Experiment 11: Resistors in Series and Parallel - Theory & Concepts

Experiment 11: Resistors in Series and Parallel

Learning Objectives

Define node, branch, loop, junction, series connection, parallel connection, equivalent resistance, and conductance.
Use circuit topology rather than visual appearance to identify series and parallel relationships.
Explain and apply Kirchhoff's junction rule and loop rule.
Derive equivalent-resistance formulas for series and parallel resistor networks.
Analyze voltage division, current division, and power distribution.
Reduce mixed series-parallel networks systematically.
Predict the effects of open circuits, short circuits, added resistors, meter loading, and resistor tolerance.
Use an interactive simulation to compare theory with laboratory measurements.

Series and parallel connections are the basic building blocks of direct-current circuits. The important distinction is not whether components look side-by-side or end-to-end, but whether they share the same current path or the same pair of nodes. This lesson develops the network rules from conservation of charge and conservation of energy before applying them in the laboratory.

Target Learning Outcome

Analyze resistor networks using topology, Ohm's law, Kirchhoff's rules, equivalent resistance, voltage division, current division, and power relationships.

1. Circuit Topology and Essential Terms

Node

A node is a set of points joined by ideal conductors and therefore treated as having the same electric potential.

Branch

A branch is a single current path between two nodes and may contain one or more circuit elements.

Junction

A junction is a node where three or more branches meet, allowing current to divide or combine.

Loop

A loop is any closed path through a circuit that begins and ends at the same node without interruption.

Series Connection

Components are in series when the same current must pass through them because their shared node has no other branch connected to it.

Parallel Connection

Components are in parallel when both of their terminals connect to the same pair of nodes, so they have the same voltage.

Equivalent Resistance

Equivalent resistance $R_{\text{eq}}$ is the single resistance that draws the same total current from the same applied voltage as the original network.

Conductance

Conductance $G$ is the reciprocal of resistance. Parallel-network calculations are often interpreted as the addition of conductances.

Topology checks

Series test: the same current is forced through the components.
Parallel test: the components share the same two nodes and therefore the same voltage.
A circuit drawing may be stretched, rotated, or rearranged without changing topology.
A shared point does not prove series connection if another branch also joins that point.

2. Conservation Laws in Electric Circuits

Kirchhoff's Junction Rule

Kirchhoff's junction rule states that the total current entering a node equals the total current leaving it. It follows from conservation of electric charge.

Kirchhoff's junction rule

The algebraic sum of currents at a node is zero when a consistent sign convention is used.

\sum I_{\text{in}}=\sum I_{\text{out}}

\sum I=0

Variables

Symbol	Description	Unit
$I$	branch current	A

Kirchhoff's Loop Rule

Kirchhoff's loop rule states that the algebraic sum of potential changes around any closed loop is zero. It follows from conservation of energy.

Kirchhoff's loop rule

Voltage rises and drops around a complete loop must balance.

\sum \Delta V=0

Variables

Symbol	Description	Unit
$\Delta V$	potential change across a circuit element	V

Physical meaning of Kirchhoff's rules

Current cannot accumulate indefinitely at an ordinary junction, and a charge returning to its starting point after one complete loop must have zero net change in electrical potential energy per unit charge.

3. Resistors in Series

Series-circuit behavior

In a series path, the same current passes through every resistor. The source voltage is divided among the resistors, and the individual voltage drops add to the source voltage.

Equivalent resistance of series resistors

Series resistances add because the same current passes through each component.

R_{\text{eq}}=R_1+R_2+\cdots+R_n

Variables

Symbol	Description	Unit
$R_{\text{eq}}$	equivalent series resistance	Ω
$R_1, R_2, \ldots, R_n$	individual resistances	Ω

Derivation of the series formula

Kirchhoff's loop rule gives $V=V_1+V_2+\cdots+V_n$ . Because the same current $I$ passes through every resistor, substituting $V_i=IR_i$ gives $V=I(R_1+R_2+\cdots+R_n)$ . Therefore the equivalent resistance is the sum.

Voltage-divider relation

Series voltage divides in proportion to resistance.

V_i=V_{\text{s}}\frac{R_i}{R_{\text{eq}}}

Variables

Symbol	Description	Unit
$V_i$	voltage across resistor i	V
$V_{\text{s}}$	source voltage	V
$R_i$	selected series resistance	Ω

Series-network size check

The equivalent resistance of positive series resistors must be greater than every individual resistance in the series path.

