Semiconductor Devices

Rectifier: A rectifier is a device (circuit) that converts alternating current (ac) into direct current (dc). It uses the one-way conduction of p-n junction diodes.

Full wave rectifier using two diodes

                         D1
             A o--------|>|--------+
               )                   |
     )||(      )                   |   K (+)
 AC  )||(      ) C (centre tap)    |
 in  )||(      o------[ R_L ]------+
     )||(      )                   |
               )                   |
             B o--------|>|--------+
                         D2
  primary   secondary   (current in R_L always flows K -> C)

Working (centre-tap full wave rectifier, two diodes $D_1$ and $D_2$):

1. The ac input is applied to the primary of a transformer. The secondary has a centre tap C, so the ends A and B are always opposite in polarity with respect to C.
2. Positive half cycle: A is positive and B is negative with respect to C. $D_1$ is forward biased and conducts; $D_2$ is reverse biased and does not conduct. Current flows A → $D_1$ → K → $R_L$ → C.
3. Negative half cycle: B is positive and A is negative. Now $D_2$ conducts and $D_1$ is off. Current flows B → $D_2$ → K → $R_L$ → C.
4. In both half cycles the current through $R_L$ flows in the same direction (K to C). So the output is unidirectional (pulsating dc) for the full cycle. This is full wave rectification.

 Input (ac)                      Output across R_L (full wave)
 V                               V
 ^   __        __                ^   __    __    __    __
 |  /  \      /  \               |  /  \  /  \  /  \  /  \
 +-/----\----/----\---> t        | /    \/    \/    \/    \
 |       \__/      \__/          +--------------------------> t
          (D1 on)(D2 on)             D1    D2    D1    D2

(a) Rectification

Rectification is the process of converting alternating current (ac) into direct current (dc), i.e. into a current that flows in one direction only. A p-n junction diode can do this because it conducts only when forward biased.

(b) Full wave rectifier using two diodes

                         D1
             A o--------|>|--------+
               )                   |
     )||(      )                   |   K (+)
 AC  )||(      ) C (centre tap)    |
 in  )||(      o------[ R_L ]------+
     )||(      )                   |
               )                   |
             B o--------|>|--------+
                         D2
  primary   secondary   (current in R_L always flows K -> C)

Working (centre-tap full wave rectifier, two diodes $D_1$ and $D_2$):

1. The ac input is applied to the primary of a transformer. The secondary has a centre tap C, so the ends A and B are always opposite in polarity with respect to C.
2. Positive half cycle: A is positive and B is negative with respect to C. $D_1$ is forward biased and conducts; $D_2$ is reverse biased and does not conduct. Current flows A → $D_1$ → K → $R_L$ → C.
3. Negative half cycle: B is positive and A is negative. Now $D_2$ conducts and $D_1$ is off. Current flows B → $D_2$ → K → $R_L$ → C.
4. In both half cycles the current through $R_L$ flows in the same direction (K to C). So the output is unidirectional (pulsating dc) for the full cycle. This is full wave rectification.

 Input (ac)                      Output across R_L (full wave)
 V                               V
 ^   __        __                ^   __    __    __    __
 |  /  \      /  \               |  /  \  /  \  /  \  /  \
 +-/----\----/----\---> t        | /    \/    \/    \/    \
 |       \__/      \__/          +--------------------------> t
          (D1 on)(D2 on)             D1    D2    D1    D2

(c) If one diode becomes functionless

If one diode stops working (open circuit), current flows through $R_L$ only during the half cycle in which the other, good diode is forward biased. In the other half cycle there is no output. So the circuit works as a half wave rectifier: the output is present in alternate half cycles only, and the average output voltage falls to half of its full-wave value.

