{"id":16310,"date":"2026-01-12T18:51:44","date_gmt":"2026-01-12T13:21:44","guid":{"rendered":"https:\/\/johnsonfrancis.org\/techworld\/?p=16310"},"modified":"2026-01-12T18:51:54","modified_gmt":"2026-01-12T13:21:54","slug":"delving-deeper-into-resonance-and-impedance","status":"publish","type":"post","link":"https:\/\/johnsonfrancis.org\/techworld\/delving-deeper-into-resonance-and-impedance\/","title":{"rendered":"Delving Deeper Into Resonance and Impedance!"},"content":{"rendered":"<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/Oo446VBQ5Kg?si=naLHZ3SYCho3WZLT\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n<p class=\"wp-block-paragraph\">While the physics of antenna resonance can be visualized like a tuning fork, the underlying &#8220;engine&#8221; of radio theory is <strong>Complex Algebra<\/strong>. But you don\u2019t need to be an expert in maths to use these formulas, yet you do need to understand three core mathematical concepts: <strong>The Imaginary Unit (j)<\/strong>, <strong>The Impedance Equation<\/strong>, and the <strong>Reflection Coefficient<\/strong>. Actually the imaginary unit is &#8216;i&#8217;. But in electrical discussions as<strong> <\/strong>it is likely to be confused with current, &#8216;j&#8217; is used instead. It is the square root of -1. If any real number is multiplied by it, the product is an imaginary number. Let us go back to our higher secondary school days and recollect the equations!  Practical measurement using the NanoVNA is also illustrated, which of course is more important for the amateur radio operator.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. The Core Equation: Complex Impedance (Z)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">In a DC circuit, resistance is simple (R). In an AC circuit (like radio), we use <strong>Impedance (Z)<\/strong>, which is a complex number (also called a 2D number!).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Z = R + jX<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>R (Resistance):<\/strong> The real part. Energy that is actually radiated or turned into heat.<\/li>\n\n\n\n<li><strong>j:<\/strong> The &#8220;imaginary unit&#8221; (square root of -1). In radio, it represents a 90-degree phase shift between voltage and current. It means that voltage and current do not rise and fall simultaneously when there is inductance or capacitance, unlike in a purely resistive circuit. There is difference in timing by one fourth of a cycle period. One full cycle is 360 degrees.<\/li>\n\n\n\n<li><strong>X (Reactance):<\/strong> The &#8220;imaginary&#8221; part. Energy that is stored in magnetic or electric fields but not radiated.\n<ul class=\"wp-block-list\">\n<li><strong>X<strong><sub>L<\/sub><\/strong> = 2\u03c0fL<\/strong> (Inductive reactance: increases with frequency).<\/li>\n\n\n\n<li><strong>X<strong><sub>C<\/sub><\/strong> = 1\/(2<strong>\u03c0<\/strong>fC)<\/strong> (Capacitive reactance: decreases with frequency).<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">2. The Math of Resonance<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Resonance<\/strong> is the specific mathematical state where the &#8220;imaginary&#8221; parts cancel each other out to zero.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">At Resonance: X<sub>L<\/sub> &#8211; X<sub>C<\/sub> = 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">When this happens, the equation Z = R + jX simplifies to Z = R + j0. The antenna looks like a pure resistor to the radio.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Mathematical Result:<\/strong> Voltage and current are perfectly in phase.<\/li>\n\n\n\n<li><strong>The &#8220;But&#8221;:<\/strong> Even if jX is 0, R could still be 10 \u03a9 or 500 \u03a9. Resonance doesn&#8217;t mean it&#8217;s 50 \u03a9; it just means it&#8217;s &#8220;pure.&#8221;<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">3. The Math of Impedance Matching (SWR)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Impedance Matching<\/strong> uses the <strong>Reflection Coefficient (\u0393)<\/strong> to see how much energy &#8220;bounces back&#8221; because the load (Z<sub>L<\/sub>) doesn&#8217;t match the source (Z<sub>O<\/sub>, usually 50 \u03a9).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u0393<\/strong> = (Z<sub>L<\/sub>&#8211; Z<sub>O<\/sub>)\/(Z<sub>L<\/sub>+ Z<sub>O<\/sub>)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The <strong>SWR<\/strong> (Standing Wave Ratio) is then derived from the magnitude of that reflection (<strong>\u0393<\/strong>):<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">SWR = (1 + \u0393)\/(1 &#8211; \u0393)<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>If Z<sub>L<\/sub> = 50 \u03a9:<\/strong> Then <strong>\u0393<\/strong> = (50- 50)\/(50+ 50) = 0. SWR is 1:1, as per the above equation. Perfect match when both load or antenna impedance and source or radio impedance are the same!<\/li>\n\n\n\n<li><strong>If <strong>Z<sub>L<\/sub><\/strong> is resonant but 10 <strong>\u03a9<\/strong>:<\/strong> Then <strong>\u0393<\/strong> = (10- 50)\/(10+ 50) = -0.66. The SWR would be 5:1 by the same equation. Even though it&#8217;s resonant, the math says the match is terrible. Something similar to that occurs in a multi-element Yagi antenna. As the number of elements increase, the impedance of the resonant dipole radiator decreases from the typical 70 \u03a9 of a dipole to a low value. That is why you use a gamma match for the Yagi antenna!<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Summary Table for the Mathematically Minded<\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><td><strong>Concept<\/strong><\/td><td><strong>Mathematical Condition<\/strong><\/td><td><strong>Result<\/strong><\/td><\/tr><\/thead><tbody><tr><td><strong>Resonance<\/strong><\/td><td>X<sub>total<\/sub> = 0<\/td><td>Pure resistance; no reactive power.<\/td><\/tr><tr><td><strong>Impedance Match<\/strong><\/td><td>Z<sub>load<\/sub> = Z<sub>source<\/sub><\/td><td>Zero reflection; maximum power transfer.<\/td><\/tr><tr><td><strong>Perfect Combo<\/strong><\/td><td>Z<sub>load<\/sub> = 50 + j0<\/td><td>Maximum efficiency AND no reflections.<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Let me hope that anyone with a minimal knowledge of mathematics up to the higher secondary school level would have been able to follow the discussion.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>While the physics of antenna resonance can be visualized like a tuning fork, the underlying &#8220;engine&#8221; of radio theory is Complex Algebra. But you don\u2019t need to be an expert in maths to use these formulas, yet you do need to understand three core mathematical concepts: The Imaginary Unit (j), The Impedance Equation, and the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":16311,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[],"class_list":["post-16310","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-amateur-radio-ham-radio"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.8 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Delving Deeper Into Resonance and Impedance! - Johnson&#039;s Techworld<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/johnsonfrancis.org\/techworld\/delving-deeper-into-resonance-and-impedance\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Delving Deeper Into Resonance and Impedance! - Johnson&#039;s Techworld\" \/>\n<meta property=\"og:description\" content=\"While the physics of antenna resonance can be visualized like a tuning fork, the underlying &#8220;engine&#8221; of radio theory is Complex Algebra. 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