to match the impedance of a source to a load by adjusting the turns ratio. The turns ratio (N) is given by:

N = sqrt(ZL / ZS)

where ZL is the load impedance and ZS is the source impedance. Transformers are simple and effective, but are limited by their bandwidth and insertion loss.

LC Matching Network

An LC network consisting of inductors and capacitors can be used to match arbitrary impedances over a narrow frequency range. There are several topologies, such as L, T, and π networks, each with different trade-offs in terms of bandwidth, loss, and component values.

Stub Matching

A stub is a section of transmission line connected in parallel or series with the main line. By adjusting the length and characteristic impedance of the stub, a wide range of impedances can be matched. Stubs are commonly used in microwave circuits and antennas.

Impedance in Practice

To illustrate the concept of impedance, let’s consider a few practical examples:

Example 1: Resistor-Capacitor (RC) Circuit

An RC circuit consists of a resistor (R) in series with a capacitor (C). The impedance of this circuit is given by:

Z = R – jXC

where XC = 1 / (2πfC)

At low frequencies, the capacitive reactance is high and the impedance is dominated by the resistor. At high frequencies, the capacitive reactance is low and the impedance is dominated by the capacitor.

Frequency (Hz)	Capacitance (F)	Resistance (Ω)	Reactance (Ω)	Impedance (Ω)
100	1e-6	1000	-1592	1000 – j1592
1000	1e-6	1000	-159	1000 – j159
10000	1e-6	1000	-16	1000 – j16

As frequency increases, the capacitive reactance decreases and the impedance becomes more resistive.

Example 2: Resistor-Inductor (RL) Circuit

An RL circuit consists of a resistor (R) in series with an inductor (L). The impedance of this circuit is:

Z = R + jXL

where XL = 2πfL

At low frequencies, the inductive reactance is low and the impedance is dominated by the resistor. At high frequencies, the inductive reactance is high and the impedance is dominated by the inductor.

Frequency (Hz)	Inductance (H)	Resistance (Ω)	Reactance (Ω)	Impedance (Ω)
100	0.1	100	63	100 + j63
1000	0.1	100	628	100 + j628
10000	0.1	100	6283	100 + j6283

As frequency increases, the inductive reactance increases and the impedance becomes more reactive.

Example 3: Antenna Impedance Matching

An antenna with an impedance of 100 Ω is connected to a 50 Ω transmission line. To match the impedances, a quarter-wave transformer can be used. The characteristic impedance of the transformer is given by:

Z0 = sqrt(ZL * Z0)

where ZL is the load (antenna) impedance and Z0 is the line impedance.

In this case, Z0 = sqrt(100 * 50) = 70.7 Ω

A quarter-wave section of 70.7 Ω transmission line is inserted between the antenna and main line. At the design frequency, this provides a perfect match and maximum power transfer.

Conclusion

Impedance is a fundamental concept in electrical engineering that describes the total opposition to current flow in an AC circuit. It is a complex quantity that includes both resistance and reactance, and varies with frequency.

Defining and controlling impedance is essential for ensuring signal integrity, power efficiency, and electromagnetic compatibility in a wide range of applications, from audio systems to antennas.

Impedance can be measured using various instruments such as bridges, network analyzers, and time-domain reflectometers. It can be matched using transformers, LC networks, and transmission line stubs.

By understanding and applying the principles of impedance, engineers can design better circuits and systems that meet the increasingly demanding requirements of modern technology.

Frequently Asked Questions

What is the difference between impedance and resistance?

Resistance is the opposition to current flow in a DC circuit, while impedance is the total opposition to current flow in an AC circuit. Impedance includes both resistance and reactance, which varies with frequency.

What is the difference between impedance and reactance?

Reactance is the imaginary part of impedance, representing the opposition to current flow by inductors and capacitors. Impedance is the complex sum of resistance and reactance.

What is the unit of impedance?

The unit of impedance is the ohm (Ω), same as resistance. However, impedance is a complex quantity, so it is often expressed as a magnitude and phase angle, e.g. 100 Ω ∠ 30°.

What is the purpose of impedance matching?

The purpose of impedance matching is to maximize power transfer and minimize reflections between a source and load. When the impedances are matched, maximum power is delivered to the load and reflections are minimized, improving efficiency and signal integrity.

What is the relationship between impedance and frequency?

Impedance varies with frequency due to the frequency-dependent behavior of reactance. Inductive reactance increases with frequency, while capacitive reactance decreases. The impedance of a circuit can be plotted as a function of frequency to show its variation.