{"id":80,"date":"2025-10-28T11:35:08","date_gmt":"2025-10-28T04:35:08","guid":{"rendered":"https:\/\/phamquangvuong.com\/?p=80"},"modified":"2025-10-29T07:23:08","modified_gmt":"2025-10-29T00:23:08","slug":"mach-noi-tiep-va-mach-song-song-phan-tich-vi-du-thuc-te","status":"publish","type":"post","link":"https:\/\/phamquangvuong.com\/en\/series-parallel-circuits-analysis-practical-examples\/","title":{"rendered":"Series & Parallel Circuits \u2014 Analysis & Practical Examples"},"content":{"rendered":"

<\/p>\n

1. Introduction<\/h2>\n

Electrical circuits are typically formed by combining components (resistors, capacitors, inductors, sources) in two fundamental topologies: series<\/strong> and parallel<\/strong>. Understanding the characteristics of each topology enables you to analyze circuits, compute equivalent resistance, voltage drops and current distribution \u2014 essential tasks for design and troubleshooting.<\/p>\n

2. Series Circuits<\/h2>\n

In a series circuit, components are connected end-to-end so that the same current flows through every element (I_total = I1 = I2 = …). The supply voltage is divided among the resistances proportionally to their values.<\/p>\n

Equivalent resistance<\/h3>\n
R_total = R1 + R2 + R3 + ...<\/code><\/pre>\n
Voltage drop across a resistor<\/h3>\n
Since the same current I flows through each resistor:<\/p>\n
V_k = I \u00d7 R_k<\/code><\/pre>\n
Example<\/h3>\n
With a 12 V source and resistors R1 = 100 \u03a9, R2 = 200 \u03a9 in series:<\/p>\n
R_total = 100 + 200 = 300 \u03a9\r\nI = V \/ R_total = 12 \/ 300 = 0.04 A (40 mA)\r\nV_R1 = I \u00d7 R1 = 4 V\r\nV_R2 = I \u00d7 R2 = 8 V<\/code><\/pre>\n
3. Parallel Circuits<\/h2>\n
In a parallel circuit, elements are connected across the same two nodes; therefore, the voltage across each branch is equal (V_total = V1 = V2 = …). The total current is the sum of branch currents.<\/p>\n
Equivalent resistance<\/h3>\n
1 \/ R_total = 1 \/ R1 + 1 \/ R2 + 1 \/ R3 + ...\r\nFor two resistors:\r\nR_eq = (R1 \u00d7 R2) \/ (R1 + R2)<\/code><\/pre>\n
Current division<\/h3>\n
I1 = V \/ R1\r\nI2 = V \/ R2\r\nI_total = I1 + I2<\/code><\/pre>\n
Example<\/h3>\n
A 10 V source feeds two parallel resistors R1 = 100 \u03a9 and R2 = 200 \u03a9:<\/p>\n
I1 = 10 \/ 100 = 0.10 A\r\nI2 = 10 \/ 200 = 0.05 A\r\nI_total = 0.15 A\r\nR_eq = 1 \/ (1\/100 + 1\/200) = 66.67 \u03a9<\/code><\/pre>\n
\"Voltage<\/a>
Figure:<\/em> Voltage divider \u2014 visualizing voltage drop across series resistors.<\/figcaption><\/figure>\n
\"Current
Figure:<\/em> Current division in parallel resistors.<\/figcaption><\/figure>\n
4. Quick calculation tips<\/h2>\n
    \n
  • Add resistances for series connections.<\/li>\n
  • Use reciprocal formula for parallel networks; for convenience, reduce pairwise.<\/li>\n
  • For mixed series-parallel networks, reduce sub-networks step by step to a single equivalent.<\/li>\n<\/ul>\n
    5. Practical applications<\/h2>\n
    The voltage divider<\/strong> is a common use of series resistors \u2014 it creates a scaled voltage reference for sensors, ADC inputs, or bias networks:<\/p>\n
    V_out = V_in \u00d7 (R2 \/ (R1 + R2))  <\/code><\/pre>\n
    Example: V_in = 12 V, R1 = 1 k\u03a9, R2 = 2 k\u03a9 \u2192 V_out = 8 V.<\/p>\n
    6. Practice problems<\/h2>\n
    Problem 1:<\/strong> Compute the total resistance for this network: R1 = 220 \u03a9 in parallel with the series pair (R2 = 330 \u03a9 + R3 = 470 \u03a9). (Hint: compute R2 + R3 first, then parallel with R1.)<\/p>\n
    Problem 2:<\/strong> Design a divider to produce 3.3 V from a 12 V source to feed a sensor drawing ~5 mA. Choose R1 and R2 so that the divider’s output impedance is low enough (e.g., divider impedance \u2264 1\/10 of sensor input impedance).<\/p>\n
    7. Practical notes<\/h2>\n
      \n
    • Always verify power dissipation (P = I\u00b2R or P = V\u00b2\/R) across resistors to avoid overheating.<\/li>\n
    • When using a voltage divider for low-impedance loads, include a buffer (op-amp follower) to prevent loading errors.<\/li>\n
    • Simulate circuits with LTspice, Falstad, or Proteus before building physical prototypes.<\/li>\n<\/ul>\n
      8. Conclusion<\/h2>\n
      Mastering series and parallel analysis is fundamental for circuit design and troubleshooting. The ability to reduce complex networks into simpler equivalents makes solving real-world electrical problems manageable and reliable.<\/p>\n
      References<\/h3>\n
        \n
      • Paul Horowitz & Winfield Hill, The Art of Electronics<\/em>, 3rd Edition, Cambridge University Press, 2015.<\/li>\n
      • Thomas L. Floyd, Electronics Fundamentals<\/em>, Pearson.<\/li>\n<\/ul>\n
        Keywords:<\/strong> series circuit, parallel circuit, equivalent resistance, voltage divider, current division, basic electronics<\/p>","protected":false},"excerpt":{"rendered":"
        1. Introduction Electrical circuits are typically formed by combining components (resistors, capacitors, inductors, sources) in two fundamental topologies: series and parallel. Understanding the characteristics of each topology enables you to analyze circuits, compute equivalent resistance, voltage drops and current distribution \u2014 essential tasks for design and troubleshooting. 2. Series Circuits In a series circuit, components […]<\/p>\n","protected":false},"author":1,"featured_media":93,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[14],"tags":[],"class_list":["post-80","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-dien-tu-co-ban"],"_links":{"self":[{"href":"https:\/\/phamquangvuong.com\/en\/wp-json\/wp\/v2\/posts\/80","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/phamquangvuong.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/phamquangvuong.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/phamquangvuong.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/phamquangvuong.com\/en\/wp-json\/wp\/v2\/comments?post=80"}],"version-history":[{"count":2,"href":"https:\/\/phamquangvuong.com\/en\/wp-json\/wp\/v2\/posts\/80\/revisions"}],"predecessor-version":[{"id":94,"href":"https:\/\/phamquangvuong.com\/en\/wp-json\/wp\/v2\/posts\/80\/revisions\/94"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/phamquangvuong.com\/en\/wp-json\/wp\/v2\/media\/93"}],"wp:attachment":[{"href":"https:\/\/phamquangvuong.com\/en\/wp-json\/wp\/v2\/media?parent=80"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/phamquangvuong.com\/en\/wp-json\/wp\/v2\/categories?post=80"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/phamquangvuong.com\/en\/wp-json\/wp\/v2\/tags?post=80"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}