 V across R_L
 ^    __            __            __
 |   /  \          /  \          /  \
 |  /    \        /    \        /    \
 +-/------\------/------\------/------\------> t
   0      T/2    T            2T
          |<-gap->|  (no output in alternate half cycles)

(a) P-N junction diode

When a p-type semiconductor is joined with an n-type semiconductor in a single crystal, the device formed is called a p-n junction diode. It conducts current easily when forward biased (p side to positive terminal) and almost no current when reverse biased, so it allows current in one direction only.

   p  ----|>|----  n
 anode (+)    cathode (-)

(b) Junction diode as a full wave rectifier

                         D1
             A o--------|>|--------+
               )                   |
     )||(      )                   |   K (+)
 AC  )||(      ) C (centre tap)    |
 in  )||(      o------[ R_L ]------+
     )||(      )                   |
               )                   |
             B o--------|>|--------+
                         D2
  primary   secondary   (current in R_L always flows K -> C)

Working (centre-tap full wave rectifier, two diodes $D_1$ and $D_2$):

1. The ac input is applied to the primary of a transformer. The secondary has a centre tap C, so the ends A and B are always opposite in polarity with respect to C.
2. Positive half cycle: A is positive and B is negative with respect to C. $D_1$ is forward biased and conducts; $D_2$ is reverse biased and does not conduct. Current flows A → $D_1$ → K → $R_L$ → C.
3. Negative half cycle: B is positive and A is negative. Now $D_2$ conducts and $D_1$ is off. Current flows B → $D_2$ → K → $R_L$ → C.
4. In both half cycles the current through $R_L$ flows in the same direction (K to C). So the output is unidirectional (pulsating dc) for the full cycle. This is full wave rectification.

 Input (ac)                      Output across R_L (full wave)
 V                               V
 ^   __        __                ^   __    __    __    __
 |  /  \      /  \               |  /  \  /  \  /  \  /  \
 +-/----\----/----\---> t        | /    \/    \/    \/    \
 |       \__/      \__/          +--------------------------> t
          (D1 on)(D2 on)             D1    D2    D1    D2

(c) Truth table of AND gate ($Y = A \cdot B$)

The output is 1 only when both inputs are 1.

a) Depletion layer becomes thinner in forward bias

• In forward bias, the p side is connected to the positive terminal and the n side to the negative terminal of the battery.
• The applied voltage $V$ sets up an electric field across the junction that is opposite to the internal field of the barrier.
• Holes in the p region and electrons in the n region are pushed towards the junction. They enter the depletion layer and neutralise some of the immobile ions there.
• So the region of uncovered charges becomes narrower and the barrier potential falls from $V_0$ to $(V_0 - V)$.

Hence the width of the depletion layer decreases and majority carriers can cross the junction, giving a large forward current.

b) Truth table

A NOR gate with its two inputs tied together works as a NOT gate:

$$Y_1 = \overline{A + A} = \overline{A}, \qquad Y_2 = \overline{B + B} = \overline{B}$$

The third NOR gate gives

$$Y = \overline{Y_1 + Y_2} = \overline{\overline{A} + \overline{B}} = A \cdot B \quad \text{(De Morgan's theorem)}$$

c) Gate obtained

The output is 1 only when both A and B are 1, so the combination works as an AND gate ($Y = A \cdot B$).

a) Rectification and full wave rectification

Rectification is the process of converting ac into dc (current in one direction only).

                         D1
             A o--------|>|--------+
               )                   |
     )||(      )                   |   K (+)
 AC  )||(      ) C (centre tap)    |
 in  )||(      o------[ R_L ]------+
     )||(      )                   |
               )                   |
             B o--------|>|--------+
                         D2
  primary   secondary   (current in R_L always flows K -> C)

Working (centre-tap full wave rectifier, two diodes $D_1$ and $D_2$):

1. The ac input is applied to the primary of a transformer. The secondary has a centre tap C, so the ends A and B are always opposite in polarity with respect to C.
2. Positive half cycle: A is positive and B is negative with respect to C. $D_1$ is forward biased and conducts; $D_2$ is reverse biased and does not conduct. Current flows A → $D_1$ → K → $R_L$ → C.
3. Negative half cycle: B is positive and A is negative. Now $D_2$ conducts and $D_1$ is off. Current flows B → $D_2$ → K → $R_L$ → C.
4. In both half cycles the current through $R_L$ flows in the same direction (K to C). So the output is unidirectional (pulsating dc) for the full cycle. This is full wave rectification.

