Impedance, symbolized by the letter Z, is the total opposition a circuit presents to the flow of alternating current (AC). This concept extends the idea of resistance found in direct current (DC) circuits to include frequency-dependent effects. Like resistance, impedance is measured in Ohms (\(\Omega\)). Calculating impedance is fundamental to analyzing any AC circuit, as it dictates the relationship between voltage and current.
Defining Resistance and Reactance
Impedance is composed of two primary components that oppose current flow: Resistance (R) and Reactance (X). Resistance is the non-frequency-dependent opposition that converts electrical energy into heat and remains constant regardless of the AC signal’s frequency. This opposition is primarily contributed by resistors.
Reactance (X) is the opposition to AC flow caused by energy storage elements like inductors and capacitors. Because these components store energy in magnetic or electric fields, their opposition changes significantly with the frequency of the AC signal. Reactance is divided into Inductive Reactance (\(X_L\)) from inductors and Capacitive Reactance (\(X_C\)) from capacitors.
Calculating Inductive and Capacitive Reactance
Calculating the individual reactances precedes finding the total circuit impedance. Inductive Reactance (\(X_L\)) is the opposition offered by an inductor and increases directly with frequency. The formula for calculating this value is \(X_L = 2\pi fL\), where \(f\) is the frequency in Hertz (Hz) and \(L\) is the inductance in Henrys (H). This direct relationship means that a higher frequency causes the inductor to offer greater opposition to current flow.
Capacitive Reactance (\(X_C\)) is the opposition offered by a capacitor and exhibits an inverse relationship with frequency. The formula to calculate this value is \(X_C = 1 / (2\pi fC)\), where \(C\) represents the capacitance in Farads (F). As the frequency increases, the denominator grows larger, causing the overall \(X_C\) value to decrease.
Calculating Total Impedance in Series Circuits
In a series circuit containing a resistor, an inductor, and a capacitor (an RLC circuit), the total impedance (\(Z_T\)) is calculated by combining the resistance and the net reactance. Resistance dissipates energy while reactance stores it, meaning they cannot be simply added together arithmetically. Instead, they must be combined orthogonally using a vector approach, represented by the complex impedance formula \(Z = R + j(X_L – X_C)\). The term \(j\) signifies the imaginary axis, indicating that reactive components are 90 degrees out of phase with the resistive component.
To find the magnitude of the total impedance, which is the value measured in Ohms, the Pythagorean theorem is applied to the resistance and the net reactance. The magnitude formula is \(Z = \sqrt{R^2 + (X_L – X_C)^2}\). For example, if a circuit has a resistance of \(3\ \Omega\) and a net reactance of \(4\ \Omega\), the total impedance is \(5\ \Omega\).
Calculating Total Impedance in Parallel Circuits
Calculating total impedance in parallel RLC circuits requires a mathematical approach distinct from the series method. The difficulty arises because the currents through parallel branches containing reactive components are out of phase and cannot be summed directly. Therefore, it is simpler to work with the reciprocal concept known as Admittance (Y) instead of impedance (Z).
Admittance, measured in Siemens (S), is the reciprocal of impedance (\(Y = 1/Z\)) and represents the ease with which a circuit allows current to flow. In a parallel configuration, the total admittance (\(Y_T\)) is found by summing the individual admittances of each branch. The total admittance is calculated as \(Y_T = Y_R + Y_L + Y_C\), where \(Y_R\), \(Y_L\), and \(Y_C\) are the complex admittances of the resistor, inductor, and capacitor branches. Once \(Y_T\) is calculated, the total impedance (\(Z_T\)) is found by taking the reciprocal, \(Z_T = 1/Y_T\).