Electric Circuits: Series and Parallel Configurations Explained

Phys 102 – Lecture 7
Series and parallel circuits
1
Today we will...
Phys. 102, Lecture 7, Slide 2
Learn about 
electric circuits
Circuits with a battery, wires, and resistors
Circuits with a battery, wires, and capacitors
Analyze circuits
Take a complex-looking circuit like...
...and turn into a simple-looking circuit like
Recall from last time...
Phys. 102, Lecture 7, Slide 3
Electric potential difference across circuit element is its “voltage”
V
element
Should be “
Δ
V”, but we’ll usually drop the “
Δ
 
Circuits connect elements with wires, which we treat as 
ideal
conductors
Kirchhoff loop rule
Phys. 102, Lecture 7, Slide 4
+
A charge making a complete loop around a circuit must return to the
same electric potential (“height”) at which it started
Battery
(pump)
Resistor 
R
(bumpy hill)
Sum of electric potential differences
(voltages) around circuit loop is zero
ε
R
Series components
Phys. 102, Lecture 7, Slide 5
Two components are said to be 
in series
 when they are connected
end-to-end by a 
single
 wire
+
ε
R
1
R
2
R
1
R
2
A.
 
V
1
 > 
V
2
B.
 
V
1
 = 
V
2
C.
 
V
1
 < 
V
2
ACT: CheckPoint 1.2
+
R
1
 = 1 
Ω
R
2
 = 10 
Ω
ε
Consider a circuit with two resistors
R
1
 and 
R
2
 in series. Compare the
voltages across the resistors:
ACT: Capacitors in series
Phys. 102, Lecture 7, Slide 7
+
ε
Consider a circuit with two 
capacitors
C
1
 and 
C
2
 in series. Compare the
voltages across the capacitors:
A.
 
V
1
 > 
V
2
B.
 
V
1
 = 
V
2
C.
 
V
1
 < 
V
2
C
1
 = 1 
μ
F
C
2
 = 10 
μ
F
Equivalent resistance & capacitance
Phys. 102, Lecture 7, Slide 8
Circuit behaves the same as if 
series
 components were replaced
by a 
single, equivalent
 component
R
1
R
2
Resistors
Capacitors
=
=
R
eq
C
1
C
2
C
eq
Calculation: vascular resistance
Phys. 102, Lecture 7, Slide 9
Pressure difference
Blood flow
Vascular resistance
emf
Electric current
Electrical resistance
The circulatory system is analogous to an electric circuit
The circulatory system consists of different types of vessels in series
with different resistances to flow
*Numbers represent accurate relative values
Artery
Arteriole
Capillary
Venule
Vein
Calculation: vascular resistance
Phys. 102, Lecture 7, Slide 10
ε
heart
 
= 120 V
R
A
 = 20 
Ω
R
a
 = 50 
Ω
R
c
 
= 20 
Ω
R
v
 
= 6 
Ω
Calculate the current 
I
 through the vascular circuit and the voltages
across the different types of vessels
R
V
 = 4 
Ω
Artery
Arteriole
Capillary
Venule
Vein
Simplify
Expand
Kirchhoff junction rule
Junction
Phys. 102, Lecture 7, Slide 11
Charges flowing through a junction split. By conservation of charge,
the sum of currents into a junction equals the sum of currents out of
a junction
I
1
Parallel components
Phys. 102, Lecture 7, Slide 12
Components are said to be 
in parallel
 
when both ends are
connected to each other, forming a loop containing only them
+
ε
R
1
R
2
ACT: Parallel or series?
Phys. 102, Lecture 7, Slide 13
+
Consider the circuit to the right. Which
of the following statements is true?
A.
 
C
1
 & 
C
4 
are in series
B.
 
C
2
 & 
C
4 
are in parallel
C.
 
C
1
 & 
C
3 
are in parallel
ε
C
2
C
4
C
1
C
3
R
1
 = 1 
Ω
ACT: Resistors in parallel
+
Consider a circuit with two
resistors 
R
1 
and 
R
2 
in parallel.
Compare 
I
1
, the current through
R
1
, to 
I
2
, the current through 
R
2
:
R
2
 = 10 
Ω
ε
A.
 
I
1
 > 
I
2
B.
 
I
1
 = 
I
2
C.
 
I
1
  < 
I
2
Phys. 102, Lecture 7, Slide 14
ACT: CheckPoint 2.3
R
1
+
R
2
ε
Now we add a switch 
S
. What
happens to the current out of the
battery
 when the switch is closed?
A.
 
I
battery
 increases
B.
 
I
battery
 remains the same
C.
 
I
battery
 decreases
DEMO
S
Phys. 102, Lecture 7, Slide 15
Equivalent resistance & capacitance
Phys. 102, Lecture 7, Slide 16
Circuit behaves the same as if 
parallel 
components were replaced
by a 
single, equivalent
 component
R
1
R
2
Resistors
Capacitors
=
=
R
eq
C
1
C
2
C
eq
Calculation: vascular resistance
Phys. 102, Lecture 7, Slide 17
In previous calculation, capillaric resistance accounts for ~20% of
total vascular resistance, yet capillaries are the thinnest blood
vessels, and should have the 
highest 
resistance. Why?
Calculation: cardiovascular system
Phys. 102, Lecture 7, Slide 18
 
 
The human cardiovascular system consists of two circuits:
pulmonary
 circulation which carries blood though the lungs,
and 
systemic
 circulation which carries blood to the organs
Calculate current through each
component of circulatory system
R
pulm
 = 12 
Ω
, 
R
brain
 = 1 k
Ω
,
R
body
 = 160 
Ω
, 
ε
heart
 = 120 V
The organs of the body are connected
in parallel in the systemic circuit
Simple circuit model*: 
*Numbers represent accurate relative values
ACT: analyzing circuits
Phys. 102, Lecture 7, Slide 19
Which of the following circuit is different than the others?
I
II
III
A.
 Circuit I
B.
 Circuit II
C.
 Circuit III
D.
 All three are equivalent
E.
 All three are different
+
+
Calculation: circulatory system
Phys. 102, Lecture 7, Slide 20
+
R
pulm
 = 12 
Ω
, 
R
brain
 = 1 k
Ω
, 
R
body
 = 160 
Ω
, 
ε
heart
 = 120 V
Calculate current through each component of circulatory system
R
brain
 & 
R
body
 are in parallel
R
pulm
 & 
R
syst
 are in series
Simplify
Simplify
R
brain
R
pulm
R
body
ε
heart
Calculation: circulatory system
Phys. 102, Lecture 7, Slide 21
Expand
R
pulm
 & 
R
syst
 are in series
R
pulm
 = 12 
Ω
, 
R
brain
 = 1 k
Ω
, 
R
body
 = 160 
Ω
, 
ε
heart
 = 120 V
Calculate current through each component of circulatory system
R
tot
ε
heart
Calculation: circulatory system
Phys. 102, Lecture 7, Slide 22
Expand
Expand
R
brain
 & 
R
body
 are in parallel
R
pulm
 = 12 
Ω
, 
R
brain
 = 1 k
Ω
, 
R
body
 = 160 
Ω
, 
ε
heart
 = 120 V
Calculate current through each component of circulatory system
R
tot
ε
heart
Summary of today’s lecture
Two basic principles:
Kirchhoff loop rule
Voltages around circuit loop sum to zero (based on
conservation of energy)
Kirchhoff junction rule
Currents into a circuit branch equal currents out (based
on conservation of charge)
Phys. 102, Lecture 7, Slide 23
Summary of today’s lecture
Series
 components
Currents are the same
Voltages add
Resistors
   
Capacitors
Parallel
 components
Voltages are the same
Currents add
Resistors
   
Capacitors
Don’t mix equations!
Phys. 102, Lecture 7, Slide 24
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In this lecture on electric circuits, we delve into series and parallel circuits featuring batteries, wires, resistors, and capacitors. Key concepts discussed include Kirchhoff's loop rule, series components, and equivalent resistance and capacitance. Learn how to simplify complex circuits and analyze voltages in series and parallel setups.


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  1. Phys 102 Lecture 7 Series and parallel circuits 1

  2. Today we will... Learn about electric circuits Circuits with a battery, wires, and resistors Circuits with a battery, wires, and capacitors Analyze circuits Take a complex-looking circuit like... ...and turn into a simple-looking circuit like + + Phys. 102, Lecture 7, Slide 2

  3. Recall from last time... Electric potential difference across circuit element is its voltage Velement Should be V , but we ll usually drop the + R C Batteries pump charges Provide emf for charges Resistors regulate current Dissipate power Capacitors store charge Store energy = Q C V = = battery V V IR C R Circuits connect elements with wires, which we treat as ideal conductors Phys. 102, Lecture 7, Slide 3

  4. Kirchhoff loop rule A charge making a complete loop around a circuit must return to the same electric potential ( height ) at which it started Resistor R (bumpy hill) + VR R + Battery (pump) = 0 V Sum of electric potential differences (voltages) around circuit loop is zero Phys. 102, Lecture 7, Slide 4

  5. Series components Two components are said to be in series when they are connected end-to-end by a single wire I1 R1 V1 R1 + I2 R2 V2 R2 Phys. 102, Lecture 7, Slide 5

  6. ACT: CheckPoint 1.2 Consider a circuit with two resistors R1 and R2 in series. Compare the voltages across the resistors: R1 = 1 + R2 = 10 A. V1 > V2 B. V1 = V2 C. V1 < V2

  7. ACT: Capacitors in series Consider a circuit with two capacitors C1 and C2 in series. Compare the voltages across the capacitors: C1 = 1 F + C2 = 10 F A. V1 > V2 B. V1 = V2 C. V1 < V2 Phys. 102, Lecture 7, Slide 7

  8. Equivalent resistance & capacitance Circuit behaves the same as if series components were replaced by a single, equivalent component Resistors Capacitors C1 C2 Ceq R1 R2 Req I1 I2 Ieq Q1 Q1 Q2 Q2 Qeq Qeq = = V1 V2 Veq V1 V2 Veq = = + = = = = + I I I V V V Q Q Q V V V 1 2 1 2 1 2 1 2 eq eq eq eq 1 1 C 1 = + = + R R R 1 2 eq C C 1 2 eq Phys. 102, Lecture 7, Slide 8

  9. Calculation: vascular resistance The circulatory system is analogous to an electric circuit Pressure difference Blood flow Vascular resistance emf Electric current Electrical resistance The circulatory system consists of different types of vessels in series with different resistances to flow heart= 120 V* Vein Venule Capillary Arteriole Artery RV = 4 Rv= 6 Rc= 20 Ra= 50 RA = 20 *Numbers represent accurate relative values Phys. 102, Lecture 7, Slide 9

  10. Calculation: vascular resistance Calculate the current I through the vascular circuit and the voltages across the different types of vessels heart= 120 V heart Simplify Vein Venule Capillary Arteriole Artery Expand RV = 4 Rv= 6 Rc= 20 Ra = 50 RA = 20 Req Phys. 102, Lecture 7, Slide 10

  11. Kirchhoff junction rule Charges flowing through a junction split. By conservation of charge, the sum of currents into a junction equals the sum of currents out of a junction I2 I2 I1 + + I3 + + + + + Junction I3 I1 = I I in out Phys. 102, Lecture 7, Slide 11

  12. Parallel components Components are said to be in parallel when both ends are connected to each other, forming a loop containing only them I I2 I1 + V2 R1 R2 V1 I1 I I2 Phys. 102, Lecture 7, Slide 12

  13. ACT: Parallel or series? C2 Consider the circuit to the right. Which of the following statements is true? + C1 C4 C3 A. C1 & C4 are in series B. C2 & C4 are in parallel C. C1 & C3 are in parallel Phys. 102, Lecture 7, Slide 13

  14. ACT: Resistors in parallel Consider a circuit with two resistors R1 and R2 in parallel. Compare I1, the current through R1, to I2, the current through R2: + R1 = 1 R2 = 10 A. I1 > I2 B. I1 = I2 C. I1 < I2 Phys. 102, Lecture 7, Slide 14

