Fundamentals of Electric Circuits: Series, Parallel, Power, Kirchhoff’s Rules, and RC Circuits

Fundamentals of Circuits

Circuit Symbols and Diagrams

Understanding electric circuits begins with recognizing the standard symbols used to represent various components. Circuit diagrams (schematics) use these symbols to show how components are connected, making analysis and construction easier.

• Capacitor: Two parallel lines represent a capacitor, which stores electric charge and energy.

• Switch: A device that can open or close a circuit, controlling current flow.

• Battery: Represented by alternating long (positive) and short (negative) lines.

• Lightbulb: Indicates a resistive load that emits light.

• Resistor: A zigzag line, representing a component that resists current.

• Wire: Straight lines connect components, representing conducting paths.

• Junction: Where three or more wires meet, allowing current to split or combine.

• Other symbols: Include ammeter, voltmeter, variable resistor, ground, and fuse.

Circuit diagrams use these symbols to represent real circuits, making it easier to analyze and communicate circuit design.

Resistors in Series and Parallel

Series and Parallel Connections

Resistors can be connected in two fundamental ways: series and parallel. The way they are connected affects the total (equivalent) resistance, current, and voltage distribution in the circuit.

• Series: Components are connected end-to-end, so the same current flows through each. The total resistance is the sum of individual resistances.

• Parallel: Components are connected across the same two points, providing multiple paths for current. The voltage across each is the same, but the total current is the sum of the branch currents.

Identifying Series and Parallel: If you can move from point A to B without passing a junction, the components are in series. If there is a choice of paths, they are in parallel.

Resistors in Series

• Current: Same through all resistors.

• Voltage: Total voltage is the sum of voltage drops: $V = V_1 + V_2 + V_3$

• Equivalent Resistance: $R_{eq} = R_1 + R_2 + R_3$

• Ohm's Law: $V = IR$

Note: The equivalent resistance in series is always greater than any individual resistor.

Resistors in Parallel

• Voltage: Same across all resistors.

• Current: Total current is the sum of branch currents: $I = I_1 + I_2 + I_3$

• Equivalent Resistance: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

• Ohm's Law: $I = \frac{V}{R}$

Note: The equivalent resistance in parallel is always less than any individual resistor.

Summary Table: Series vs. Parallel

Property	Series	Parallel
Current	Same through all	Adds: $I = I_1 + I_2 + ...$
Voltage	Adds: $V = V_1 + V_2 + ...$	Same across all
Equivalent Resistance	$R_{eq} = R_1 + R_2 + ...$	$\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$

Example: Resistor Ladder

To find the equivalent resistance of a complex network, reduce the circuit step by step, replacing series and parallel groups with their equivalents until only one resistor remains.

Electric Power in Circuits

Power Dissipation

Electric power is the rate at which energy is transferred or converted. In resistors, power is dissipated as heat.

• General formula: $P = IV$

• Using Ohm's Law: $P = I^2R = \frac{V^2}{R}$

Choose the formula that matches the known quantities in the problem.

Example: Household Heater

• Given: $I = 15.0\,A$, $V = 120\,V$

• Power: $P = IV = 15 \times 120 = 1800\,W = 1.8\,kW$

• Energy for 30 days, 3 hours/day: $E = P \times t = 1.8\,kW \times 90\,h = 162\,kWh$

• Cost at $10.5$ cents/kWh: $162 \times 0.105 = \$17$

Example: Light Bulbs in Series and Parallel

Two 100 Ω bulbs connected to a 24 V battery:

• Series: $R_{eq} = 200\,\Omega$, $I = \frac{24}{200} = 0.12\,A$, $P = (0.12)^2 \times 100 = 1.44\,W$ per bulb

• Parallel: $R_{eq} = 50\,\Omega$, $I = \frac{24}{50} = 0.48\,A$, $P = \frac{24^2}{100} = 5.76\,W$ per bulb

Bulbs in parallel are brighter because each receives the full voltage.

