Ohm's Law & DC Circuit Fundamentals

OHM’S LAW

  • 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).

Worked Examples (Ohm’s Law)

  • 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

DC CIRCUITS

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

SERIES CIRCUIT

  • 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).

Voltage in Series

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

Current in Series

  • Same current flows through every element because charge has only one route.
  • Formula: IT = I1 = I2 = I3 = \dots = I_n

Resistance in Series

  • 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.

Example 1 – Figure 3-7

  • 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 ✔️

Example 2 – Figure 3-8

  • 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.

Series Summary

  • RT = \sum Rn
  • I_T is common to all parts.
  • V_T divides proportionally to resistance values.

PARALLEL CIRCUIT

  • 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).

Voltage in Parallel

  • 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 in Parallel

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

Resistance in Parallel

  • 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.

Example 1 – Figure 4-1

  • 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

Example 2 – Figure 4-6

  • 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

Parallel Summary

  • V_T common across all branches.
  • IT = \sum In (current splits).
  • \frac{1}{RT} = \sum \frac{1}{Rn}.

COMPARATIVE INSIGHTS & REAL-WORLD RELEVANCE

  • 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.

ETHICAL & PRACTICAL CONSIDERATIONS

  • 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.

REFERENCES

  • 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.