# Zener Diodes and Voltage Regulation by nikeborome

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```									                                                                         ELE5PRA Engineering Practice 2007

DC Power Supplies and Zener Regulation
2.1.1 Single phase rectifier                                             vs
v s  Vm sin t
Vm
v D   -
           2        t
vo    Vm
Vm
Vo dc  
                  
                       vo    RL                vD                     2
                       
             2
vr
Vm    vr
Vm  Vo dc 
           2
Figure 3.1.0 (a) Single Phase Diode Rectifier

The diode converts ac to dc and the average output voltage Vo dc  is found in the
following equation.
             
1             1              Vm
Vodc      
2 0
vo dt      Vm sin t    0.318Vm
2 0

The rms output voltage is
1                                                1
 1  2          2  1  2                       2 V
Vo dc     v o dt     Vm sin 2 td t   m  0.5Vm
 2 0                 2 0                        2
The pulsating output contains ripple the ripple is expressed as
v s  Vo dc   Vm sin t  Vo dc  for 0  t  

vr  
 -Vo dc 
                                     for   t  2
The rms ripple voltage can be calculated to give
1                                1
 2                         2           2  2
Vr rms    V 2 o dc   V 2 o dc    Vo dc       1  1.21Vo dc 
 4                                      4     
The ripple factor is therefore
Vr rms  1.21Vodc 
RF %                                  100  121%
Vodc             Vodc 
The effectiveness of the rectifier is measured by the rectification efficiency
Po dc     Vo dc  I o dc      Vm  2 R 4
                                           40.5%
Po ac  Vo rms  I o rms  Vm 22 R  2
Rectifiers are generally supplied through a transformer from a fixed ac input voltage of
120Vrms, figure 3.1.1 below.

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D1

io
                                                                   

RL
vp                                                                  vo
vs

                                                                   


Figure 3.1.1a  : Rectifier with input
transfomer

2.1.2 Single Phase rectifier as a battery charger
v s  Vm sin 

                t
1    2              2
R      D1 io
vo     vs  E
                                             Vm  E

                    t
                     v s  Vm sin 
vp                                  E        1    2              2
                     
                                             Vm
Vm  E
io
Vm  E
     R              t
1    2             2

Figure 3.1.2 (a) Single Phase Transfomer Diode Rectifier as a battery charger

There are many uses for the half wave rectifier of figure 3.1.1. Besides ac to dc
conversion a half wave rectifier can be used for peak detection if used with a capacitor
filter or as a battery charger. The figure in 3.1.2 shows an example of the single phase
rectifier used as a battery charger.

2.1.3 Single Phase Full Wave Center Tapped Rectifier

The average dc voltage in a half wave rectifier is low as shown earlier. The full wave
rectifier is designed to give a higher dc content, double the one in a half wave rectifier.
Referring to figure 3.1.3 below, the output voltage is
Vm sin ωt 0  ωt  π
vo  
- Vm sin ωt π  ωt  2π
The average voltage is therefore

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             
2             2              2Vm
Vo dc        
2 0
vo dt      Vm sin t    0.636Vm
2 0

v s  Vm sin 

  t
              2
D1 io         vo
                           Vm
             vs
                             R               Vo dc 
vp                                                                    t
                        E                            2

             vs                      vD
          2   t

vD  PIV
PIV
 2Vm
vr                   Vm  Vdc

  t
 Vo dc                          2

Figure 3.1.3 (a) Single Phase transfomer center tapped Diode Rectifier

The rms output voltage is
1                                       1
 2          2  2  2                2 V
Vo dc       v 2 o dt     Vm sin 2 td t   m  0.707Vm
 2 0            2 0                    2

The ripple content in this waveform is
v s  Vo dc   Vm sin ωt  Vo dc  for 0  ωt  π

vr  
 -Vm sin ωt -Vo dc 
                                      or π  ωt  2π
f
The ripple voltage can be expressed from
1                             1
 2                        2            2  2
Vr rms    V 2 o dc   V 2 o dc    Vo dc      1  0.483Vo dc 
 8                                      8    
The rectification efficiency
2Vm   R
2
Podc      Vo dc  I o dc                        8
                                    2  81%
Poac  Vo rms  I o rms       Vm 2 R 
2

Twice that of a half wave rectifier. The PIV is 2Vm.
The second type of rectifier is a bridge rectifier shown in the figure below. The analysis
is the same as the center tapped rectifier yielding exactly the same results.