  15. ACT: CheckPoint 2.3 Now we add a switch S. What happens to the current out of the battery when the switch is closed? S + R1 R2 A. Ibattery increases B. Ibattery remains the same C. Ibattery decreases DEMO Phys. 102, Lecture 7, Slide 15

  16. Equivalent resistance & capacitance Circuit behaves the same as if parallel components were replaced by a single, equivalent component Capacitors Resistors V1 V1 C1 R1 I1 Q1 Q1 Ceq Req Ieq Qeq Qeq = C2 = R2 I2 Q2 Q2 Veq Veq V2 V2 + = = = + = = = I I I V V V Q Q Q V V V 1 2 1 2 1 2 1 2 eq eq eq eq 1 1 R 1 R = + = + C C C 1 2 eq R 1 2 eq Phys. 102, Lecture 7, Slide 16

  17. Calculation: vascular resistance In previous calculation, capillaric resistance accounts for ~20% of total vascular resistance, yet capillaries are the thinnest blood vessels, and should have the highest resistance. Why? Vein Artery Venule Arteriole Capillary Phys. 102, Lecture 7, Slide 17

  18. Calculation: cardiovascular system The human cardiovascular system consists of two circuits: pulmonary circulation which carries blood though the lungs, and systemic circulation which carries blood to the organs Rbrain The organs of the body are connected in parallel in the systemic circuit Rpulm Simple circuit model*: Rpulm = 12 , Rbrain = 1 k , Rbody = 160 , heart = 120 V + heart *Numbers represent accurate relative values Rbody Calculate current through each component of circulatory system Phys. 102, Lecture 7, Slide 18

  19. ACT: analyzing circuits Which of the following circuit is different than the others? I II III + + + A. Circuit I B. Circuit II C. Circuit III D. All three are equivalent E. All three are different Phys. 102, Lecture 7, Slide 19

  20. Calculation: circulatory system Calculate current through each component of circulatory system Rpulm = 12 , Rbrain = 1 k , Rbody = 160 , heart = 120 V Rpulm Rpulm Simplify Simplify heart heart heart + + + Rbody Rsyst Rtot Rbrain Rbrain & Rbody are in parallel 1 1 R R R = = syst brain body V V V = + syst brain body I I I Rpulm & Rsyst are in series 1 = + = + R R R tot pulm syst syst brain body = + = V pulm V = pulm I syst V = I tot heart heart I syst Phys. 102, Lecture 7, Slide 20

  21. Calculation: circulatory system Calculate current through each component of circulatory system Rpulm = 12 , Rbrain = 1 k , Rbody = 160 , heart = 120 V Rpulm Expand heart heart + + Rtot Rsyst Rpulm & Rsyst are in series = + tot pulm R R R syst = + = V pulm V syst V tot heart = = I I heart I pulm syst Phys. 102, Lecture 7, Slide 21

  22. Calculation: circulatory system Calculate current through each component of circulatory system Rpulm = 12 , Rbrain = 1 k , Rbody = 160 , heart = 120 V Rpulm Rpulm Expand Expand heart heart heart + + + Rbody Rbrain Rtot Rsyst Rbrain & Rbody are in parallel 1 1 R R R 1 = + syst brain body = = syst V brain V body V = + I brain I body I syst Phys. 102, Lecture 7, Slide 22

  23. Summary of todays lecture Two basic principles: Kirchhoff loop rule Voltages around circuit loop sum to zero (based on conservation of energy) = 0 V Kirchhoff junction rule Currents into a circuit branch equal currents out (based on conservation of charge) = I I in out Phys. 102, Lecture 7, Slide 23

  24. Summary of todays lecture Series components Currents are the same Voltages add Resistors = = I I I 1 2 eq = + V V V 1 2 eq R 1 1 C 1 = + = + R R 2 Capacitors 1 eq C C 1 2 eq Parallel components = = V V V Voltages are the same Currents add Resistors R 1 2 eq = + 1 I I I 1 2 eq 1 1 R = + C C C = + R Capacitors 1 2 eq 1 2 eq Don t mix equations! Phys. 102, Lecture 7, Slide 24

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