AC parallel circuits

Learning objectives

• AC parallel circuit
• Solving AC circuits using complex numbers
• Resonance Parallel circuit
• Rectification and smoothing of AC

Parallel AC circuits

As shown in in previous chapters, several types of circuits in AC can be built, which will be analysed differently. Using phasor diagrams can become very complicated with parrallel networks containing more than two branches, in these cases, using complex numbers may be appreciated

AC circuits with resistance only

$Z=\frac{V_R\angle 0^o}{I_R\angle 0^o}$

AC circuits with inductance only

$Z=\frac{V_L\angle 90^o}{I_L\angle 0^o}=X_L\angle 90^o =\jmath X_L$

AC circuits with capacitor only

$Z=\frac{V_C\angle -90^o}{I_C\angle 0^o}=X_C\angle -90^o =-\jmath X_C$

RL series AC circuits

RC series AC circuit

RLC series AC circuit

Series Resonnance

The RLC circuit has total impedance : $Z=R+ \jmath(X_L-X_C)$ If$ (X_L=X_C)$, by the equation above, minimal impedance Z is obtained with Z=R. It is said that the circuit is at resonnance

Genral series circuit

In AC circuit containing several impedances connected in series (Z₁, Z₂…), the total equivalent impedance Z_T is given by: Z_T=Z₁+ Z₂…

Parallel AC circuits

It is in Parallel AC circuits that the power of complex numbers can fully be apprecited. Some key concepts need to be defined before starting to solve AC parallel circuits.

Admittance, conductance and susceptance

It is the reciprocal of impedance Z $Y=\frac{I}{V}$ Unit: Siemen As Impedance is defined with a real and imaginary part (Z=R+jX), Admittance can be similarly defined as Y=G+jB with G as conductance and B as susceptance.

Parallel AC networks

This is very similar to direct current, except that resistance is replaced with impedance Z. As with resistance, the total impedance of a circuit containing an impedance in each branch add up as follows: $\frac{1}{Z_T}=\frac{1}{Z_1} + \frac{1}{Z_2} …$ The problems with AC circuit are now easily solved with complex form, exactly as if solving a parallel circuit with DC (same current divider equations applicable, with impedance instead of resistance).

Parallel Resonnance Circuit

The total admittance of an L-R-C parallel circuit is given by $Y= G+\jmath(B_C-B_L)=\frac{1}{R}+\jmath(\omega C-\frac{1}{\omega L})$ The circuit is at resonance when the imaginary part is zero, in other words: $\omega C= \frac{1}{\omega L}$. This result in minial admittace $Y_r=\frac{1}{R}$

A more practical network contains a coil of inductance L with resistance R in parallel with a pure capacitance C. The following equation is thus obtained: $ Y= \frac{R}{R^2 + (\omega L)^2}+\jmath(\omega C -\frac{\omega L}{R^2 +(\omega L)^2}) $

Since at resonance, the imaginary part is zero, omega is thus written as follows: $\omega_R = \sqrt{\frac{1}{LC}-\frac{R^2}{L}}$

Rectification and smoothing

The rectification process is a process of obtaining unidirectional currents and voltage from alternating currents and voltages. Automatic switching in circuits is achieved using diodes. A diode is a two terminal electronic component with an asymmetric transfer characteristic with:

• Low (ideally zero) resistance to current flow in one direction and,
• high (ideally infinite) resistance in the other direction.

There are two main rectifications circuits: half wave and full wave circuits.

Smoothing is the process of removing the worst of the output waveform variations. To smooth out the pulsations a large capacitor C, is connected across the output of the rectifier, the effect is to maintain the output voltage at a level which is very near to the peak of the output waveform.