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Voltage and Current Division. 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

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Objective of lecture
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


Voltage dividers
Voltage Dividers

Resistors in series share the same current

Vin


Voltage dividers1
Voltage Dividers

Resistors in series share the same current

From Kirchoff’s Voltage Law and Ohm’s Law :

+

V1

-

+

V2

_

Vin


Voltage dividers2
Voltage Dividers

Resistors in series share the same current

From Kirchoff’s Voltage Law and Ohm’s Law :

+

V1

-

+

V2

_

Vin


Voltage division
Voltage Division

The 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.


Voltage division1
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, Req.

    • The largest value resistor has the largest voltage.



Example 11

Voltage across R1 is: across R2

Voltage across R2 is:

Check: V1 + V2 should equal Vtotal

Example 1

+

V1

-

+

V2

_

8.57 sin(377t)V + 11.4 sin(377t) = 20 sin(377t) V


Example 2
Example 2 across R2

+

V1

-

+

V2

-

+

V3

-

  • Find the voltages listed in the

    circuit to the right.


Example 2 con t
Example 2 ( across R2con’t)

+

V1

-

+

V2

-

+

V3

-

Check: V1 + V2 + V3 = 1V


Symbol for parallel resistors
Symbol for Parallel Resistors across R2

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.


Current division
Current Division across R2

All resistors in parallel share the same voltage

+

Vin

_


Current division1
Current Division across R2

All resistors in parallel share the same voltage

From Kirchoff’s Current Law and Ohm’s Law :

+

Vin

_


Current division2
Current Division across R2

All resistors in parallel share the same voltage

+

Vin

_


Current division3
Current Division across R2

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.

+

Vin

_


Current division4
Current Division across R2

+

Vin

_


Current division5
Current Division across R2

The current associated with one resistor R1in parallel with one other resistor is:

  • The current associated with one resistor Rmin parallel with two or more resistors is:

where Itotalis the total of the currents entering the node shared by the resistors in parallel.


Current division6
Current Division across R2

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


Example 3
Example 3 across R2

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


Example 3 con t
Example 3 (con’t) across R2

Check: I1 + I2 + I3 = Iin


Example 4
Example 4 across R2

I1

  • 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

_

I2

I3

+

Vp

_


Example 4 con t
Example 4 (con’t) across R2

I1

  • First, calculate the total voltage applied to the network of resistors.

    • This is the addition of two voltage sources in series.

+

V1

_

+

Vtotal

_

I2

I3

+

Vp

_


Example 4 con t1
Example 4 (con’t) across R2

I1

  • Second, calculate the equivalent resistor that can be used to replace the parallel combination of R2 and R3.

+

V1

_

+

Vtotal

_

+

Vp

_


Example 4 con t2
Example 4 (con’t) across R2

I1

  • To calculate the value for I1, replace the series combination of R1 and Req1 with another equivalent resistor.

+

Vtotal

_


Example 4 con t3
Example 4 (con’t) across R2

I1

+

Vtotal

_


Example 4 con t4
Example 4 (con’t) across R2

I1

  • To calculate V1, use one of the previous simplified circuits where R1 is in series with Req1.

+

V1

_

+

Vtotal

_

+

Vp

_


Example 4 con t5
Example 4 (con’t) across R2

I1

  • 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.

+

V1

_

+

Vtotal

_

+

Vp

_


Example 4 con t6
Example 4 (con’t) across R2

I1

  • Finally, use the original circuit to find I2 and I3.

+

V1

_

I2

I3

+

Vp

_


Example 4 con t7
Example 4 (con’t) across R2

I1

  • Lastly, the calculation for I3.

+

V1

_

I2

I3

+

Vp

_


Summary
Summary across R2

  • The 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:


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