2Objective of LectureExplain mathematically how a voltage that is applied to resistors in series is distributed among the resistors.Chapter 2.5 in Fundamentals of Electric CircuitsChapter 5.7 Electric Circuit FundamentalsExplain mathematically how a current that enters the a node shared by resistors in parallel is distributed among the resistors.Chapter 2.6 in Fundamentals of Electric CircuitsChapter 6.7 in Electric Circuit FundamentalsWork through examples include a series-parallel resistor network (Example 4).Chapter 7.2 in Fundamentals of Electric Circuits
3Resistors in series share the same current Voltage DividersResistors in series share the same currentVin
4Resistors in series share the same current Voltage DividersResistors in series share the same currentFrom Kirchoff’s Voltage Law and Ohm’s Law :+V1-V2_Vin
5Resistors in series share the same current Voltage DividersResistors in series share the same currentFrom Kirchoff’s Voltage Law and Ohm’s Law :+V1-V2_Vin
6Voltage DivisionThe voltage associated with one resistor Rn in a chain of multiple resistors in series is: or where Vtotal is the total of the voltages applied across the resistors.
7Voltage DivisionThe percentage of the total voltage associated with a particular resistor is equal to the percentage that that resistor contributed to the equivalent resistance, Req.The largest value resistor has the largest voltage.
8Example 1Find the V1, the voltage across R1, and V2, the voltage across R2.+V1-V2_
9Example 1 + V1 - V2 _ Voltage across R1 is: Voltage across R2 is: Check: V1 + V2 should equal Vtotal+V1-V2_8.57 sin(377t)V sin(377t) = 20 sin(377t) V
10Example 2 Find the voltages listed in the circuit to the right. + V1 -
12Symbol for Parallel Resistors To make writing equations simpler, we use a symbol to indicate that a certain set of resistors are in parallel. Here, we would write R1║R2║R3 to show that R1 is in parallel with R2 and R3. This also means that we should use the equation for equivalent resistance if this symbol is included in a mathematical equation.
13Current Division All resistors in parallel share the same voltage + Vin_
14From Kirchoff’s Current Law and Ohm’s Law : Current DivisionAll resistors in parallel share the same voltageFrom Kirchoff’s Current Law and Ohm’s Law :+Vin_
15Current Division All resistors in parallel share the same voltage + Vin_
16Current DivisionAlternatively, you can reduce the number of resistors in parallel from 3 to 2 using an equivalent resistor. If you want to solve for current I1, then find an equivalent resistor for R2 in parallel with R3.+Vin_
18Current DivisionThe current associated with one resistor R1 in parallel with one other resistor is:The current associated with one resistor Rm in parallel with two or more resistors is:where Itotal is the total of the currents entering the node shared by the resistors in parallel.
19Current DivisionThe largest value resistor has the smallest amount of current flowing through it.
20Example 3Find currents I1, I2, and I3 in the circuit to the right.
27Example 4 (con’t)To calculate V1, use one of the previous simplified circuits where R1 is in series with Req1.I1+V1_+Vtotal_+Vp_
28Example 4 (con’t) To calculate Vp: Note: rounding errors can occur. It is best to carry the calculations out to 5 or 6 significant figures and then reduce this to 3 significant figures when writing the final answer.I1+V1_+Vtotal_+Vp_
29Example 4 (con’t) Finally, use the original circuit to find I2 and I3. +V1_I2I3+Vp_
30Example 4 (con’t) Lastly, the calculation for I3. I1 + V1 _ I2 I3 + Vp
31SummaryThe equations used to calculate the voltage across a specific resistor Rn in a set of resistors in series are:The equations used to calculate the current flowing through a specific resistor Rm in a set of resistors in parallel are: