Divider Circuits and Kirchoff’s Laws

04/21/23 D.N 2

Divider Circuits and Kirchoff’s Laws

• Voltage divider circuits

• Kirchhoff's Voltage Law (KVL)

• Current divider circuits

• Kirchhoff's Current Law (KCL)

04/21/23 D.N 3

Voltage Divider Circuits

• Let's analyze a simple series circuit, determining the voltage drops across individual resistors:

04/21/23 D.N 4

Voltage Divider Circuits (cont’d)

• From the given values of individual resistances, we can determine a total circuit resistance, knowing that resistances add in series:

04/21/23 D.N 5

Voltage Divider Circuits (cont’d)

• From here, we can use Ohm's Law (I=E/R) to determine the total current, which we know will be the same as each resistor current, currents being equal in all parts of a series circuit:

04/21/23 D.N 6

Voltage Divider Circuits (cont’d)

• Now, knowing that the circuit current is 2 mA, we can use Ohm's Law (E=IR) to calculate voltage across each resistor:

04/21/23 D.N 7

Voltage Divider Circuits (cont’d)

• It should be apparent that the voltage drop across each resistor is proportional to its resistance, given that the current is the same through all resistors. Notice how the voltage across R2 is double that of the voltage across R1, just as the resistance of R2 is double that of R1.

04/21/23 D.N 8

Voltage Divider Circuits (cont’d)

04/21/23 D.N 9

Voltage Divider Circuits (cont’d)

• The ratio of individual resistance to total resistance is the same as the ratio of individual voltage drop to total supply voltage in a voltage divider circuit. This is known as the voltage divider formula, and it is a short-cut method for determining voltage drop in a series circuit without going through the current calculation(s) of Ohm's Law.

04/21/23 D.N 10

Voltage Divider Circuits (cont’d)

• Using this formula, we can re-analyze the example circuit's voltage drops in fewer steps:

04/21/23 D.N 11

Voltage Divider Circuits (cont’d)

• REVIEW: • Series circuits proportion, or divide, the total

supply voltage among individual voltage drops, the proportions being strictly dependent upon resistances: ERn = ETotal (Rn / RTotal)

• A potentiometer is a variable-resistance component with three connection points, frequently used as an adjustable voltage divider.

04/21/23 D.N 12

Kirchoff’s Voltage Law

• Let's take another look at our example series circuit, this time numbering the points in the circuit for voltage reference:

04/21/23 D.N 13

Kirchoff’s Voltage Law (cont’d)

• If we were to connect a voltmeter between points 2 and 1, red test lead to point 2 and black test lead to point 1, the meter would register +45 volts. Typically the "+" sign is not shown, but rather implied, for positive readings in digital meter displays.

04/21/23 D.N 14

Kirchoff’s Voltage Law (cont’d)

• However, for this lesson the polarity of the voltage reading is very important and so we will show positive numbers explicitly:

04/21/23 D.N 15

Kirchoff’s Voltage Law (cont’d)

• When a voltage is specified with a double subscript (the characters "2-1" in the notation "E2-1"), it means the voltage at the first point (2) as measured in reference to the second point (1). A voltage specified as "Ecg" would mean the voltage as indicated by a digital meter with the red test lead on point "c" and the black test lead on point "g": the voltage at "c" in reference to "g".

04/21/23 D.N 16

Kirchoff’s Voltage Law (cont’d)

04/21/23 D.N 17

Kirchoff’s Voltage Law (cont’d)

• If we were to take that same voltmeter and measure the voltage drop across each resistor, stepping around the circuit in a clockwise direction with the red test lead of our meter on the point ahead and the black test lead on the point behind, we would obtain the following readings.

04/21/23 D.N 18

Kirchoff’s Voltage Law (cont’d)• We should already be familiar with the general principle for series

circuits stating that individual voltage drops add up to the total applied voltage, but measuring voltage drops in this manner and paying attention to the polarity (mathematical sign) of the readings reveals another facet of this principle: that the voltages measured as such all add up to zero:

04/21/23 D.N 19

Kirchoff’s Voltage Law (cont’d)

• This principle is known as Kirchhoff's Voltage Law (discovered in 1847 by Gustav R. Kirchhoff, a German physicist), and it can be stated as such: "The algebraic sum of all voltages in a loop

must equal zero"

04/21/23 D.N 20

Kirchoff’s Voltage Law (cont’d)

• By algebraic, we mean accounting for signs (polarities) as well as magnitudes. By loop, we mean any path traced from one point in a circuit around to other points in that circuit, and finally back to the initial point. In the above example the loop was formed by following points in this order: 1-2-3-4-1. It doesn't matter which point we start at or which direction we proceed in tracing the loop; the voltage sum will still equal zero.

