Impedance refers to the combination of resistance and reactance, in an AC circuit. It obstructs the flow of electrons within an electrical circuit and affects the current generated .The letter Z mathematically symbolizes impedance and has its unit as ohms, denoted by the Omega(Ω) symbol.

Mathematically, Impedance can be written as:

Z = R + j*X, where

Z = Impedance.

R = Resistance – Real part of the value.

j = Imaginary part of the value.

X = Inductance or Capacitance

A standard RLC circuit.

Inductance and Capacitance

There are two primary types of reactance:

a. Capacitive Reactance – X_C : When a capacitor is connected to a circuit with an AC supply, there is no change in the capacitor voltage or current. The potential difference across the capacitor is dependent on the AC power supply. The current is maximum when the potential difference is at zero.

The capacitive reactance for a capacitor with capacitance C connected in the circuit along with the AC power supply is given as: Where,

X_C – Capacitive Reactance.

f – Frequency of AC power supply.

C – Capacitance.

X_C = 1/ 2π*f*C

Graph of a capacitive circuit.

b. Inductive Reactance – X_L: In inductive reactance, the current across an inductor changes when a potential difference develops across it. The potential difference and rate of change of current are proportional to each other. The inductive reactance for an inductor with inductance L connected in the circuit along with the AC power supply is given as:

Where,

X_L – Inductive Reactance.

L – Inductance of Inductor.

f – Frequency of the Alternating Current.

X_L = 2π*f*L .

Graph of an Inductive Circuit.

Calculating Impedance

1. Impedance is the pure Ohmic resistance of a circuit and is a complex number consisting only of a real part being the actual AC resistance value, ( R ), and a zero imaginary part, ( 0j ). Thus, Ohm’s Law can be used in circuits containing an AC resistance to calculate these voltages and currents.
2. Impedance is the combination of the resistance and the reactance present in an AC circuit and can be broken down into resistance and reactance.
3. Impedance is represented by the letter Z and is present in the circuit only when both capacitor and inductor are present or when capacitor, inductor, and resistance are present.
4. It is mathematically stated as Z = √R² + ( XL – XC)² and has a unit of Ohm, which is denoted by the symbol, Omega(Ω).

Phasors and Complex Impedance

A phasor is represented by a constant complex number in the exponential form and represents the complex amplitude (magnitude and phase) of a sinusoidal function of time. Phasors are used to simplify computations involving sinusoids, where they can often reduce a differential equation problem to an algebraic one.

The impedance of a circuit element can be defined as the ratio of the phasor voltage across the element to the phasor current through the element, as determined by the relative amplitudes and phases of the voltage and current.

In the analysis of three-phase AC power systems, a set of phasors is defined as the three complex cube roots of unity, graphically represented as unit magnitudes at angles of

a. 0 degrees.

b. 120 degrees.

c. 240 degrees.

By treating AC circuit quantities as phasors, balanced circuits can be simplified, and unbalanced circuits can be treated as an algebraic combination of symmetrical components, greatly simplifying the work required in the calculations of voltage drop, power flow, and short-circuit currents.

The phase angle is often given in degrees, and the magnitude is in RMS value rather than the peak amplitude of the sinusoid. Digital instruments are used to measure the phasors that represent the transmission system voltages at widespread points in a transmission network. The differences among the phasors indicate the power flow and system stability.

Applications

a. Impedance is crucial in AC circuit analysis because it enables us to calculate the AC voltage and current, allowing us to map their behaviour within a circuit, which is necessary when designing and troubleshooting electrical systems.

b. Impedance matching refers to adjusting the impedance of a source and a significant load to match it, maximizing the power transfer between the two. This is observed in various appliances such as radio frequency (RF) communications, audio systems, and power transmission.

c. The impedance of capacitors and inductors in the circuit is dependent on the frequency of the AC Circuit. As a result, capacitive impedance decreases with an increase in the frequency while the inductive impedance will increase. This characteristic can be utilized to design filters that selectively allow specific frequency ranges to pass through, allowing us to shape the output signal as desired.

Key Points of Impedance in AC Circuits

Impedance measurements can vary depending on the method and the environmental conditions during measurement. To obtain stable values, it is essential to measure impedance in controlled environments and a controlled set-up.

a. A wide range of stable sine waves or a singular wave with a stable frequency should be used. The careful consideration of the measurement frequency is necessary, to match the characteristics of the target device or circuit. Factors such as poor connections in cables should be rectified at once, as well.

b. The temperature at the time of measurement and the capacitance of the probes should be steady and not fluctuate.

c. Considering the thermal characteristics of the target device is key to obtaining a stable, steady and accurate reading.

Conclusion

Impedance is the resistance provided by an AC circuit and is responsible for blocking the flow of electrons and thus, affecting the current generated by an AC circuit. It is the Ohmic resistance of a circuit and is a complex number consisting only of a real part being the actual AC resistance value, ( R ), and a zero imaginary part, ( 0j ). Ohm’s Law can be used to calculate the voltages and currents flowing in AC circuits.

Frequently Asked Questions on Impedance in AC Circuits – FAQ’s

Define Impedance and give its formula and unit.

Impedance is the cumulative resistance provided to an AC circuit and is responsible for obstructing the flow of electrons, thereby limiting the current produced.

It is denoted by the symbol Z where:

Z = √R² + (X_L – X_C)² and it’s unit is Ohm, denoted by Omega(Ω).

If the frequency of the circuit is increased, how will X_C and X_L change ?

On increasing the frequency of the AC circuit, the capacitive reactance (X_C) will decrease and the inductive reactance (X_L) will increase.

What are the magnitude of the angles taken in a 3-phase AC system ?

The angles taken in a 3-phase AC system are:

a. 0 degrees.

b. 120 degrees.

c. 240 degrees.