4. Resistors in Parallel

Parallel-circuit behavior

In a parallel network, every branch has the same voltage. The source current divides among the branches, and the branch currents add to the total current.

Equivalent resistance of parallel resistors

Parallel conductances add because every branch has the same voltage.

\frac{1}{R_{\text{eq}}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots+\frac{1}{R_n}

Variables

Symbol	Description	Unit
$R_{\text{eq}}$	equivalent parallel resistance	Ω
$R_1, R_2, \ldots, R_n$	branch resistances	Ω

Two-resistor parallel shortcut

This product-over-sum relation applies only to two parallel resistors.

R_{\text{eq}}=\frac{R_1R_2}{R_1+R_2}

Variables

Symbol	Description	Unit
$R_1$	first branch resistance	Ω
$R_2$	second branch resistance	Ω

Equal resistors in parallel

The equivalent of n identical parallel resistors is one resistance divided by n.

R_{\text{eq}}=\frac{R}{n}

Variables

Symbol	Description	Unit
$R$	value of each identical resistor	Ω
$n$	number of identical parallel branches	dimensionless

Derivation of the parallel formula

Kirchhoff's junction rule gives $I=I_1+I_2+\cdots+I_n$ . Because every branch has the same voltage $V$ , substituting $I_i=V/R_i$ gives $I=V(1/R_1+1/R_2+\cdots+1/R_n)$ . Comparing with $I=V/R_{\text{eq}}$ gives the reciprocal formula.

Parallel-network size check

The equivalent resistance of positive parallel branches must be less than the smallest branch resistance because adding a branch creates an additional path for current.

5. Current Division

Current Divider

A current divider is a parallel network in which total current separates among branches according to their conductances.

Current division for two parallel resistors

Branch current is inversely related to the resistance of that branch.

I_1=I_{\text{total}}\frac{R_2}{R_1+R_2}

I_2=I_{\text{total}}\frac{R_1}{R_1+R_2}

Variables

Symbol	Description	Unit
$I_1$	current through resistor 1	A
$I_2$	current through resistor 2	A
$I_{\text{total}}$	current entering the parallel pair	A

Current-divider interpretation

The lower-resistance branch carries the larger current. For equal branch resistances, current divides equally. Current division is governed by conductance, so a branch with twice the conductance carries twice the current.

6. Power in Series and Parallel Networks

Power in a resistor

Equivalent forms are selected according to the known quantities.

P=VI=I^2R=\frac{V^2}{R}

Variables

Symbol	Description	Unit
$P$	power dissipated by a resistor	W
$V$	voltage across the resistor	V
$I$	current through the resistor	A
$R$	resistance	Ω

Power comparison principles

In series, every resistor carries the same current, so $P_i=I^2R_i$ and the larger resistance dissipates more power.
In parallel, every branch has the same voltage, so $P_i=V^2/R_i$ and the smaller resistance dissipates more power.
Total source power equals the sum of resistor powers in an ideal network.

Power-rating check

Equivalent-resistance calculations alone do not prove that a circuit is safe. Calculate the power of each resistor and keep it below the manufacturer's rating.

7. Mixed Series-Parallel Networks

Mixed Network

A mixed network contains both series and parallel relationships and must usually be reduced in stages.

Systematic network reduction

STEP-BY-STEP

Label all nodes and identify components sharing the same pair of nodes.
Reduce the innermost clear series or parallel group.
Redraw the circuit after each reduction.
Continue until one equivalent resistance remains.
Calculate total source current using Ohm's law.
Work backward through the reductions to determine branch voltages and currents.
Verify junction currents, loop voltages, and total power.

Visual-layout trap

Components drawn beside each other are not automatically parallel, and components drawn in a row are not automatically series. Connectivity and nodes determine the relationship.

8. Open Circuits, Short Circuits, and Network Changes

Open Circuit

An open circuit is a broken path with extremely large effective resistance, so current through that path is essentially zero.

Short Circuit

A short circuit is an unintended or idealized path with very small resistance that can carry excessive current.

Failure and modification effects

Change	Series network	Parallel network
Add a resistor	$R_{\text{eq}}$ increases	$R_{\text{eq}}$ decreases
Remove or open one resistor	entire path stops conducting	only that branch stops conducting
Short one resistor	its voltage becomes nearly zero	may bypass a branch or source and cause excessive current

Physical reason parallel resistance decreases

Adding a parallel resistor does not add obstruction; it adds another path. The total current drawn at the same voltage increases, so the equivalent resistance $V/I_{\text{total}}$ decreases.

