Electrical Impedances - In Parallel
Electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied. The term complex impedance may be used interchangeably.
Quantitatively, the impedance of a two-terminal circuit element is the ratio of the complex representation of a 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 AC circuits, and possesses both magnitude and phase, unlike resistance, which has only magnitude. When a circuit is driven with direct current (DC), there is no distinction between impedance and resistance; the latter can be thought of as impedance with zero phase angle.
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.
Impedance is 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 form |Z|∠θ. However, cartesian complex number representation is often more powerful for circuit analysis purposes.
The reciprocal of impedance is admittance, whose SI unit is the siemens, formerly called mho.
The term impedance was coined by Oliver Heaviside in July 1886. Arthur Kennelly was the first to represent impedance with complex numbers in 1893.
The introduction of the concept of impedance in AC circuit is justified by the fact that there are two additional impeding mechanisms to be taken into account besides the normal resistance of DC circuits: the induction of voltages in conductors self-induced by the magnetic fields of currents (inductance), and the electrostatic storage of charge induced by voltages between conductors (capacitance). The impedance caused by these two effects is collectively referred to as reactance and forms the imaginary part of complex impedance whereas resistance forms the real part.
Impedance is defined as the frequency domain ratio of the voltage to the current. In other words, it is the voltage–current ratio for a single complex exponential at a particular frequency ω.
For a sinusoidal current or voltage input, the polar form of the complex impedance relates the amplitude and phase of the voltage and current. In particular:
The magnitude of the complex impedance is the ratio of the voltage amplitude to the current amplitude;
the phase of the complex impedance is the phase shift by which the current lags the voltage.
Quantitatively, the impedance of a two-terminal network is represented as a complex quantity Z, defined in Cartesian form.
The total impedance of many simple networks of components can be calculated using the rules for combining impedances in series and parallel. The rules are identical to those used for combining resistances, except that the numbers in general will be complex numbers. In the general case, however, equivalent impedance transforms in addition to series and parallel will be required.
For components connected in parallel, the voltage across each circuit element is the same; the ratio of currents through any two elements is the inverse ratio of their impedances.
Hence the inverse total impedance is the sum of the inverses of the component impedances as shown here.Related formulas
|Zeq||total impedance (ohm)|
|R1||resistance 1 (ohm)|
|X1||reactance 1 (ohm)|
|R2||resistance 2 (ohm)|
|X2||reactance 2 (ohm)|
|R3||resistance 3 (ohm)|
|X3||reactance 3 (ohm)|
|R4||resistance 4 (ohm)|
|X4||reactance 4 (ohm)|
|R5||resistance 5 (ohm)|
|X5||reactance 5 (ohm)|
|R6||resistance 6 (ohm)|
|X6||reactance 6 (ohm)|