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Objective of Lecture Explain mathematically how a voltage that is applied to resistors in series is distributed among the resistors. Chapter 2.5 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 2.6 in Fundamentals of Electric Circuits Chapter 6.7 in Electric Circuit Fundamentals Work through examples include a series-parallel resistor network (Example 4). Chapter 7.2 in Fundamentals of Electric Circuits

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Voltage Dividers Resistors in series share the same current V in

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Voltage Dividers Resistors in series share the same current From Kirchoff’s Voltage Law and Ohm’s Law : + V1 - + V2 _ V in

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Voltage Dividers Resistors in series share the same current From Kirchoff’s Voltage Law and Ohm’s Law : + V1 - + V2 _ V in

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Voltage Division The voltage associated with one resistor R n in a chain of multiple resistors in series is: or where V total is the total of the voltages applied across the resistors.

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Voltage 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, R eq. The largest value resistor has the largest voltage.

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Example 1 Find the V1, the voltage across R1, and V2, the voltage across R2. + V1 - + V2 _

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Example 1 Voltage across R1 is: Voltage across R2 is: Check: V1 + V2 should equal V total + V1 - + V2 _ 8.57 sin(377t)V + 11.4 sin(377t) = 20 sin(377t) V

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+ V1 - + V2 - + V3 - Example 2 Find the voltages listed in the circuit to the right.

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Example 2 (con’t) Check: V1 + V2 + V3 = 1V + V1 - + V2 - + V3 -

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Symbol 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.

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Current Division All resistors in parallel share the same voltage + V in _

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Current Division All resistors in parallel share the same voltage From Kirchoff’s Current Law and Ohm’s Law : + V in _

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Current Division All resistors in parallel share the same voltage + V in _

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Current Division Alternatively, 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. + V in _

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Current Division + V in _

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Current Division The current associated with one resistor R 1 in parallel with one other resistor is: The current associated with one resistor R m in parallel with two or more resistors is: where I total is the total of the currents entering the node shared by the resistors in parallel.

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Current Division The largest value resistor has the smallest amount of current flowing through it.

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Example 3 Find currents I1, I2, and I3 in the circuit to the right.

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Example 3 (con’t) Check: I1 + I2 + I3 = Iin

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Example 4 The circuit to the right has a series and parallel combination of resistors plus two voltage sources. Find V1 and Vp Find I1, I2, and I3 + V1 _ + Vp _ I1 I2I3

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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. + V1 _ + Vp _ I1 I2I3 + Vtotal _

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Example 4 (con’t) Second, calculate the equivalent resistor that can be used to replace the parallel combination of R2 and R3. + V1 _ + Vp _ I1 + Vtotal _

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Example 4 (con’t) To calculate the value for I1, replace the series combination of R1 and R eq1 with another equivalent resistor. I1 + Vtotal _

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Example 4 (con’t) I1 + Vtotal _

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Example 4 (con’t) To calculate V1, use one of the previous simplified circuits where R1 is in series with R eq1. + V1 _ + Vp _ I1 + Vtotal _

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Example 4 (con’t) To calculate V p : 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. + V1 _ + Vp _ I1 + Vtotal _

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Example 4 (con’t) Finally, use the original circuit to find I2 and I3. + V1 _ + Vp _ I1 I2I3

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Example 4 (con’t) Lastly, the calculation for I3. + V1 _ + Vp _ I1 I2I3

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Summary The equations used to calculate the voltage across a specific resistor R n in a set of resistors in series are: The equations used to calculate the current flowing through a specific resistor R m in a set of resistors in parallel are:

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