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*Fixed Bias Circuit:*

The fixed-bias circuit gives relatively straightforward and simple introduction to transistor dc bias analysis. Although network uses npn transistor, equations and calculations apply evenly well to pnp transistor configuration just by changing all current direction and voltage polarities. Current direction of is really current directions, and voltages are stated by standard double-subscript notation. For dc analysis network can be isolated from indicated ac level by replacing capacitor with the open circuit equivalent. Additionally, dc supply V_{cc} can be separated in two supplies (for analysis purpose only) as shown in figure to allow a separation of input and output circuits. It also decreases linkage between two to base current I_{B}. Separation is surely valid; V_{CC} is connected directly to R_{B} and R_{C}.

**Forward Bias Base Emitter**:

Consider first base-emitter circuit loop. Writing Kirchhoff's voltage equation in clockwise direction for loop, we get

+V_{CC} - I_{B}R_{B} - V_{BE} = 0

Note polarity of voltage drop across R_{B} as established by indicated direction of I_{B}. Solving equation for current I_{B} will result in following

I_{B}= (V_{CC}- V_{BE})/R_{B}

As supply voltage V_{CC} and base-emitter voltage V_{BE} are constant, selection of base resistor, R_{B}, sets level of base current for operating point.

**Collector-Emitter Loop:**

Collector-emitter section of network with indicated direction of current I_{C} and resulting polarity across R_{C}. Magnitude of collector current is related directly to I_{B} through

I_{C} = βI_{B}

It is interesting to note that as base current is controlled by level of R_{B} and I_{C} is related to I_{B} by constant β, magnitude of I_{C} is not function of resistance R_{C}. Change R_{C} to any level and it will not affect level of I_{B} or I_{C} as long as we remain in active region of device. Though, level of R_{C} will determine magnitude of V_{CE} that is significant parameter. Applying Kirchhoff's voltage law in clockwise direction around indicated closed loop will result in following:

V_{CE} + I_{C}R_{C} - V_{CC} = 0 and V_{CE }= V_{CC} - I_{C}R_{C}

It defines that voltage across collector-emitter region of the transistor in fixed-bias configuration is supply voltage less the drop across RC.

V_{CE} = V_{C }- V_{E}

Where V_{CE }is voltage from collector to emitter and V_{C }and V_{E} are voltages from collector and emitter to ground respectively. But in this situation, as V_{E} = 0V, we have

V_{CE}= V_{C}

In addition, as V_{BE}= V_{B}- V_{E }and V_{E} = 0V, then V_{BE}= V_{B}

*Emitter-Stabilized Bias Circuit:*

The dc bias network contains emitter resistor to enhance stability level over that of fixed-bias configuration. Analysis will be performed by first examining base-emitter loop and then using result to investigate collector-emitter loop.

Base-emitter loop of network can be redrawn as shown. Writing Kirchhoff's voltage law around indicated loop in clockwise direction will result in given equations:

+V_{CC} - I_{B}R_{B }- V_{BE} - I_{E}R_{E} = 0

Recall that I_{E}=(β +1)I_{B}

Substituting equation will result in

V_{CC}- I_{B}R_{B}- V_{BE}-(β+1)I_{B}R_{E}=0

Expanding and multiplying through by (-1) we have

I_{B}R_{B}+β+1R_{E}-V_{CC}+ V_{BE}=0

And solving for gives

I_{B}= V_{CC}- V_{BE}R_{B}+(β+1)R_{E}

Collector-Emitter Loop:

Writing Kirchhoff's voltage law for indicated loop in clockwise direction will result in

+I_{E}R_{E} + V_{CE} + I_{C}R_{C} - V_{CC} = 0

Substituting I_{E} ≈ I_{C} and regrouping terms provides

V_{CE} - V_{CC} + I_{C}(R_{C} +R_{E}) = 0 and

V_{CE} = V_{CC} - I_{C}(R_{C} + R_{E})

