Ohm's Law & DC Circuit Fundamentals

Statement: In any electrical circuit, the current ( $I$ ) flowing is
- Directly proportional to the applied voltage ( $V$ )
- Inversely proportional to the circuit resistance ( $R$ )
- Expressed mathematically by Ohm’s equation: $I = \frac{V}{R}$ (or equivalently $V = IR$ , $R = \frac{V}{I}$ )
Variable definitions
- $V$ = voltage across the circuit in volts (V)
- $I$ = current through the circuit in amperes (A)
- $R$ = resistance of the circuit in ohms (Ω)
Significance
- Fundamental law linking the three primary electrical quantities; basis for analyzing all resistive networks.
- Provides intuitive sense: doubling $V$ doubles $I$ (if $R$ unchanged); doubling $R$ halves $I$ (if $V$ unchanged).

Example 1 – Electric Iron
- Data: $I = 2\,A$ , $V = 120\,V$
- Resistance: $R = \frac{V}{I} = \frac{120}{2} = 60\,\Omega$
Example 2 – Toaster Element
- Data: $R = 20\,\Omega$ , $V = 110\,V$
- Current: $I = \frac{V}{R} = \frac{110}{20} = 5.5\,A$
Example 3 – Unknown Voltage
- Data: $R = 8\,\Omega$ , $I = 10\,A$
- Voltage: $V = IR = (10)(8) = 80\,V$
Example 4 – Figure 2.8 Circuit (Large Resistance)
- Data: $V = 30\,V$ , $R = 5\times10^{3}\,\Omega$
- Current: $I = \frac{30}{5\times10^{3}} = 6\times10^{-3}\,A = 6\,mA$
Example 5 – Figure 2.9 Circuit (Milliamps Given)
- Data: $I = 3\times10^{-3}\,A$ , $R = 10\times10^{3}\,\Omega$
- Voltage: $V = IR = (3\times10^{-3})(10\times10^{3}) = 30\,V$

Direct-current (DC) circuits contain sources and loads where current flows in one constant direction.
Two fundamental connection arrangements: series and parallel.

Definition: All devices connected sequentially so there is only one path for current.
Practical note: Any open component stops current everywhere (e.g.lown fuse).

The applied source voltage is split into individual voltage drops across each component.
Formula: $VT = V1 + V2 + V3 + \dots + V_n$

Total resistance equals the arithmetic sum of all resistances.
Formula: $RT = R1 + R2 + R3 + \dots + R_n$
- Implication: Adding another resistor always increases $R_T$ and therefore decreases circuit current.

Given: $R1 = 2\,\Omega$ , $R2 = 3\,\Omega$ , $R3 = 7\,\Omega$ , $VT = 240\,V$
Total resistance: $R_T = 2 + 3 + 7 = 12\,\Omega$
Total current: $IT = \frac{VT}{R_T} = \frac{240}{12} = 20\,A$
Individual voltage drops (all share same $I_T$ )
- $V1 = IT R_1 = (20)(2) = 40\,V$
- $V2 = IT R_2 = (20)(3) = 60\,V$
- $V3 = IT R_3 = (20)(7) = 140\,V$
- Check: $40 + 60 + 140 = 240\,V = V_T$ ✔️

Given: $R1 = 2\,\Omega$ , $R2 = 6\,\Omega$ , $R3 = 2\,\Omega$ , $VT = 120\,V$
$R_T = 2 + 6 + 2 = 10\,\Omega$
$I_T = \frac{120}{10} = 12\,A$
- Each resistor carries 12 A; voltage drops can be found if needed: $V1 = 24\,V$ , $V2 = 72\,V$ , $V_3 = 24\,V$ .

Definition: Components are connected so that there are multiple independent paths for current; nodes share common voltage.
Useful when loads must operate at same voltage while drawing different currents (e.g.lighting strings, household receptacles).

Each branch experiences the full source voltage.
Formula: $VT = V1 = V2 = V3 = \dots = V_n$
- Safety note: Never place a component rated for lower voltage in a parallel network supplied by higher voltage.

Current divides among branches according to individual resistances (Ohm’s law in each branch).
Formula: $IT = I1 + I2 + I3 + \dots + I_n$

Reciprocal addition formula: $\frac{1}{RT} = \frac{1}{R1} + \frac{1}{R2} + \frac{1}{R3} + \dots + \frac{1}{R_n}$
- Result: $RT$ is always less than the smallest single branch resistance; adding a branch decreases $RT$ , increasing total current.

Given: $R1 = 3\,\Omega$ , $R2 = 6\,\Omega$ , $R_3 = 8\,\Omega$
$\frac{1}{R_T} = \frac{1}{3} + \frac{1}{6} + \frac{1}{8} = \frac{8}{24} + \frac{4}{24} + \frac{3}{24} = \frac{15}{24}$
$R_T = \frac{24}{15} = 1.6\,\Omega$

Given: Branch 1 current $I1 = 3\,A$ through $R1 = 40\,\Omega$ , identical branch 2.
Each branch voltage: $V1 = I1 R_1 = (3)(40) = 120\,V$
Because voltages are equal in parallel, $VT = V1 = 120\,V$

Series vs Parallel trade-offs
- Series increases reliability risk (one open breaks entire path).
- Parallel maintains operation when one branch fails, but draws more total current.
Household wiring: Outlets are wired in parallel so each appliance receives full line voltage.
Measurement instrumentation: Ammeters connect in series (to measure same current); voltmeters connect in parallel (to sample same voltage).
Power calculation reminder: Electrical power $P = VI = I^2 R = \frac{V^2}{R}$ ; combining with Ohm’s law aids thermal design of resistors, heaters, etc.

Component ratings must never be exceeded to prevent fire, shock, or equipment damage.
Engineers must design with margins (derating) and incorporate protection (fuses, circuit breakers).
Understanding basic Ohm/series/parallel rules is foundational for later topics: Kirchhoff’s laws, network theorems, AC phasors, semiconductor biasing.

Alexander, C. K., & Sadiku, M. N. (2013). Fundamentals of Electric Circuits (5th ed.). McGraw-Hill.
Kubala, T. (2009). Electricity 1: Devices, Circuits, and Materials (9th ed.). DEL MAR CENGAGE Learning.