what is power dissipation in digital electronics formula , Power Dissipation Capability ?

**Performance Quantities of Power Amplifiers**

As mentioned previously, the prime objective for a power amplifier is to obtain maximum output power. since, a transistor, like any other electronic devices has voltage, current and power dissipation limits therefore, the criteria for a power amplifier are collector efficiency, distortion and power dissipation capability.

**(1) Collector Efficiency**

The main criterion for a power amplifier is not the power gain rather it is the maximum AC output power. now an amplifier converts DC power from supply into AC output power. therefore, the ability of a power amplifier to convert DC power from supply into AC output power is a measure of its effectiveness. this is known as collector effcieney and may be de defined as

The ratio of AC output power to the zero signal power (i.e., DC power) supplied by the battery of a power amplifier is known as collector efficiency.

Collector efficiency means as to how well an amplifier converted DC power from the battery into AC output power. for instance, if the DC power supplied by the battery is 10 W and AC output power is 2 W. then collector efficiency is 20% the greater the collector efficiency, the larger is the AC power output. it is obvious that for power amplifiers. maximum collector efficiency is the desired goal.

**(2) Distortion **

The change of output wave shape from the input wave shape of an amplifier is known as distortion.

A transistor like other electronic devices, is essentially a non-linear device. therefore, whenver a signal is applied to the input of the transistor the output signal is not exactly like the input signal i.e., distortion occurs. distortion is not a problem for small signals (i.e., voltage amplifiers), since transistor is a linear device for small variations about the operating point. however, a power amplifier handles large signals and therefore , the problem of distortion immediately arises. for the comparison of two power amplifiers, the one which amplifiers.

**(3) Power Dissipation Capability**

The ability of a power transistor to dissipate heat is known as power dissipation capability .

As started before, a power transistor handles large currents and heats up during operation. as any temperature change influences the operation of transistor, therefore, the transistor must dissipate this heat to its surroundings.

To achieve this, generally a heat sink (a metal case) is attached to a power transistor case. the increased surface area allows heat to escape easily and keeps the case temperature of the transistor within permissible limits.

**Classification of Power Amplifiers**

Thransistor power amplifiers handle large signals. many of them are driven so hard by the input large signal that collector current is either cut-off or is in the saturation region during a large portion of the input cycle.therefore, such amplifiers are generlly classified according to their mode of operatio, i.e., the portion of the input cycle during which the collector current is expectes to flow. on this baiss, they are classified as

(1) Class A power amplifier

(2) Class B power amplifier

(3) Class C power amplifier

**(1) Class A Power Amplifier**

If the collector current flows at all times during the full cycle the signal, the power amplifier is known as class A power amplifier.

Obviously for this to happen, the power amplifier must be biased in such a way that no part of the signal is cut-off. fig 2 (a) shows the circuit of a class A power amplifier. note that collector has a transformer as the load which is most common for all classes of power amplifiers. the use of transformer permits impedance matching, resulting in the transference of maximum power to the load e.g., loudspeaker.

Fig. 2 (b) shows the class A operation in terms of AC load line. the operating point Q is so selected that collector current flows at all times throughout the full cycle of the applied signal. as the output wave shape is exactly similar to the input wave shape therefore, such amplifiers have least distortion. however. they have the disadvantages of low power output and low collector efficiency (about 35%).

**(2) Class B Power Amplifier**

If the collector current flows only during the positive half-cycle of the input signal, it is called a class B power amplifier.

In class B operation, the transistor bias is so adjusted that zero signal, collector current is zero, i.e., no biasing circuit is needed at all. during the positive half-cycle of the signal. the input circuit is forward biased and hence collector current flows. however, during the **negative half-cycle of the signal, the input circuit is reverse biased and no collector current **flows. fig. 3 shows the class B operation in terms of AC load line. obviously, the operation point Q shall be located at collector cut-off voltage. it is easy to see that output from a class B amplifier is amplified half-wave rectification. in class B amplifier the negative half-cycle of the signal is out-off and hence, a severe distortion occurs. however, class B amplifiers provide higher power output and collector efficiency (50-60%) such amplifiers are mostly used for power amplification in push-pull arrangement. in used an arrangement, two transistors are used in class B operation. one transistor amplifies the positive half-cycle of the signal while the other amplifiers negative half-cycle.

**(3) Class C Power Amplifier**

If the collector current flows for less than half-cycle of the input signal, it is called class C power amplifier.

In class C amplifier, the base is given some negative bias so that collector current does not flow just when the positive half-cycle of signal starts. such amplifiers are never used for power amplification. however, they are used as tuned amplifies i.e., to amplify a narrow band of frequencies near the resonant frequency.