 Input (ac)                      Output across R_L (full wave)
 V                               V
 ^   __        __                ^   __    __    __    __
 |  /  \      /  \               |  /  \  /  \  /  \  /  \
 +-/----\----/----\---> t        | /    \/    \/    \/    \
 |       \__/      \__/          +--------------------------> t
          (D1 on)(D2 on)             D1    D2    D1    D2

b) Truth table of the digital circuit

A NAND gate with both inputs tied together works as a NOT gate:

$$A' = \overline{A \cdot A} = \overline{A}, \qquad B' = \overline{B \cdot B} = \overline{B}$$ $$Y = \overline{A' \cdot B'} = \overline{\overline{A} \cdot \overline{B}} = A + B$$

The circuit as a whole works as an OR gate.

a) NAND gate

A NAND gate is an AND gate followed by a NOT gate. Boolean expression: $Y = \overline{A \cdot B}$.

  A ----+-----\
        |      )o---- Y = (A.B)'
  B ----+-----/
     (AND shape with a small bubble at the output)

The output is 0 only when both inputs are 1.

b) Full wave rectification using two p-n junction diodes

Rectification is the conversion of ac into dc. A diode conducts only in forward bias, so two diodes can be arranged to use both halves of the ac cycle.

                         D1
             A o--------|>|--------+
               )                   |
     )||(      )                   |   K (+)
 AC  )||(      ) C (centre tap)    |
 in  )||(      o------[ R_L ]------+
     )||(      )                   |
               )                   |
             B o--------|>|--------+
                         D2
  primary   secondary   (current in R_L always flows K -> C)

Working (centre-tap full wave rectifier, two diodes $D_1$ and $D_2$):

1. The ac input is applied to the primary of a transformer. The secondary has a centre tap C, so the ends A and B are always opposite in polarity with respect to C.
2. Positive half cycle: A is positive and B is negative with respect to C. $D_1$ is forward biased and conducts; $D_2$ is reverse biased and does not conduct. Current flows A → $D_1$ → K → $R_L$ → C.
3. Negative half cycle: B is positive and A is negative. Now $D_2$ conducts and $D_1$ is off. Current flows B → $D_2$ → K → $R_L$ → C.
4. In both half cycles the current through $R_L$ flows in the same direction (K to C). So the output is unidirectional (pulsating dc) for the full cycle. This is full wave rectification.

 Input (ac)                      Output across R_L (full wave)
 V                               V
 ^   __        __                ^   __    __    __    __
 |  /  \      /  \               |  /  \  /  \  /  \  /  \
 +-/----\----/----\---> t        | /    \/    \/    \/    \
 |       \__/      \__/          +--------------------------> t
          (D1 on)(D2 on)             D1    D2    D1    D2

(d) ac into dc

A diode conducts only when it is forward biased. So it lets one half of the alternating input through and blocks the other half, giving a current in one direction only (dc).

(a)(i) Formation of the potential barrier

When the p-type and n-type materials are joined, electrons diffuse from the n-side (higher electron concentration) to the p-side, and holes diffuse the other way. Near the junction the electrons and holes recombine, leaving behind immobile positive donor ions on the n-side and negative acceptor ions on the p-side. This region, with no free carriers, is the depletion layer. The ions set up an electric field directed from n to p, which opposes further diffusion. The potential difference across the layer (about 0.7 V for Si, 0.3 V for Ge) is the potential barrier.

(a)(ii) Large current in forward bias

In forward bias (p to the positive terminal) the applied voltage opposes the barrier, narrowing the depletion layer and lowering the barrier. Majority carriers (holes from p and electrons from n) cross the junction in large numbers, so a large current flows.

