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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Voltage and Current Division
Resistors in series share the same current
Vin
Resistors in series share the same current
From Kirchoff’s Voltage Law and Ohm’s Law :
+
V1

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

+
V2
_
Vin
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.
Find the V1, the voltage across R1, and V2, the voltage across R2.
+
V1

+
V2
_
Voltage across R1 is:
Voltage across R2 is:
Check: V1 + V2 should equal Vtotal
+
V1

+
V2
_
8.57 sin(377t)V + 11.4 sin(377t) = 20 sin(377t) V
+
V1

+
V2

+
V3

circuit to the right.
+
V1

+
V2

+
V3

Check: V1 + V2 + V3 = 1V
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.
All resistors in parallel share the same voltage
+
Vin
_
All resistors in parallel share the same voltage
From Kirchoff’s Current Law and Ohm’s Law :
+
Vin
_
All resistors in parallel share the same voltage
+
Vin
_
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
_
+
Vin
_
The current associated with one resistor R1in parallel with one other resistor is:
where Itotalis the total of the currents entering the node shared by the resistors in parallel.
Find currents I1, I2, and I3 in the circuit to the right.
Check: I1 + I2 + I3 = Iin
I1
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V1
_
I2
I3
+
Vp
_
I1
+
V1
_
+
Vtotal
_
I2
I3
+
Vp
_
I1
+
V1
_
+
Vtotal
_
+
Vp
_
I1
+
Vtotal
_
I1
+
Vtotal
_
I1
+
V1
_
+
Vtotal
_
+
Vp
_
I1
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
_
I1
+
V1
_
I2
I3
+
Vp
_
I1
+
V1
_
I2
I3
+
Vp
_