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Introduction to Engineering14 chapters | 122 lessons

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Instructor:
*Kip Ingram*

Kip holds a PhD in Engineering from The University of Texas at Austin and was an occasional substitute lecturer in engineering classes at that institution.

In this lesson, you will learn that a voltage divider is a set of series-connected resistors with an input voltage applied to the entire string. Each resistor develops an output voltage across its terminals that is a fraction of the input voltage. These fractions depend only on the resistance values.

Meet Alice and Bob, professional basket weavers. Alice has been at it a long time and can weave three baskets an hour, while Bob is less experienced and can only weave one per hour. The market for baskets is volatile and uncertain, so our intrepid weavers are busier some weeks than others. Company policy, however, dictates that Alice and Bob have equal opportunity, so they both work the same number of hours each and every week.

Knowing this, we can show that Alice will always produce 75% of their combined weekly output:

Ratio = Output(Alice) / ( Output(Alice) + Output(Bob) )

= Hours*Rate(Alice) / ( Hours*Rate(Alice) + Hours*Rate(Bob) )

= Rate(Alice) / ( Rate(Alice) + Rate(Bob) )

= 3 / (3 + 1) = 0.75

We see that this result depends only on each worker's individual production rate and does not depend on any other 'market' factors. The 'equal hours' assumption is key - if this assumption didn't hold we would have to use different values for the various Hours factors in the second line above, and they wouldn't cancel out to get us to step 3.

It turns out that voltage dividers operate according to very similar principals. Step one to understanding voltage dividers is understanding resistors.

A **resistor** is an electrical component that maintains a fixed ratio between the voltage V across its terminals and the current I that flows through it. You can arrange to control either voltage or current, but the resistor then insists on what the other one must be.

Mathematically, resistors obey **Ohm's Law**:

V = I * R

Where V is the terminal voltage (in volts), I is the current (in amperes), and R is the resistance (in ohms). If we divide both sides of this equation by I:

V/I = R

We see that R is in fact the ratio of voltage to current, and that one ohm is equivalent to one volt per ampere. For each ampere of current, an R-ohm resistor develops R volts across its terminals.

That matches up nicely with our basket weaving example if we choose to let volts represent baskets and amperes represent hours. Then, resistance has to represent production rate, since its units, volts/ampere, are equivalent to baskets/hour. So, in this equivalence scheme, an R-ohm resistor is equivalent to a worker who produces R baskets/hour. We have a 'model' of the basket weaving work:

Baskets --> volts

Hours --> amperes

Baskets/hour --> volts/ampere

Production rate --> resistance

We can represent Alice (who weaves three baskets an hour) with a three-ohm resistor Ra, and Bob (who weaves one basket an hour) with a one-ohm resistor Rb. Alice and Bob work equal hours, so Ra and Rb must carry equal current. We achieve this by wiring them in series. We can model a production run (say 100 baskets) by applying an appropriate voltage to the series combination.

Analysis of this circuit shows us that it takes Alice and Bob 25 hours to fulfill this order, and that Alice makes 75 baskets and Bob makes 25 baskets. Because Alice always makes 75% of any amount of output, while Bob always makes 25%, the two of them working together function as a 'basket divider.' So we shouldn't be surprised that the series resistors used to model them function as a voltage divider.

A **voltage divider** is a set of resistors (usually two) connected in series. When a voltage is applied to the series string, the resistors divide the voltage, such the voltage across each resistor is a well-defined fraction of the input voltage:

Vi = Vin * ( Ri / Rtotal )

Where Rtotal is the sum of all of the resistances in the string: Rtotal = R1 + R2 + ... + Rn.

In general, the string can include any number of resistors, but the basic analysis is similar to that shown in the basket model illustration:

I = V / R = Vin / Rtotal

Vi = I * Ri = (Vin / Rtotal) * Ri = Vin * ( Ri / Rtotal )

Voltage dividers are often used to create bias in electronic circuits. **Bias** refers to establishing average operating conditions. For example, in a transistor amplifier, it's often desirable to have the input terminal 'rest' halfway between the power supply rails, so that an active input signal can swing it in either direction.

Can you apply the lessons learned so far to see why the voltage on pin 1 of the transistor will be Vcc / 2?

**Voltage dividers** use series-connected resistors to produce lower voltages that are proportional to the input voltage. The voltage ratios created in this manner depend only on the value of the resistances comprising the divider. Voltage dividers can be analyzed using Ohm's law multiple times, first to calculate the (shared) current flowing through the divider resistors, and then again for each resistor to calculate the voltage developed across its terminals. Voltage dividers have many applications in electronics circuits, with one of the most common being **bias** circuits used to establish the average voltages that appear in various circuit locations.

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15 in chapter 9 of the course:

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Introduction to Engineering14 chapters | 122 lessons

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