#### Theory of Static Characteristics II and Base Overdrive Factor

Simple Transistor Inverter:

Figure: Simple Bipolar Transistor Inverter

With Vi = VL = 0V; IB = 0, IC = 0; Vo = VCC = VH

Input LO          Output HI            logic inverter action

Beneath such conditions, the transistor is cut-off or just “OFF”. This is equal to it acting similar to an open switch as shown in figure below, where the state of switch is controlled by the input to the base. As switch does not permit the collector current to flow, then the output voltage is pulled up to supply the rail by collector resistor RC. In actuality, the transistor is not an ideal switch and some leakage current flows via it making it look like a very big resistance >> 100kΩ however which does not noticeably lower the HI output voltage.

Figure: The Transistor Inverter in OFF Switch Mode

On other hand,

Vi = VH = VCC, IB = (VCC - VBE)/RB, IC = (VCC - VCESAT)/RC = ICMAX and VO = VCESAT = VL

Input HI               Output LO    :    logic inverter action

Under such conditions, with IB > IC max/βF, the transistor is driven to saturation and is termed to as “ON”. This is equal to it acting similar to a closed switch as shown in figure below. In this case, the switch permits maximum collector current to flow and hence all the supply voltage is dropped across the resistor, RC, and the output voltage drops close to zero. In practice, there will be certain small voltage drop among the emitter and collector of transistor. This is generally less than 0.1V however at worst case is taken as 0.2V and is termed as VCE sat.

Figure: The Transistor Inverter in ON Switch Mode

Note that, as VCE sat is close to zero in the saturation mode of operation then the power dissipated by transistor, that is given as P = VCE x IC, is much small. The vital thing is to make sure that adequate base current is provided with the input voltage HI for assurance and sustain the transistor operating in saturation region.

Base Overdrive Factor:

Base overdrive factor is the measure of how far into saturation a transistor is driven that is, how much the base current is above that which is essential to bring the transistor to the edge of saturation (that is, the boundary between saturation and forward active region).

Base Overdrive Factor, σu = (Actual base current)/(Base current to reach edge of saturation)

The base current needed to reach the edge of saturation is just the base current needed to initially bring the collector current to its maximum value. This is ICMAXF.

Then,

σu = IB/(IC maxF)  = [(Vcc - VBESAT)/RB]/[(VCC - VCESAT)/βFRC]

If VCC >> VBESAT, VCESAT then this approximates to:

σu = (VCC/RB)/(VCCFRC) = βFRC/RB

Usually, a σu of 5 is employed to permit for temperature variations in βF and as well the fact that βF itself is decreased whenever a transistor is operating in the saturation region. This too permits for manufacturing variations in βF that can be as much as 3:1 for the discrete transistor. The degree of overdrive as well becomes decreased whenever a load is joined to the circuit.

Base Charge in Saturation:

Since the base current of transistor is raised from zero, the collector current increases proportionately in the forward active region. The surplus minority charge concentration present in the base region as well rises proportionately. Whenever the transistor enters the saturation region, though, the collector current remains constant at its maximum value, ICMAX, however the minority charge concentration continues to raise proportionately as the base current is further raised and the transistor is overdriven well into saturation region as shown in figure below. The surplus minority charge stored in the base increases as the base current is raised in the forward active region. Slope of this profile reaches an utmost at the edge of saturation. On being overdriven, the charge continues to increase in the base however the slope of the profile remains constant at the value reached at the edge of saturation as can be seen in figure below. It is noted that the slope of minority carrier profile which finds out the value of diffusion current.

Figure: Increase in base charge with Transistor Overdriven

Figure: Charge Profile in the base region with Transistor in Saturation

Physical Mechanism in Saturation:

In forward active mode, most of the electrons injected from the emitter to the base make their way to collector. With a reasonable reverse bias on base-collector junction, there is a substantial voltage fall across the transistor from collector to base.

Since the base current is raised, the collector current as well increases and with the load resistor RC, in the collector circuit, the voltage drop across transistor VCE, drops. Finally, with further raise in the base current, VCE drops to permit the base-collector junction to become forward biased. This states the onset of saturation of transistor when both junctions become forward biased (figure is as shown below).

Whenever the base collector junction becomes forward biased, it starts to conduct as well. In this case, holes are injected from the base into collector and as well electrons are injected from collector to the base. The holes crossing from base into the collector region comprise the primary component of the base overdrive current. On reaching the collector, such holes can rejoin with electrons reaching the collector region from emitter. This as well offsets any tendency for the collector current to mount with rising bias on the base-emitter junction. This condition leads to a significant increase in the minority carrier concentration in the base as seen in the charge profiles in figure below. The base can be considered in saturation as containing a forward component of minority carrier charge flow and a reverse component of minority carrier charge flow due to base overdrive.

Figure: Transistor Physical Operation in Saturation

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