(b) Truth table

$Y_1 = A + B$ (OR) and $Y_2 = \overline{Y_1}$ (NOT), so the circuit acts as a NOR gate.

(a) A p-n junction conducts easily when forward biased (low resistance) and almost not at all when reverse biased (very high resistance). This one-way conduction allows it to convert AC to DC.

(b) Construction: A centre-tapped transformer has its secondary ends connected to the p-sides of two diodes $D_1$ and $D_2$. The n-sides of both diodes are joined together, and the load resistor $R_L$ is connected between this joint and the centre tap.

Working: During the positive half cycle, the upper end of the secondary is positive, so $D_1$ is forward biased and conducts while $D_2$ is reverse biased and does not. Current flows through $R_L$ in one direction. During the negative half cycle, the lower end is positive, so $D_2$ conducts and $D_1$ is off. The current again flows through $R_L$ in the same direction. Hence the output across $R_L$ is unidirectional (pulsating DC) in both half cycles, with a frequency twice that of the input. A filter can then smooth it.

(c) NAND gate truth table:

(a) Characteristic used in rectification

A diode conducts (low resistance) when it is forward biased and almost does not conduct (very high resistance) when it is reverse biased. This one-way (unidirectional) conduction is used in rectification.

(b) Circuit diagram of a full wave rectifier (centre-tap type)

                         D1
             A o--------|>|--------+
               )                   |
     )||(      )                   |   K (+)
 AC  )||(      ) C (centre tap)    |
 in  )||(      o------[ R_L ]------+
     )||(      )                   |
               )                   |
             B o--------|>|--------+
                         D2
  primary   secondary   (current in R_L always flows K -> C)

In each half cycle one diode conducts, and the current through $R_L$ is always from K to C.

(c) Truth table

From the circuit:

$$C = \overline{A}, \quad D = \overline{B}, \quad E = A \cdot \overline{B}, \quad F = \overline{A} \cdot B, \quad G = \overline{E + F}$$

$G$ is 1 when both inputs are the same and 0 when they differ. So the whole circuit works as an XNOR (equivalence) gate: $G = A\cdot B + \overline{A}\cdot\overline{B}$.

(i) Symbol of a p-n junction diode

        p          n
  o-----|>|---------o
  anode (+)    cathode (-)

The arrow points from p (anode, positive end) to n (cathode, negative end). This is the direction of conventional current in forward bias. The bar marks the cathode.

(ii) Completed bridge rectifier

Label the four nodes T (top), B (bottom), L (left) and R (right). The source is connected across T and B, and $R_L$ is connected across R and L. The right node R is the positive output.

   +----------------------o T
   |                    /   \
   |       D3 (L -> T) /     \ D1 (T -> R)
  (~) V_in            /       \
   |               L o         o R ----+---- V_out (+)
   |                  \       /        |
   |       D2 (L -> B) \     / D4 (B -> R)
   |                    \   /         R_L
   +----------------------o B          |
                                       |
                   L o ----------------+   (same node L)

   (arrow "X -> Y" means anode at X, cathode at Y)

• Half cycle with T positive: current T → $D_1$ → R → $R_L$ → L → $D_2$ → B → source.
• Half cycle with B positive: current B → $D_4$ → R → $R_L$ → L → $D_3$ → T → source.

In both half cycles current flows through $R_L$ in the same direction (R to L), giving full wave rectification.

(iii) One diode damaged (open in both directions)

Each half cycle needs a pair of diodes in series ($D_1$, $D_2$ for one half and $D_4$, $D_3$ for the other). If any one diode is open, the path for its half cycle is broken, so no current flows through $R_L$ in that half cycle. The other pair still conducts in the other half cycle. The circuit then works as a half wave rectifier: output appears only in alternate half cycles and the average output is halved.

 V across R_L
 ^    __            __            __
 |   /  \          /  \          /  \
 |  /    \        /    \        /    \
 +-/------\------/------\------/------\------> t
   0      T/2    T            2T
          |<-gap->|  (no output in alternate half cycles)