How do you handle complex impedances and admittances in phasor circuits with capacitors and inductors?

1 Complex impedance and admittance

Complex impedance (Z) and admittance (Y) are two ways of expressing the opposition or the ease of current flow in an AC circuit. They are both composed of a real part and an imaginary part, which are related to the resistance (R) and the reactance (X) of the circuit element. The complex impedance is defined as Z = R + jX, where j is the imaginary unit and X is positive for inductors and negative for capacitors. The complex admittance is the reciprocal of the complex impedance, that is, Y = 1/Z = G + jB, where G is the conductance and B is the susceptance.

2 Phasor diagrams and circuit analysis

A phasor is a vector that rotates at a constant angular frequency and has a length equal to the peak value of a sinusoidal wave. The angle of the phasor represents the phase difference between the wave and a reference wave, usually the voltage source. Phasor diagrams are useful tools for visualizing and analyzing AC circuits, as they allow you to add and subtract complex numbers geometrically. To draw a phasor diagram, you need to convert the time-domain expressions of the current and voltage into phasor-domain expressions using Euler's formula. Then, you can use Kirchhoff's laws, Ohm's law, and the impedance or admittance formulas to find the unknown values in the circuit.

3 Series and parallel circuits

One of the advantages of using phasors is that you can apply the same rules for series and parallel circuits as you would for DC circuits, as long as you use complex impedances or admittances instead of resistances. For series circuits, the total impedance is the sum of the individual impedances, and the total current is the same for all elements. For parallel circuits, the total admittance is the sum of the individual admittances, and the total voltage is the same for all elements. You can also use the voltage divider and current divider rules to find the voltage or current across or through any element in the circuit.

4 Power factor and power triangle

Another important concept in AC circuits is the power factor, which is the ratio of the real power (P) to the apparent power (S) in the circuit. The real power is the average power delivered to the resistive load, while the apparent power is the product of the rms voltage and current. The power factor can also be expressed as the cosine of the angle between the voltage and current phasors, which is called the phase angle. The power factor indicates how efficiently the circuit converts the input power into useful output power. A low power factor means that there is a large phase difference between the voltage and current, which implies a high reactive power (Q), which is the power stored and released by the capacitors and inductors. The power triangle is a graphical representation of the relationship between P, Q, and S, where P is the horizontal side, Q is the vertical side, and S is the hypotenuse.

5 Impedance matching and resonance

In some applications, such as signal transmission or amplification, it is desirable to have a high power transfer between the source and the load. This can be achieved by matching the impedance of the source and the load, that is, by making them equal or conjugate. Impedance matching can be done by adding a matching network, which is a combination of capacitors and inductors that can adjust the impedance of the load to match the source. Impedance matching can also result in resonance, which is a condition where the reactance of the circuit is zero or the impedance is purely resistive. Resonance occurs when the frequency of the source is equal to the natural frequency of the circuit, which depends on the values of the capacitors and inductors. Resonance can increase the voltage or current in the circuit to a maximum value, which can be useful or harmful depending on the situation.

6 Practical tips and examples

To effectively handle complex impedances and admittances in phasor circuits with capacitors and inductors, you should practice some skills and follow some tips. For example, use a calculator or software that can perform complex arithmetic and trigonometry, and decide whether to use polar or rectangular forms of complex numbers. Additionally, scientific or engineering notation can help to avoid rounding errors and simplify the notation. To ensure accuracy, you should use standard values and units for capacitors and inductors such as microfarads, nanofarads, henrys, or millihenrys. Phasor diagrams can be used to check your results and understand the phase relationships in the circuit. Finally, it is beneficial to use examples from textbooks, online resources, or your own experiments to test your knowledge and apply the concepts.

7 Here’s what else to consider

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