**Expression for Collector Efficiency**

For comparing power amplifiers, collector efficiency is the main criterion. the greater the collector efficiency, the better is power amplifier.

Now, collector efficiency n = AC power output / DC power input

= P_{AC} /P_{DC}

_{ }P_{DC} = V_{CC}I_{C}

P_{AC} = V_{CE}I_{C}

Where V_{CE} is the rms value of signal output voltage and I_{C} is the rms value of output signal current. in terms of peak-to-peak values (which are often convenient values in load-line work) the AC power output can be expressed as

P_{AC} = [(0.5 x 0.707)V_{CE (P-P)}] [(0.5 x 0.707) I_{C} _{(P-P)}]

= V_{CE(P-P)} x I_{C(P-P)} / 8

Collector n = V_{CE(P-P)} x I_{C} _{(P-P)} / 8V_{CC}I_{C}

**Maximum Collector Efficiency of Series-Fed Class A Amplifier**

Fig. 4 (a) shows a series-fed class A amplifier. this circuit is seldom used for power amplification due to its poor collector efficiency. nevertheless, it will help the reader to understad the class A operation. the DC load line of circuit is shown in fig. 4(a) when an AC signal is applied to the amplifier, the output current and voltage will vary about the operation ponit Q. in order to achieve the maximum symmetrical swing of current and voltage (to achieve maximum output power) the Q point should be located at the centre of DC load line. in that case, operating point is I_{C} = V_{CC} /2R_{C} and V_{CE} = V_{CC} /2

Maximum V_{CE(P-P)} = V_{CC}

Maximum I_{C(P-P) }= V_{CC} / R_{C}

Maximum AC output power,

P_{O(max) } = v_{ce (p-p) }x I_{C} _{(P-P) }/ 8

= V_{CC} x V_{CC} /R_{C} /8 = V_{CC}^{2} /8R_{C}

DC Power supplied,

P_{DC} = V_{CC}I_{C} = V_{CC} (V_{CC}/2R_{C}) = V^{2}_{CC} /2R_{C}

Maximum collector efficiency n = P_{O(max)} /P_{DC} x 100

V^{2}_{CC}/8R_{C} / V^{2}_{CC}/2R_{C} x 100 = 25%

Thus, the maximum collector efficiency of a class A series fed amplifier is 25% in actual practice, the collector efficiency is far less than this value.

**Maximum Collector Efficiency of Transformer Coupled Class A Power Amplifier**

In calss A power amplifier, the load can be either connected directy in the collector or it can be transformer coupled. the latter method is often preferred for two main reasons. first, transfoemer coupling permits impedance matching and secondly, it keeps the DC power loss small because of the small resistance of the transformer primary winding.

**Boolean Algebra**

In boolean algebra the binary digits are utilized to represent the two levels that occurs within digital logic circuits. A binary 1 represent a high level and a binary 0 will represent a low level in boolean equations and in terms of voltages there are two types of logic systems

(a) Positve level logic system

(b) Negative level logic system

**(a) Positive Level Logic System**

Out of the given two voltage levels, the more positive value is assumed as logic 1 and the more negative level as logic 0.

**For example**

**Logic 0 Logic 1**

0V 5V

-2V +3V

-7V -2V

+2V +7V

**(b) Negative Level Logic System**

Out of the given two voltage levels, the more negative value is assumed as logic 1 and the more positive level as 0.

**For example**

** Logic 1 Logic 0**

0V 5V

-2V +3V

-7V -2V

+2V +7V

**Boolean Addition**

The basic rules for boolean addition are as follows

0 + 0 = 0

0 + 1 =1

1 + 0 = 1

1 + 1 = 1

Boolean addition is same as the OR operation. it differs from binary addition.

**Boolean Multiplication**

Multiplication in boolean algebra follows the same basic rules govering binary multiplication.

0.0 = 0

0.1 = 0

1.0 = 0

1.1 = 1

Boolean multiplication is same as the AND operation.

**Logic Expressions **

**Not Operation**

The operation of an inverter (NOT circuit) can be represented with symbol as follows

This expressin states that the output is the complement of the input

A (input)

A (output)

**AND Operation**

The operation of two input AND gate can be expressed in equation form as follows

**OR Operation**

The operation of a two input OR gate can be expressed in equation form as follows

**Rules and Laws Boolean Algebra**

Three of the basic laws of the boolean algebra are same as in ordinary algebra the commutative laws, the associative laws and the distributive law.

