WEBVTT NOTE Reactance, Impedance, and Resonance 1 00:00:00.000 --> 00:00:02.640 Reactance, precisely. 2 00:00:02.640 --> 00:00:06.484 Question. What is reactance? 3 00:00:07.384 --> 00:00:15.277 Answer. Opposition to the flow of alternating current caused by capacitance or inductance. 4 00:00:15.877 --> 00:00:33.671 Note that the pool asks the same thing three ways, "opposition to AC in an inductor", "opposition to AC in a capacitor", "what is reactance", and the answer is reactance every time. 5 00:00:33.671 --> 00:00:49.597 It is measured in ohms and represented by the letter X. The two formulas, and the behaviour that follows: X L equals 2 pi f L X C equals (1) divided by (2 pi f C) 6 00:00:49.597 --> 00:00:54.495 Question. How does an inductor react to AC? 7 00:00:55.395 --> 00:01:02.755 Answer. As the frequency of the applied AC increases, the reactance increases. 8 00:01:03.355 --> 00:01:08.515 Question. How does a capacitor react to AC? 9 00:01:09.415 --> 00:01:17.495 Answer. As the frequency of the applied AC increases, the reactance decreases. 10 00:01:18.095 --> 00:01:36.322 Frequency is in the numerator for the inductor and the denominator for the capacitor. Everything else about tuned circuits, filters, and matching follows from that one asymmetry. 11 00:01:36.322 --> 00:01:50.452 Worked: a 1 µH inductor at 7.15 MHz has X_L = 2π times 7.15 times 10 to the power of 6 times 1 times 10 to the power of minus 6 = 45 ohms. 12 00:01:50.452 --> 00:02:05.095 A 100 pF capacitor at 14.2 MHz has X_C = 1 / (2π times 14.2 times 10 to the power of 6 times 100 times 10 to the power of minus 12) = 112 ohms. 13 00:02:05.095 --> 00:02:17.895 Neither is negligible, which is why stray inductance and capacitance matter at RF in a way they never do at audio. Impedance. 14 00:02:17.895 --> 00:02:21.815 Question. What is impedance? 15 00:02:22.715 --> 00:02:26.999 Answer. The opposition to AC current flow. 16 00:02:27.599 --> 00:02:39.178 That is the general definition, and it holds whether the opposition is resistance, reactance, or both. Ohm's law with Z in place of R. 17 00:02:39.178 --> 00:02:44.858 Question. What is the term for the inverse of impedance? 18 00:02:45.758 --> 00:02:48.314 Answer. Admittance. 19 00:02:48.914 --> 00:03:08.617 Admittance (Y) is to impedance what conductance is to resistance, the reciprocal. It simplifies parallel circuits for the same reason conductance does: admittances add in parallel. Impedance matching. 20 00:03:08.617 --> 00:03:27.403 Question. Which of the following devices can be used for impedance matching at radio frequencies? A. A transformer. B. A Pi-network. C. A length of transmission line. D. All these choices are correct. 21 00:03:28.303 --> 00:03:31.823 Answer. All these choices are correct. 22 00:03:32.423 --> 00:03:41.319 Three different mechanisms, one purpose: A transformer transforms impedance by the square of its turns ratio. 23 00:03:41.319 --> 00:03:50.786 A Pi-network uses reactances to transform between impedances, and is the classic output network in a tube amplifier. 24 00:03:50.786 --> 00:04:06.537 A length of transmission line transforms impedance along its length; a quarter-wave section is the standard trick, transforming Z to Z 0 to the power of 2/Z. Resonance. There is a diagram here. 25 00:04:06.537 --> 00:04:20.982 Impedance magnitude plotted against frequency for a series RLC circuit. The curve falls steeply to a minimum equal to the resistance at the resonant frequency, then rises again. 26 00:04:20.982 --> 00:04:27.103 Below resonance the circuit is capacitive; above resonance it is inductive. 27 00:04:27.103 --> 00:04:32.863 Question. What occurs in an LC circuit at resonance? 28 00:04:33.763 --> 00:04:39.636 Answer. Inductive reactance and capacitive reactance cancel. 