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v s  Vm sin 

  t
             2
io                                            vo

D3                   D1                 Vm
Vo dc 
                                                                                                                   t
                                                                                E                             2
vp                Vm sin t                                                   vD
                                                    D2
                                                                              vr                  Vm  Vdc
                                  D4         R                                                    t
 Vo dc                           2


Figure 3.1.4 (a) Single Phase Full Wave Bridge Rectifier

2.1.4 Output Filters for Rectifiers

The rectifier output contains a dc component as well as cosine components at various
frequencies. The magnitudes of these are called harmonics. The desired output is pure dc.
Filters are used to smooth out the output. DC filters are C-Filters, L-filters and LC filters.
The C filters are used in integrated circuits and the last two filters are more suitable for
use in high power applications such as dc power supplies.

2.1.5 L-Filters

An inductor is a storage device that mains constant current through the load so that the
variations in output voltage are minimized.

io

                                              D3                   D1


vp                       Vm sin t
                                                             D2

                                                             D4

L

R
Figure 3.1.5 : Bridge Rectifier with an L - Filter

At ripple frequency the inductance has a high impedance and this causes the load current
ripple to be reduced.
Z  RL  jnL  RL  nL  n
2         2

Where

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 n  tan 1 nL R L 
4Vm                                         cos nt   n
io t   I o dc  
1

 n  1n  1
RL  nL 
2                  2
2, 4, 6

Where
2Vm    Vodc 
I o dc                 
RL         RL
Taking the first two harmonics and ignoring the higher order harmonics
I r2rms   I o 2rms   I o 4rms 
2             2

2                                                        2
                                                                                                   
1      4Vm                                 1   1      4Vm                                         1
.I r2rms                                                                                                            ..........
...
2                                          3   2                                                 15 
                                                                                
1                                                       1
  RL  2L                                    RL  4L                                       
2         2                                      2          2
                                                                                                   
2                                                       2

2.1.6 C-filters

A capacitor is a storage device that maintains a constant voltage. The capacitor is
connected across the load to minimize the output ripple.
io

                                                         D3                    D1


vp                             Vm sin t
                                                                                 D2

D4      C             R



vo              Vc max                                          Vc min 

a 
t
                 2                  3           4
t2
vT          t1
T 2                                              VT  peak 
b 
t
                                                      4
2                  3

is

c 
t

Figure 3.1.6 : Bridge Rectifier with an C - Filter

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t1 charging time of C, t2 discharge time of C, figure above. Period is the sum of t1 and t2
such that
T
 for a full waverectifier
t1  t 2   2
T for a half waverectifier

But t2>>t1 so that
T
 for a full waverectifier
t2   2
T for a half waverectifier


Time interval is redefined such that from t = 0 to the beginning of interval 1.

Vm
io e  t t1  RLC     for t 1  t 1  t 2 
RL
The instantaneous output voltage during the discharge period is
vo  R L io  Vm e  t t1  RLC
The peak to peak ripple voltage is then

Vr  pp   vo t  t1   vo t  t1  t 2   Vm  Vm e t2   RLC

 Vm 1  e t2   RLC

  x
Since e   1  x
            t 2  Vm t 2  Vm
Vm 1  1 
             
 R C  2 fR C for a full waverectifier
            RL C 
Vr  pp  
L        L

 Vm
 fR C                          for half waverectifier
 L

The average output then becomes

    V                 Vm    V 4 fRLC  1
Vm  r  pp  Vm          m              for a full wave rectifier
      2             4 fRLC     4 fRLC
Vo dc   
V  Vm  Vm 2 fRLC  1
 m 2 fRLC
for half wave rectifier
                     2 fRLC

Assuming a sine wave peak ripple then the rms is

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Vr  pp     Vm
                  for a full waverectifier
 2 2 4 2 fRL C
Vr rms    
 Vm

 2 2 fR C           for half waverectifier

           L

The ripple factor RF is therefore
Vr rms    Vm         4 fRL C                                       1
RF                                     
Vo dc  4 2 fRL C Vm 4 fRL C  1                       2 4 fRL C  1
       1
 2 4 fR C  1 for a full waverectifier


L

       1
for half waverectifier
 2 2 fRL C  1


2.1.7 Zener Diodes and Voltage Regulation

Zener diodes operate in the reverse break down region. In this region the diode current
increases rapidly with small increases in voltage. The I-V characteristic of the zener is
shown in figure 3.1.7. The voltage at the knee is V ZK this is the voltage where the zener
enters the breakdown region. The zener voltage is the breakdown voltage at a specific test
current I Z  I ZT . I Z m ax is current that the zener can withstand and still remain within
permissible limits for power dissipation. I Z m in  is slightly below the knee on the
characteristic curve.

