Rectifier:- 

• A Rectifier is a circuit which converts A.C voltage into pulsating D.C voltage. There are three types of rectifiers. They are

1.Halfwave Rectifier.

2.Center-tapped Full wave Rectifier.

1. Full wave Bridge Rectifier.

Full wave Bridge Rectifier:-

• Bridge rectifier is a full wave rectifier circuit which uses four diodes and avoids center tap on the secondary winding of the transformer.

• During the positive half cycle the point A becomes positive. The diodes D1 and D2 will be forward biased while D3 and D4 will be reverse biased.During this cycle the current flows through D1, RL and D2. 

• During the negative half cycle the point B becomes positive. The diodes D3 and D4 will be forward biased while D1 and D2 will be reverse biased.During this cycle the current flows through D3, RL and D4. 

• The input and output waveforms of bridge rectifier is as follows

Expressions for Idc :-

• The average or dc value of the alternating current is obtained by integration. To find the average value of an alternating wave form , determine the area under the curve over one cycle and divide it by its base.
• Let us consider the out put waveform given below

• Let$i_L = I_m Sin wt$
• Consider one cycle of the load current iL from 0 to π. Thus the average value is calculated as follows

$I_{av} = I_{dc}= \dfrac{1}{\pi}\int_0^\pi {Im Sinwt \ d(wt)} \\= \dfrac{I_m}{\pi}[-coswt]_0^\pi \\= \dfrac{I_m}{\pi}[-cos\pi -(-cos0)] \\= \dfrac{I_m}{\pi}[+1+1]\\=\dfrac{2I_m}{\pi}$

• The expression for average voltage is given as follows

$V_{av} = I_{av}\times R_L $

• Substituting the value of Iav we have

$V_{av} = I_{av}\times R_L =\dfrac{2I_mR_L }{\pi}$

Expressions for Irms:-

• The rms value of the alternating current is obtained by taking square root of mean sqaure of the current over a complete cycle. Thus the rms value is calculated as follows
• $I_{rms}=\sqrt{\dfrac {1}{2\pi}\int_0^{2\pi}{i^2_L \ d(wt)}}$
• $I_{rms}^2=\dfrac {1}{2\pi}\int_0^{2\pi}{i^2_L \ d(wt)} \\=\dfrac {1}{2\pi}\int_0^{2\pi}{I^2_mSin^2 \ d(wt)} \\= \dfrac {1}{2\pi}\int_0^{2\pi}{I^2_mSin^2 \ d(wt)}\\= \dfrac {I^2_m}{2\pi}\int_0^{2\pi}{Sin^2 \ d(wt)}$

• The wave form for 0 to π is the same as the  wave form for π to 2π. So we double the interation from 0 to π.

  Thus

• $I_{rms}^2= \dfrac {I^2_m}{2\pi}\int_0^{2\pi}{Sin^2wt \ d(wt)} \\ = \dfrac {2I^2_m}{2\pi}\int_0^{\pi}{Sin^2wt \ d(wt)} \\= \dfrac {I^2_m}{\pi}\int_0^{\pi}{[{{1-cos2wt}\over 2}] \ d(wt)} \\=\dfrac{I^2_m}{2\pi}[[wt]_0^\pi-{[{sin2wt\over 2}}]_0^\pi]\\=\dfrac{I^2_m}{2\pi}[\pi-0] \\=\dfrac{I^2_m}{2}$

• Therefore $I_{rms}=\sqrt{\dfrac{I^2_m}{2}}=\dfrac{I_m}{\sqrt{2}}$

Ripple Factor:-

• Ripple factor is a measure of ripples present in the output of a rectifier. Ripple factor of any rectifier is given by the below expression.
• $Ripple factor = \sqrt{\left [\dfrac{I_{RMS}}{I_{DC}}\right ]^2 -1}$
• For a full wave rectifier$I_{rms}=\dfrac{I_m}{\sqrt{2}}$and$I_{dc}= \dfrac{2I_m}{\pi}$
• $Ripple factor \ of \ a \ FWR = \sqrt{\left [\dfrac{{\dfrac{I_M}{\sqrt 2}}}{\dfrac{2I_M}{\pi}}\right ]^2 -1} = \sqrt{\dfrac{\pi^2}{8} -1} = 0.48$

Rectifier efficiency:-

• Rectifier efficiency is defined as the ratio of output dc power to input ac power.

$\eta = \frac {output \ P_{dc}}{input \ P_{ac}}$

• Neglecting reistance of the diode and resistance of secondary winding.
• The input AC power is given by

$P_{ac}=I^2_{rms}R_L = \left[{\dfrac {I_m}{\sqrt 2}}\right]^2R_L = \dfrac{I^2_m}{2} R_L$

• The otuput DC power is given by

$P_{dc}=I^2_{dc}R_L = \dfrac{4I^2_m}{\pi^2}R_L$

• Rectifier efficiency of full wave rectifier is given by

$\eta = \dfrac{P_{dc}}{P_{ac}}=\left[\dfrac{\dfrac{4I^2_m}{\pi^2}R_L}{\dfrac{I^2_m}{2}R_L}\right] = \dfrac{8}{\pi^2} = 0.81$

• Efficiency of Full wave Rectifier in percentage η=81%