Introduction to Electrical Impedance
– Complex numbers were first used in circuit analysis by Johann Victor Wietlisbach in 1879.
– The term ‘impedance’ was coined by Oliver Heaviside in July 1886.
– Arthur Kennelly published an influential paper on impedance in 1893.
– Charles Proteus Steinmetz generalized impedance to all AC circuits in the same year.
– Impedance in AC circuits includes the effects of inductance and capacitance.
– Inductance causes voltage induction in conductors, while capacitance stores charge induced by voltage.
– Reactance refers to the combined effects of inductance and capacitance.
– Resistance forms the real part of complex impedance.
– Impedance is represented as a complex quantity (Z) with magnitude and phase characteristics.
– Impedance can be represented in polar form as |Z|e^(jθ).
– The magnitude (|Z|) represents the ratio of voltage difference amplitude to current amplitude.
– The argument (θ) gives the phase difference between voltage and current.
– In Cartesian form, impedance is defined as Z = R + jX.
– Conversion between polar and Cartesian forms follows the rules of complex numbers.
– Sinusoidal voltage and current waves are represented as complex-valued functions of time.
– Impedance is the ratio of voltage to current, denoted as Z = V/I.
– The magnitude equation is Ohm’s law applied to voltage and current amplitudes.
– The phase relationship is defined by the difference in phase angles.
– Impedance allows for the extension of DC circuit analysis results to AC circuits.
– Phasors are constant complex numbers representing the complex amplitude of a sinusoidal function of time.
– Phasors simplify computations involving sinusoids in AC circuits.
– Impedance can be defined as the ratio of phasor voltage to phasor current.
– Phasors reduce differential equation problems to algebraic ones.
– Phasors cancel out the factor e^(jωt) in the impedance definition.
Deriving Device-Specific Impedances
– Impedance can be derived for resistors, capacitors, and inductors.
– The derivations assume sinusoidal signals, which can be approximated as a sum of sinusoids through Fourier analysis.
– For resistors, Ohm’s law relates the voltage and current.
– The ratio of AC voltage amplitude to alternating current (AC) amplitude across a resistor is the resistance.
– The AC voltage leads the current across a resistor by 0 degrees.
Resistance and Reactance
– Resistance and reactance determine the magnitude and phase of impedance.
– The magnitude of impedance is the square root of the sum of the squares of resistance and reactance.
– The phase of impedance is determined by the arctan of reactance divided by resistance.
– In many applications, only the magnitude of impedance is significant, and the phase is not critical.
– Resistance is the real part of impedance.
– A device with purely resistive impedance exhibits no phase shift between voltage and current.
– Resistance can be calculated as the cosine of the phase angle of impedance.
– Reactance is the imaginary part of impedance.
– A component with finite reactance induces a phase shift between voltage and current.
– A purely reactive component alternately absorbs and returns energy to the circuit.
– Capacitive reactance is inversely proportional to signal frequency.
– Inductive reactance is proportional to signal frequency and inductance.
Combining Impedances
– Impedances in series can be combined by summing their resistances and reactances.
– Impedances in parallel can be combined using the reciprocal of the sum of their reciprocals.
– The total impedance of a network is the sum of its resistive and reactive components.
– Series and parallel circuits can be analyzed using the rules for combining impedances.
– The total impedance affects the behavior of the circuit.
– Current through each circuit element is the same in a series combination.
– Total impedance is the sum of component impedances in a series combination.
– Impedance equation for series combination: Z_eq = R + jX
– Impedance equation for n components in series: Z_eq = Z_1 + Z_2 + … + Z_n
– Voltage across each circuit element is the same in a parallel combination.
– Ratio of currents through any two elements is the inverse ratio of their impedances in a parallel combination.
– Inverse total impedance is the sum of the inverses of component impedances in a parallel combination.
– Impedance equation for parallel combination: 1/Z_eq = 1/Z_1 + 1/Z_2 + … + 1/Z_n
– Impedance equation for n components in parallel: Z_eq = (Z_1 * Z_2) / (Z_1 + Z_2)
Applications and Measurement of Impedance
– Impedance matching is crucial in maximizing power transfer between circuits.
– It is used in audio systems to ensure proper speaker and amplifier matching.
– Impedance measurements are essential in testing and troubleshooting electrical circuits.
– Impedance analysis is utilized in designing transmission lines for efficient signal transmission.
– Impedance spectroscopy is a technique used in material characterization and biomedical applications.
– Impedance measurement is a practical problem in various fields.
– Measurements can be carried out at one frequency or over a range of frequencies.
– Impedance can be measured or displayed in ohms or other related values.
– Measurement requires measuring voltage and current magnitude and phase difference.
– Impedance is often measured using bridge methods or LCR meters.
– LC tank circuit has a complex impedance equation: Z(ω) = jωL / (1 – ω^2LC).
– Minimum value of 1/|Z| occurs when ω^2LC = 1.
– Fundamental resonance angular frequency is ω = 1 / √(LC).
– Impedance and admittance are generally time Source: https://en.wikipedia.org/wiki/Electrical_impedance
In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit.
Quantitatively, the impedance of a two-terminal circuit element is the ratio of the complex representation of the sinusoidal voltage between its terminals, to the complex representation of the current flowing through it. In general, it depends upon the frequency of the sinusoidal voltage.
Impedance extends the concept of resistance to alternating current (AC) circuits, and possesses both magnitude and phase, unlike resistance, which has only magnitude.
Impedance can be represented as a complex number, with the same units as resistance, for which the SI unit is the ohm (Ω). Its symbol is usually Z, and it may be represented by writing its magnitude and phase in the polar form |Z|∠θ. However, Cartesian complex number representation is often more powerful for circuit analysis purposes.
The notion of impedance is useful for performing AC analysis of electrical networks, because it allows relating sinusoidal voltages and currents by a simple linear law. In multiple port networks, the two-terminal definition of impedance is inadequate, but the complex voltages at the ports and the currents flowing through them are still linearly related by the impedance matrix.
The reciprocal of impedance is admittance, whose SI unit is the siemens, formerly called mho.
Instruments used to measure the electrical impedance are called impedance analyzers.