The equivalent circuit for a zener diode is shown in figure 3.1.7 (c) and (d). In figure (b)
the reverse characteristic can be approximated by a piecewise linear model with a fixed
voltage V ZO and an ideal resistor (see equivalent circuit). The equivalent circuit is for
v D  VZ . RZ depends on the inverse slope of the zener characteristic and is defined as
VZ               v D
RZ                  
I Z   atVZ
i D   for v D  0 and iD  0

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iD
Large voltage               Large voltage          Small voltage
large current               small current          large current
iD                                                                               I Z  min 
VZ
 VD max 
                                   VZ                 V ZK  V D  min 
I ZK                    vD
vD

IZ            I Z

iZ
I Z  max 
iZ
Figure 3.1.7
a  Symbol                          b    Zener characteristics

                                              
D1                                  D1

vD                                             vD
RD                                 RZ
                                              

VTD                          VZO  VBR

Figure 3.1.7 :
c  Forward                                     d  Reverse

Where RZ is the zener resistance. Away from the knee the zener resistance remains
fairly constant. Typically the value RZ is a few tens of ohms. At the knee RZ is around
3kΩ. The zener current iZ  i D and can be related to V ZO and RZ by
VZ  VZ 0  RZ i Z
A zener diode can be seen as offering a variable resistance whose value changes with
current so that the voltage across the terminals is constant. The zener is therefore known
as the voltage reference diode. Since RZ is very small it means that the zener voltage V Z
is almost independent of the reverse diode current. The reverse region offers a constant
voltage this makes the zener diode suitable for voltage regulation. A voltage regulator
maintains a fairly constant output even when supply voltage and the load current vary
over a wide range. A zener regulator is also known as a shunt regulator. Figure 3.1.8
shows a shunt regulator. The value of R S is chosen such that the zener operates in the
reverse region over the entire range of input voltages. The equivalent circuit is
represented by a piecewise model, figure 3.1.8(b). when v s varies the current i Z will
vary because of RZ causing a variation in the output voltage. This defined by the line
voltage given by

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v0    RZ
Line regulation                  
v S RZ  RS
If the load current i L increases then i Z ↓ because of RZ causing V out to↓. This variation
of V out is load regulation.
iS                                                    RS         iS
RS

          iL
D1              D1        iL
iZ
VZO
VZ                                          VTD
RL                                             VZ               vO
vS                                                               RL

RD             RZ

                

Figure 3.1.8 :      a  Circuit                                                  b Equivalent circuit

v0
 RZ || RS 
i L
A change in the zener voltage causes an increase in the output voltage. The variation is
caused by zener regulation.
v0        Rs
Zener regulation          
VZ 0 RZ  RS
By superposition the effective voltage is
v0         v        v
v0         VZ 0  0 v s  0 i L
VZ 0       v S      i L
Rs             RZ
           VZ 0          v S  RZ || RS i L
RZ  RS        RZ  RS

2.1.8 Design of a Regulator
The regulator is designed so that the zener can operate in the break down region under
worst case scenario. For this to happen the following conditions should be met
VS (min)  VZ 0  RZ iZ (min ) 
1.                              RS 
iZ (min)  i L (max)
VS (max)  VZ 0  RZ iZ (max) 
2.                                           RS 
iZ (max)  i L (min)
The maximum zener current in terms of the variations in VS and iL is

V 
S m in                                                             
 VZO  RZ i Z m in i Z m ax  i L m in  VS m ax  VZO  RZ i Z m ax i Z m in  i L m ax
As a rule of thumb the minimum zener current i    is limited to 10% of the maximum
Z m in

zener current i    .That is i     0.1  i   .
Z m ax                Z m in               Z m ax

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