Potential Changes and Kirchhoff’s Rules

Kirchhoff’s Loop Rule

The sum of potential differences around any closed loop is zero, reflecting energy conservation:

• $\sum V_i = 0$

• Apply sign conventions: voltage rises (battery from - to +) are positive, drops (resistor in current direction) are negative.

Kirchhoff’s Junction Rule

The sum of currents entering a junction equals the sum leaving (charge conservation):

• $\sum I_{in} = \sum I_{out}$

Solving Multi-Loop Circuits

1. Assign current directions and label them.

2. Apply the junction rule at each node.

3. Apply the loop rule for independent loops.

4. Solve the resulting system of equations for unknown currents.

Electromotive Force (emf), Terminal Voltage, and Internal Resistance

Real vs. Ideal Batteries

• Ideal battery: Maintains constant voltage regardless of current.

• Real battery: Has internal resistance $r$, so terminal voltage drops as current increases.

• emf ($\varepsilon$): The open-circuit voltage of a battery (no current drawn).

• Terminal voltage: $V_{ab} = \varepsilon - Ir$

Example: Internal Resistance Calculation

• Given: $\varepsilon = 9\,V$, $R = 1\,\Omega$, $V_{ab} = 3\,V$

• Current: $I = \frac{V_{ab}}{R} = 3\,A$

• Internal resistance: $r = \frac{\varepsilon - V_{ab}}{I} = \frac{9 - 3}{3} = 2\,\Omega$

Measuring Instruments: Ammeter, Voltmeter, Ohmmeter

Ammeter

• Measures current; connected in series.

• Should have very low resistance to minimize circuit disturbance.

• Often based on a galvanometer with a shunt resistor.

Voltmeter

• Measures voltage; connected in parallel.

• Should have very high resistance to minimize current draw.

Ohmmeter

• Measures resistance; connected across an isolated resistor (circuit off).

• Contains a battery and measures current to infer resistance.

RC Circuits: Charging and Discharging Capacitors

Charging a Capacitor

When a capacitor is connected to a battery through a resistor, it charges over time, not instantaneously. The process is governed by the time constant $\tau = RC$.

• Charge as a function of time: $q(t) = Q_{final} (1 - e^{-t/RC})$, where $Q_{final} = C\varepsilon$

• Current as a function of time: $i(t) = I_0 e^{-t/RC}$, where $I_0 = \frac{\varepsilon}{R}$

After one time constant ($t = RC$), the capacitor reaches about 63% of its final charge.

Discharging a Capacitor

When a charged capacitor is allowed to discharge through a resistor, the charge and current decrease exponentially:

• Charge: $q(t) = Q_0 e^{-t/RC}$

• Current: $i(t) = I_0 e^{-t/RC}$, where $I_0 = \frac{Q_0}{RC}$

Summary Table: RC Circuit Equations

Process	Charge $q(t)$	Current $i(t)$	Capacitor Voltage $V_C(t)$
Charging	$Q_{final}(1-e^{-t/\tau})$	$I_0 e^{-t/\tau}$	$\varepsilon(1-e^{-t/\tau})$
Discharging	$Q_0 e^{-t/\tau}$	$I_0 e^{-t/\tau}$	$V_0 e^{-t/\tau}$

Time constant: $\tau = RC$ determines how quickly the capacitor charges or discharges.

Example: RC Circuit Calculation

• Given: $C = 8\,\mu F$, $\Delta V = 30\,V$, $t = 0.693RC$

• At $t = 0.693RC$, $e^{-0.693} = 0.5$

• Charge: $q(t) = C\Delta V (1 - 0.5) = 120\,\mu C$

• Voltage across capacitor: $V_C(t) = \Delta V (1 - 0.5) = 15\,V$

Key Takeaways

• Master circuit symbols and schematic diagrams for effective communication and analysis.

• Understand the rules for combining resistors in series and parallel, and how to calculate equivalent resistance.

• Apply Ohm’s Law and power formulas to analyze circuit performance.

• Use Kirchhoff’s rules for complex circuits with multiple loops and junctions.

• Recognize the impact of real measuring instruments and internal resistance in practical circuits.

• Analyze the time-dependent behavior of RC circuits using exponential equations and the time constant.