04/21/23 D.N 21

Kirchoff’s Voltage Law (cont’d)

• To demonstrate, we can tally up the voltages in loop 3-2-1-4-3 of the same circuit:

04/21/23 D.N 22

Kirchoff’s Voltage Law (cont’d)

• REVIEW:

• Kirchhoff's Voltage Law (KVL): "The algebraic sum of all voltages in a loop must equal zero"

04/21/23 D.N 23

Current Divider Circuits

• Let's analyze a simple parallel circuit, determining the branch currents through individual resistors:

04/21/23 D.N 24

Current Divider Circuits (cont’d)

• Knowing that voltages across all components in a parallel circuit are the same, we can fill in our voltage/current/resistance table with 6 volts across the top row:

04/21/23 D.N 25

Current Divider Circuits (cont’d)

• Using Ohm's Law (I=E/R) we can calculate each branch current:

04/21/23 D.N 26

Current Divider Circuits (cont’d)

• Knowing that branch currents add up in parallel circuits to equal the total current, we can arrive at total current by summing 6 mA, 2 mA, and 3 mA:

04/21/23 D.N 27

Current Divider Circuits (cont’d)

• The final step, of course, is to figure total resistance. This can be done with Ohm's Law (R=E/I) in the "total" column, or with the parallel resistance formula from individual resistances. Either way, we'll get the same answer:

04/21/23 D.N 28

Current Divider Circuits (cont’d)

• If we were to change the supply voltage of this circuit, we find that (surprise!) these proportional ratios do not change:

04/21/23 D.N 29

Current Divider Circuits (cont’d)• Also reminiscent of voltage dividers is the fact that branch currents

are fixed proportions of the total current. Despite the fourfold increase in supply voltage, the ratio between any branch current and the total current remains unchanged:

04/21/23 D.N 30

Current Divider Circuits (cont’d)

• For this reason a parallel circuit is often called a current divider for its ability to proportion -- or divide -- the total current into fractional parts. With a little bit of algebra, we can derive a formula for determining parallel resistor current given nothing more than total current, individual resistance, and total resistance:

04/21/23 D.N 31

Current Divider Circuits (cont’d)

04/21/23 D.N 32

Current Divider Circuits (cont’d)

• The ratio of total resistance to individual resistance is the same ratio as individual (branch) current to total current. This is known as the current divider formula, and it is a short-cut method for determining branch currents in a parallel circuit when the total current is known.

04/21/23 D.N 33

Current Divider Circuits (cont’d)

• Using the original parallel circuit as an example, we can re-calculate the branch currents using this formula, if we start by knowing the total current and total resistance:

04/21/23 D.N 34

Current Divider Circuits (cont’d)

• If you take the time to compare the two divider formulae, you'll see that they are remarkably similar. Notice, however, that the ratio in the voltage divider formula is Rn (individual resistance) divided by RTotal, and how the ratio in the current divider formula is RTotal divided by Rn:

04/21/23 D.N 35

Current Divider Circuits (cont’d)

• REVIEW:

• Parallel circuits proportion, or "divide," the total circuit current among individual branch currents, the proportions being strictly dependent upon resistances: In = ITotal (RTotal / Rn)

04/21/23 D.N 36

Kirchhoff's Current Law (KCL)

• Let's take a closer look at that last parallel example circuit:

04/21/23 D.N 37

Kirchhoff's Current Law (KCL) (cont’d)

• Taking a look at the currents at each wire junction point (node) in the circuit, we should be able to see something else:

04/21/23 D.N 38

Kirchhoff's Current Law (KCL) (cont’d)

• If we were to take a closer look at one particular "tee" node, such as node 3, we see that the current entering the node is equal in magnitude to the current exiting the node:

04/21/23 D.N 39

Kirchhoff's Current Law (KCL) (cont’d)

• This holds true for any node ("fitting"), no matter how many flows are entering or exiting. Mathematically, we can express this general relationship as such:

Iexiting=Ientering

04/21/23 D.N 40

Kirchhoff's Current Law (KCL) (cont’d)

• Mr. Kirchhoff decided to express it in a slightly different form (though mathematically equivalent), calling it Kirchhoff's Current Law (KCL):

Ientering + (-Iexiting) = 0

04/21/23 D.N 41

Kirchhoff's Current Law (KCL) (cont’d)

• REVIEW:

• Kirchhoff's Current Law (KCL): "The algebraic sum of all currents entering and exiting a node must equal zero"