Single subscript voltage V_{E} is voltage from emitter to ground and is determined by

V_{E} = I_{E}R_{E}

While voltage from collector to ground can be determined from

V_{CE} = V_{C} - V_{E} and V_{C }= V_{CE} + V_{E} or V_{C} = V_{CC} - I_{C}R_{C}

The voltage at the base with respect to ground can be determined from

V_{B }= V_{CC} - I_{B}R_{B}

V_{B} = V_{BE} + V_{E}

*Voltage Divider Bias:*

The bias current I_{CQ} and voltage V_{CEQ} were function of current Gain (β) of te transistor. Though, since β is temperature sensitive, especially for silicon transistors, and the actual value of beta is usually not well defined, it will be desirable to develop a bias circuit that is less dependent, or in fact, independent of the transistor beta. If analyzed on the exact basis sensitivity to changes in beta is quite small. If circuit parameters are properly chosen, resulting level of I_{CQ} and V_{CEQ} can be almost totally independent of beta. A Q-point is stated by fixed level of I_{CQ }and V_{CEQ}. The level I_{EQ} will change with change in beta, but operating point on characteristic defined by I_{CQ }and V_{CEQ} can remain fixed if proper circuit parameters are used above, there are two methods which can be applied to examine voltage-divider configuration.

**Exact Analysis: **

The Thevenin equivalent network for network to left of base terminal can be found in given manner:

R_{Th}: Voltage source is replaced by the short-circuit equivalent.

R_{Th} = R_{1}||R_{2}

E_{Th}: Voltage source V_{CC }is returned to network and open- circuit Thevenin voltage determined as follows: Applying voltage-divider rule:

E_{Th} = V_{R2} = R_{2}V_{CC}/(R_{1} + R_{2})

Thevenin network is redrawn, and I_{BQ} can be determined by first applying Kirchhoff's voltage law in clockwise direction for loop indicated:

E_{Th} - I_{B}R_{Th} - V_{BE} - I_{E}R_{E} = 0

Substituting I_{E} = (β + 1)I_{B} and solving for I_{B} yields

I_{B} = (E_{Th} - V_{BE})/(R_{Th} + (β + 1)R_{E})

The numerator is again a difference of two voltage levels and denominator is base resistance plus the emitter resistor reflected by (β + 1) certainly very similar to equation in base-emitter loop previously treated.

Once I_{B} is known, remaining quantities of network can be found in same manner as developed for emitter-bias configuration. That is,

V_{CE} = V_{CC} - I_{C}(R_{C} + R_{E})

**Approximate Analysis:**

The input section of voltage-divider configuration can be represented by network of resistance R_{i} is equivalent resistance between base and ground for transistor with the emitter resistor R_{E}. Recall that R_{i} = (β + 1)R_{E}. If R_{i} is much larger than resistance R_{2}, current I_{B} will be much smaller than I_{2} and I_{2} will be approximately equal to I_{1}.

Voltage across R_{2} that is actually base voltage can be determined using voltage-divider rule (therefore name for configuration). That is:

V_{E} = R_{2}V_{CC}/(R_{1} + R_{2})

Since R_{1} = (β + 1)R_{E} ≈ βR_{E} the condition which will define whether approximate approach can be applied will be following:

βR_{E} ≥ 10R_{2}

In other words, if β times the value of R_{E} is at least 10 times value of R_{2}, the approximate approach can be applied with high degree of accuracy.

Once VB is determined, level of V_{E} can be calculated from

V_{E} = V_{B} - V_{BE}

And emitter current can be determined from:

I_{E} = V_{E}/R_{E}

And

I_{CQ} ≈ I_{E}

Collector-to-emitter voltage is determined by

V_{CE} = V_{CC} - I_{C}R_{C} - I_{E}R_{E}

But since I_{E} ≈ I_{C}

V_{CEQ} = V_{CC} - I_{C}(R_{C} + R_{E)}

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