**Commutative Laws**

The commutative law for addition of two variable

A + B = B + A

For multiplication

AB = BA

**Associative laws **

**For addition **A + (B + C) = (A + B) + C

**For multiplication **A (BC) = (AB) C

**Distributive law** The distributive lew is written for three variables as follows

A (B + C) = AB + AC

**Rules for Boolean Algebra**

Several basic rules that are useful in manipulating and simplifying boolean algebra expressions

- A + 0 = A
- A + 1 = 1
- A . 0 = 0
- A . 1 = A
- A + A = A
- A + A = 1
- A . A = A
- A . A = 0
- A = A
- A + AB = A
- A + AB = A + B
- (A + B) (A + C) = A + AB

**De Morgan’s Theorems**

De Morgan’s theorams that are important part of algebra are stated in equation form as :

- AB = A + B
- A + B = A . B

**Example 8. ** Apply de morgan’s theorems to the following expression :

X = A + B + C + DE

**Sol.** X = A + B + C + DE

= A . B . C + DE [A + B = A . B]

= A . B . C + (D + E) [AB = A + B]

= A . B . C + (D + E ) [D = D]

= A . B . C + (D + E)

**Duality Theorem **

relation called its dual. the steps used are

- Changing each OR sign to an AND sign.
- Changing each AND sign to an OR sign.
**Complementing all 0’s and 1’s.**

**Example 9. **Find the dual of expression A (B + C) + D

**Sol.** Given expression is

A . (B + C) + D

OR AND AND

By replacing AND – OR

and OR – AND, we get

= [A + B .C] .D

**Boolean Expressions for Gate Networks**

The form of a given boolean expression indicatea the type of gate network, it describes.

There are certain forms of boolean expressions that are more commonly used than others; the most important of these are the sum of products and product of sums form.

**Sum of Products Form**

A sum of products expression is two or more AND functions ORed together.

**For example **AB + BC

ABC + DEF

ABC + DEF + FGH + AFG

An important characteristics of the sum of products form is that the corresponding implementation is always a two level gate network.

**Example 10.** Implement the expression ABC + DE + FGHI with logic gates.

**Product of Sums Form**

The product of sume form can be thought of as the dual of the sum of products. it is the AND of two or more OR functions.

e.g., (A + B) (B + C)

(A + B + C) (E + F + G)

(A + B + C) (E + F ) (A + E)

**Example 11. **Construct the following function with logic gates.

(A + B) (C + D + E) (D + E + F + G)

**Minterm and Maxterm**

If each term in the sum of products form contains all the variables (literals), then the expression is known as standard sum of products form or canonical sum of products form and each individual term in standard sum of products form is called as minterm.

**For example** Y (A, B, C,) = AB + ABC + B

To get the standard SOP form, multiply the fist term by (C + C) and the third term (A + A) (C + C).

Y (A, B, C)= AB (C + C) + ABC + (A + A) B (C + C)

= ABC + ABC + ABC + ABC + ABC + ABC + ABC

= ABC + ABC + ABC + ABC + ABC + ABC

The total minterms are

ABC (111) = m_{7} ABC (101) m_{5} ABC (100) = m_{4}

ABC (110) = m_{6} ABC (001) = m_{1} ABC (000) = m_{0}

It can be written as a sum of minterms as follows :

Y (A, B, C) = m (0, 4, 5, 6, 7)

If each term of product of sums contains all the variables, then the expression is known as standard product of sums form or canonical product of sums form and each individual term in the standard product of sume form is called as maximum.

**For example ** Y (A, B, C) = (A + B) (A + B + C)

The first term of above equation has two variabies, variable C is missing and hence, to get the standard POS, the term (c) in the first term must be ORed.

Y (A, B, C) = (A + B + CC) (A + B + C)

= (A + B + C) (A + B + C) (A + B + C)

The total maxterms are three, which are

(A + B + C) (010) = M_{2}

(A + B + C) (011) M_{3}

(A + B + C) (001) M_{1}

It can be written as a product of maxterm :

Y (A, B, C) = M (1, 2, 3)

**Example 12.** Obtain SOP and POS expressions for the truth table of table given below.

**Sol.** From the given table, chosing minterms corresponding to Y = 1 in the truth table, we can write

Y = ABC + ABC + ABC + ABC

In terms of maxterms which correspond to Y = 0, we can write

Y = (A + B + C) (A + B +C) (A + B + C) (A + B + C)

It is observed that Y + Y = 1

So, while minterms which correspond to 1 outputs constitute SOP, the complementary minterms which corresponds to 0 outputs.

Y = ABC + ABC + ABC + ABC

Similarly, for maxterms the output corresponding to outputs would constitute y.

Y = (A + B + C) (A + B + C) (A + B + C) (A + B + C)

As per SOP expression Y of the truth table the combinational circuit is drawn ahead