Explain mathematically how a voltage that is applied to resistors in series is distributed among the resistors.
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- Elwin Esmond Young
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2 Objective of Lecture Explain mathematically how a voltage that is applied to resistors in series is distributed among the resistors. Chapter.5 in Fundamentals of Electric Circuits Chapter 5.7 Electric Circuit Fundamentals Explain mathematically how a current that enters the a node shared by resistors in parallel is distributed among the resistors. Chapter.6 in Fundamentals of Electric Circuits Chapter 6.7 in Electric Circuit Fundamentals Work through examples include a series-parallel resistor network (Example 4). Chapter 7. in Fundamentals of Electric Circuits
3 oltage Dividers esistors in series share the same current in
4 oltage Dividers in esistors in series share the same current - From Kirchoff s oltage Law and Ohm s Law : 0 in
5 oltage Dividers esistors in series share the same current From Kirchoff s oltage Law and Ohm s Law : in in in 0 - in
6 oltage Division The voltage associated with one resistor n in a chain of multiple resistors in series is: or where total is the total of the voltages applied across the resistors. total eq n n total S s s n n
7 oltage Division The percentage of the total voltage associated with a particular resistor is equal to the percentage that that resistor contributed to the equivalent resistance, eq. The largest value resistor has the largest voltage.
8 Example Find the, the voltage across, and, the voltage across. -
9 Example oltage across is: total k k 4k0 sin77t 8.57 sin77t oltage across is: total 4k k 4k0 sin77t.4 sin77t - Check: should equal total 8.57 sin(77t) =.4 sin(77t) = 0 sin(77t)
10 Example Find the voltages listed in the circuit to the right
11 Example (con t) eq eq / / / Check: = -
12 Symbol for Parallel esistors To make writing equations simpler, we use a symbol to indicate that a certain set of resistors are in parallel. Here, we would write to show that is in parallel with and. This also means that we should use the equation for equivalent resistance if this symbol is included in a mathematical equation.
13 Current Division All resistors in parallel share the same voltage in
14 Current Division All resistors in parallel share the same voltage From Kirchoff s Current Law and Ohm s Law : 0 in in in in in
15 Current Division All resistors in parallel share the same voltage in in in in
16 Current Division in Alternatively, you can reduce the number of resistors in parallel from to using an equivalent resistor. f you want to solve for current, then find an equivalent resistor for in parallel with.
17 Current Division in where eq and eq eq in
18 Current Division The current associated with one resistor in parallel with one other resistor is: total The current associated with one resistor m in parallel with two or more resistors is: m eq m total where total is the total of the currents entering the node shared by the resistors in parallel.
19 Current Division The largest value resistor has the smallest amount of current flowing through it.
20 Example Find currents,, and in the circuit to the right.
21 Example (con t) eq / 004A.8A 09 / 4004A.09A 09 / 6004A 0.77A Check: = in
22 Example 4 The circuit to the right has a series and parallel combination of resistors plus two voltage sources. Find and p Find,, and p
23 Example 4 (con t) First, calculate the total voltage applied to the network of resistors. This is the addition of two voltage sources in series. total 0.5 sin(0t) total p
24 Example 4 (con t) Second, calculate the equivalent resistor that can be used to replace the parallel combination of and. eq eq eq total p
25 Example 4 (con t) To calculate the value for, replace the series combination of and eq with another equivalent resistor. eq eq total eq eq 80
26 Example 4 (con t) total ) sin( ) sin( ) sin(0 0.5 t ma ma t t eq total
27 Example 4 (con t) To calculate, use one of the previous simplified circuits where is in series with eq. or eq 0.74 total 0.57 sin(0t) total p
28 Example 4 (con t) To calculate p: or or p p p p eq total eq eq 0.87 total 0.4 sin(0t) Note: rounding errors can occur. t is best to carry the calculations out to 5 or 6 significant figures and then reduce this to significant figures when writing the final answer. total p
29 Example 4 (con t) Finally, use the original circuit to find and. or eq 0.74mA 0.57mAsin(0t) p
30 Example 4 (con t) Lastly, the calculation for. or or eq.86ma.4masin(0t) p
31 Summary The equations used to calculate the voltage across a specific resistor n in a set of resistors in series are: n n G G n eq eq n total total The equations used to calculate the current flowing through a specific resistor m in a set of resistors in parallel are: m m G G eq m m eq total total
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