29 00:04:40.236 --> 00:04:47.436 f equals (1) divided by (2 pi the square root of (LC)) 30 00:04:53.436 --> 00:04:58.609 Nothing here is beyond the two reactance formulas you already have. 31 00:04:58.609 --> 00:05:20.691 Resonance is defined as the frequency where they are equal, so set them equal and solve: 2 pi f L equals (1) divided by (2 pi f C) 4 pi squared f squared LC equals 1 which gives f squared equals (1) divided by (4 pi squared LC) Taking the positive root because frequency is a magnitude. 32 00:05:20.691 --> 00:05:34.125 The formula is not a separate fact to memorise, it is X L equals X C rearranged, and if you forget it you can rebuild it in about fifteen seconds. A consequence worth having. 33 00:05:34.125 --> 00:05:45.243 Substitute that f back into either reactance and the 2 pi and the square roots collapse: The formula for this is written out in the lesson text. 34 00:05:45.243 --> 00:05:56.361 So at resonance both reactances equal the square root of (L divided by C), a quantity that depends on the ratio of L to C, not on their product. 35 00:05:56.361 --> 00:06:06.398 Two circuits can resonate at the same frequency (same LC) while presenting wildly different reactances (different L divided by C). 36 00:06:06.398 --> 00:06:17.979 That is the difference between a high-impedance tank and a low-impedance one, and it is why you cannot specify a tuned circuit by its frequency alone. 37 00:06:17.979 --> 00:06:26.163 Both statements above are re-proved symbolically on every build; see the derivation on the glossary entry. 38 00:06:26.163 --> 00:06:35.660 The circuit that algebra describes, with each element's contribution to the total impedance named: There is a diagram here. 39 00:06:35.660 --> 00:06:55.580 A series circuit drawn as a loop: a sinusoidal source on the left, then a resistor labelled R contributing resistance, an inductor labelled L contributing reactance plus j two pi f L, and a capacitor labelled C contributing reactance minus j over two pi f C. 40 00:06:55.580 --> 00:07:08.095 A note records that at resonance the two reactances are equal in magnitude and opposite in sign, so they cancel and the impedance is R alone. 41 00:07:08.095 --> 00:07:23.982 Written as a complex sum, the whole of the above is one line: Z equals R plus j(2 pi f L minus (1) divided by (2 pi f C)) Resonance is the frequency that kills the imaginary part. 42 00:07:23.982 --> 00:07:43.420 Everything else in this lesson, why series shorts and parallel opens, why Q sets bandwidth, is a consequence of that expression. The resonant frequency is the same for series and parallel, but the behaviour is opposite: 43 00:07:43.420 --> 00:07:52.508 Question. What happens when inductive and capacitive reactance are equal in a series LC circuit? 44 00:07:53.408 --> 00:07:59.248 Answer. Resonance causes impedance to be very low. 45 00:07:59.848 --> 00:08:15.693 Series resonance: minimum impedance. The two reactances cancel in the series total, leaving only the (usually small) resistance. Parallel resonance: maximum impedance. 46 00:08:15.693 --> 00:08:26.699 The cancellation removes the paths for current instead. If you remember one sentence: series shorts, parallel opens. 47 00:08:26.699 --> 00:08:46.340 Worked: 10 µH with 100 pF resonates at f = 1 / (2π√(10 times 10 to the power of minus 6 times 100 times 10 to the power of minus 12)) = 5.03 MHz. Check yourself. A capacitor's reactance at 7 MHz is 100 ohms. 48 00:08:46.340 --> 00:08:59.528 What is it at 14 MHz? A series LC circuit is at resonance. Is its impedance high or low, and what is the current doing? What is admittance? 49 00:09:05.528 --> 00:09:24.288 50 ohms. Capacitive reactance is inversely proportional to frequency, so doubling the frequency halves it. Low impedance, maximum current. Series resonance leaves only the resistance. The inverse of impedance..