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A single-phase, full-wave, diode rectifier is supplied from a 230V, 50Hz voltage source and uses a capacitor output filter, 1000μF with a resistor 100Ω load, as shown in Figure 11.9a. Ignoring diode voltage drop, determine i. expressions for the output voltage ii. output voltage ripple Δv0 and the % error in using the approximation equation (11.90) iii. expressions for the capacitor current iv. diode peak current v. average load voltage and current Assuming the output ripple voltage is triangular, estimate vi. average output voltage and rms output ripple voltage vii. capacitance C for of the maximum output voltage

V.S 230 V * :

f 50 Hz * :

C 1000 μF * :

R.L 100 Ω * :

T 1 f / : ms 20

V.Spk 2 sqrt V.S * : 325.3 V *

ω 2 π * f * : 314.159 Hz *

X.C 1 ω C * / : 3.1831 Ω *

Z R.L 2^X.C 2^+ sqrt : 100.051 Ω *

β π ω R.L * C * atan - : ° 91.8

The diode turn-on angle is solved iteratively from equation

α sin β sin e π α + β - (ω R.L * C * / -^* - 0 ≡

α α sin β sin e π α + β - (ω R.L * C * / -^* - 0 ≡ α solve :

α 1 el ° 66.5

The diode conduction period is

θ.C β α 1 el - : ° 25.3

The output peak-to-peak voltage ripple is given by equation:

ΔV.0 2 sqrt V.S * 1 α 1 el sin - (* : 26.94 V *

ΔV.0 V.S / % 11.7

From Table 11.2, the capacitor current is:

ω t i.C 2 sqrt V.S * ω * C * ω t * cos * :

where α1 ≤ ωt ≤ β

ω t i.C1 2 sqrt V.S * Β sin * R.L / e ω t * β * (ω R.L * C * / -^* :

where β ≤ ωt ≤ π+α1

The peak diode current is given by equation:

Id.pk 2 sqrt V.S * ω * C * α 1 el cos * 2 sqrt V.S * R.L / α 1 el sin * + : 43.7 A *

2 sqrt V.S * ω * C * α 1 el cos * 40.72 A *

is the capacitor's initial charge current

The average load voltage and current are given by equation:

VRL.ave 2 sqrt V.S * π / 1 θ.C cos - β cos - / (* : 312.28 V *

IRL.ave VRL.ave R.L / : 3.12 A *

A capacitor value to obtain 2% ripple, given by equation:

\0025\.ripple 2 % * :

C.1 12 f * R.L * \0025\.ripple * / : μF 5000

ΔV.01 V.S \0025\.ripple * : 4.6 V *

Graphing waveforms:

V.m V.S :

t v.i V.m ω t * sin * vectorize :

t v.id V.m ω t * cos * vectorize :

T ms 20

C μF 1000

R.L 100 Ω *

t v.0 ω t * sin 0 > t v.i t v.i - if 11 line :

t v.1 t v.i 0 > t v.id t v.id - if 11 line :

t 06 T * (0.0001 s * range :

δ 20 :

t.c1 δ T * (δ 5 + (T * (δ T * .0001 s * + range :

t.d 14 T * :

t t.c1 :

dc.offset .94 V.m * :

t V.charge t v.0 1 e t R.L C * / -^- (* vectorize :

t V.discharge t v.1 e t.d R.L C * / -^* vectorize :

t ms / t v.0 vectorize augment t ms / π + t V.discharge dc.offset + augment t ms / t ( V.charge augment V.m V.m ΔV.0 - 5 1 sys 1 1 sys