9. Measurement Theory and Real Components

Meter placement

Instrument	Correct placement	Ideal resistance
Ammeter	in series with the measured branch	$0$
Voltmeter	in parallel across two points	infinite
Ohmmeter	across an isolated, de-energized network	uses its own source

Loading Effect

Loading effect is the change in circuit behavior caused by the measuring instrument itself because a real meter has finite internal resistance.

Sources of disagreement between theory and measurement

Resistor tolerance, contact resistance, lead resistance, meter burden voltage, voltmeter loading, source internal resistance, resistor heating, breadboard defects, and limited instrument resolution can all shift measured values from ideal predictions.

10. Interactive Network Simulation

Series-parallel simulation activity

Switch between series and parallel modes. Before changing a control, predict the equivalent resistance, current, and power. Use the size checks to identify impossible results immediately.

Interactive engineering simulation

Ohm's Law Circuit Simulator

Switch between series and parallel resistance. Current, power, electron-flow speed, and bulb brightness update immediately.

Supply voltage12 V

1–48

Drag for exploration or enter an exact value. Press Enter to apply; Escape restores the current value.

Resistance R16 Ω

1–50

Drag for exploration or enter an exact value. Press Enter to apply; Escape restores the current value.

Resistance R24 Ω

1–50

Drag for exploration or enter an exact value. Press Enter to apply; Escape restores the current value.

Governing Formulas

Ohm's Law

I = \frac{V}{R}

Equivalent Resistance (series)

R_{eq} = R_1 + R_2

Equivalent R

10.00 Ω

Current

1.20 A

Power

14.40 W

Model scope and verification

Use the displayed units and idealizations, then verify the governing balance or compatibility equation before interpreting the result.

Simulation questions

What happens to total current when one series resistance increases?
What happens to total current when one parallel branch resistance increases?
Which arrangement has the lower equivalent resistance for the same two resistors?
How does power change when source voltage doubles?
Why does adding a parallel branch increase source current?

11. Laboratory Application

Theory-guided series and parallel investigation

STEP-BY-STEP

Measure each resistor with the circuit de-energized.
Calculate predicted equivalent resistance before construction.
Construct the series network and verify current continuity and voltage division.
Construct the parallel network and verify equal branch voltage and current division.
Measure equivalent resistance only after disconnecting the source.
Compare theoretical and measured current, voltage, resistance, and power.
Check Kirchhoff's rules using measured values.
Calculate percentage difference and explain discrepancies using realistic instrument and component effects.

Recommended theory checks

For a series network, verify $V_{\text{s}}\approx V_1+V_2+\cdots$ . For a parallel network, verify $I_{\text{total}}\approx I_1+I_2+\cdots$ . Confirm that the equivalent resistance satisfies the appropriate size rule before accepting any calculation or measurement.

Percentage difference

Compare a measured quantity with the selected theoretical reference.

\%\,\text{difference}=\left|\frac{x_{\text{measured}}-x_{\text{theoretical}}}{x_{\text{theoretical}}}\right|\times100\%

Variables

Symbol	Description	Unit
$x_{\text{measured}}$	measured value	varies
$x_{\text{theoretical}}$	predicted value	varies

Engineering applications

Series and parallel networks appear in voltage sensing, lighting circuits, bridge sensors, signal conditioning, power distribution, fault detection, instrumentation, battery management, and control systems. Real designs also require power ratings, tolerances, reliability, and protection against open and short circuits.

Key Takeaways

Series components carry the same current; parallel components share the same voltage.
Nodes and connectivity, not drawing appearance, determine circuit topology.
Kirchhoff's junction rule follows from charge conservation.
Kirchhoff's loop rule follows from energy conservation.
Series resistances add directly.
Parallel conductances add, making parallel equivalent resistance less than the smallest branch resistance.
Voltage divides in proportion to series resistance.
Current divides in proportion to branch conductance.
Total source power equals the sum of component powers in an ideal circuit.
Mixed networks should be reduced and redrawn one stage at a time.
Open and short circuits affect series and parallel networks differently.
Simulations and measurements should be checked using size relationships, Kirchhoff's rules, units, and power limits.

Previous TopicExperiment 10: Ohm's Law and Resistance - Examples and Applications

Quiz Me

Next TopicExperiment 11: Resistors in Series and Parallel - Examples and Applications

Prev Next

Quiz Me