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I am having a hard time understanding what saturation current (denoted Is in my textbook) actually means for a transistor.

For a diode, it makes perfect sense, it's the approximate maximum reverse current that will go through the diode when it's under reverse bias. (Or at least that's my understanding; correct if I'm wrong)

But what does this mean for a transistor? I see that it appears in equations all over the place but I don't understand what it means intuitively. Maybe it would help to ask if I wanted to measure saturation current of a transistor, what would I do?

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    \$\begingroup\$ Um... I've written about this at least a dozen times here. Let me go track them down..... But it's not complicated. it's the \$y\$-axis intercept of an extrapolated line. In a recent post, I wrote: "The saturation current, \$I_\text{SAT}\$, is best seen as an extrapolated \$y\$-axis intercept. I talk about it here and also here and here." (And that's not all of the places I talk about it.) It impacts \$V_\text{BE}\$. \$\endgroup\$ – jonk Feb 14 at 2:06
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    \$\begingroup\$ It is directly proportional to the active emitter-base junction area, which varies from device to device (of course.) (In a sense it a measure of how many thermally generated electron-hole pairs "wander" around across the BE junction.) \$\endgroup\$ – jonk Feb 14 at 2:30
  • \$\begingroup\$ Are you asking about a BJT or a FET? Because the word "saturation" is used to mean different things for different kinds of transistor. \$\endgroup\$ – The Photon Feb 14 at 5:53
  • \$\begingroup\$ If the question is about BJT's, see Why is the collector current at saturation, less than the collector current in the forward active mode? \$\endgroup\$ – The Photon Feb 14 at 5:55
  • \$\begingroup\$ @ThePhoton It's the \$I_\text{sat}\$ term, aka \$I_s\$ that is mentioned in the question, which mostly gives it away. That, plus the mention of the diode. \$\endgroup\$ – jonk Feb 14 at 6:02
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This is more of an intuitive answer rather than a very technical one.

Let's look at the following circuit. In layman's terms, here a power source of 5V(connected at the top) is trying to establish a current in the circuit. But a battery connected with opposite polarity(with respect to the source voltage) is trying to oppose that current.

enter image description here

Now consider this next circuit. Here the battery connected is of reduced magnitude. Hence the current in the circuit increases.

enter image description here

As you further reduce the battery voltage, the current in the circuit further increases.

enter image description here

The current goes on increasing and reaches the maximum value when the battery voltage becomes 0V.

Now consider this simple NPN Bipolar Junction transistor circuit.

enter image description here

Here voltage from Collector to Emitter is playing the role of the battery which is opposing the current (presently let's not consider what factors are really responsible for controlling this battery Voltage). The maximum current could flow in the previous circuit, when the battery voltage was at a minimum value (0V in that case). Similarly here too, the maximum current will flow through the circuit when this Collector to Emitter voltage (represented by battery) is minimum.

Now, In transistors this Collector to Emitter voltage cannot go to zero but to a certain minimum value. This minimum possible voltage value is generally given in datasheet for a transistor by the name of 'Collector-Emitter Saturation Voltage'. It is typically around 0.2V. The current which would be flowing in the circuit at this minimum Collector-Emitter Voltage is called Saturation Current. It is maximum possible current which can flow in the given circuit and hence the name Saturation Current.

So saturation current depends not only on the power input to the circuit (5V in this case) but also on the magnitude and position of the circuit elements (like Resistor) connected in the circuit. You can always apply some KCL equations and find out the saturation current for your circuit.

But how this 'Collector-Emitter Voltage' changes by changing certain circuit parameters? How can we make this 'Collector-Emitter Voltage' go down to the minimum 'Collector-Emitter Saturation Voltage'? These are the questions which would require further analysis and calculations.

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In practice, a transistor is generally considered "saturated" when increasing or decreasing the base current/gate voltage incrementally will not proportionately affect the collector/drain current. The C-E voltage of a bipolar will usually be about 0.2V under these conditions, or the D-S voltage will be small compared to its limits.

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