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Extra Course, Module 3: The Mathematics of Resonance
Covers NCVEC 2024-2028 Extra Pool, Subelement E5 (Electrical Principles). Four questions on your exam come from this material.
Ray writes one formula across the top of the whiteboard and boxes it: f = 1 / (2π√LC). "Every course so far treated resonance as a fact. This one treats it as arithmetic. Four sections: resonance with numbers, time and phase, the coordinate systems that tame complex impedance, and what RF does to real parts. This is the most math in the whole program, and every line of it drives something you own."
3.1Resonance, Quantified
The boxed formula computes the resonant frequency of any LC pair. The pool's two:
- 50 µH and 40 pF: 3.56 MHz.
- 50 µH and 10 pF: 7.12 MHz.
Notice the check: one quarter the capacitance doubled the frequency, because C sits under a square root. That relationship is worth more than the formula.
At resonance, the reactances cancel (your General definition), and what remains is telling:
- A series RLC circuit's impedance at resonance is approximately equal to the circuit resistance: the reactances vanish from the equation and R is what is left. Voltage and current are in phase.
- A parallel RLC circuit's impedance at resonance is also approximately equal to the circuit resistance, but the geometry inverts everything else: the circulating current between L and C is at maximum while the current at the circuit's input is at minimum. The tank sloshes furiously inside while sipping from the source.
Q, the quality factor, measures how furiously. For a parallel resonant circuit, Q = resistance divided by reactance. Its two consequences run in opposite directions, and both are exam material:
- Bandwidth. The half-power bandwidth of a resonant circuit is f / Q. A 7.1 MHz circuit with Q of 150: 47.3 kHz. A 3.7 MHz circuit with Q of 118: 31.4 kHz. High Q buys selectivity by narrowing the window, which is also why increasing the Q of an impedance-matching circuit decreases its matching bandwidth, the reason your mobile antenna from the General course needs retuning across one band.
- Voltage magnification. Resonance can drive the voltage across the reactances higher than the voltage applied to the whole circuit, and increasing Q in a series resonant circuit increases those internal voltages further. A modest transmitter can develop startling kilovolts across a high-Q loop capacitor. Q is free voltage gain, and free voltage gain bites.
Knowledge Check 3.1
Tap an answer. In your own words first, then check yourself.
1. What is the half-power bandwidth of a 7.1 MHz resonant circuit with a Q of 150?
2. At resonance, the input current of a parallel RLC circuit is:
3. What happens to internal voltages as the Q of a series resonant circuit increases?
E5A the easy way: all 13 questions, with the right answer
Every pool question in this group. Read the answer, let the tip make it stick.
E5A01 What can cause the voltage across reactances in a series RLC circuit to be higher than the voltage applied to the entire circuit?
Answer Resonance
Tip At resonance a series circuit rings and the reactive voltages can tower over the source. One word: resonance.
E5A02 What is the resonant frequency of an RLC circuit if R is 22 ohms, L is 50 microhenries, and C is 40 picofarads?
Answer 3.56 MHz
Tip f = 1 / (2 pi root LC), and R does not count. 50 uH with 40 pF lands on 3.56 MHz.
E5A03 What is the magnitude of the impedance of a series RLC circuit at resonance?
Answer Approximately equal to circuit resistance
Tip Series reactances cancel at resonance, so the impedance drops to just R.
E5A04 What is the magnitude of the impedance of a parallel RLC circuit at resonance?
Answer Approximately equal to circuit resistance
Tip At resonance the tank looks resistive, so its impedance is about equal to the circuit resistance.
E5A05 What is the result of increasing the Q of an impedance-matching circuit?
Answer Matching bandwidth is decreased
Tip Higher Q is sharper and narrower. More Q, less bandwidth.
E5A06 What is the magnitude of the circulating current within the components of a parallel LC circuit at resonance?
Answer It is at a maximum
Tip Inside the tank the current sloshes hardest at resonance: circulating current is at a maximum.
E5A07 What is the magnitude of the current at the input of a parallel RLC circuit at resonance?
Answer Minimum
Tip Seen from outside, a parallel tank draws the least: line current is at a minimum.
E5A08 What is the phase relationship between the current through and the voltage across a series resonant circuit at resonance?
Answer The voltage and current are in phase
Tip Reactances cancel, so it looks purely resistive: voltage and current are in phase.
E5A09 How is the Q of an RLC parallel resonant circuit calculated?
Answer Resistance divided by the reactance of either the inductance or capacitance
Tip Parallel Q is R divided by X. (Series Q is the flip, X over R.)
E5A10 What is the resonant frequency of an RLC circuit if R is 33 ohms, L is 50 microhenries, and C is 10 picofarads?
Answer 7.12 MHz
Tip Same 1 / (2 pi root LC): 50 uH with 10 pF gives 7.12 MHz.
E5A11 What is the half-power bandwidth of a resonant circuit that has a resonant frequency of 7.1 MHz and a Q of 150?
Answer 47.3 kHz
Tip Bandwidth = f / Q. 7.1 MHz divided by 150 is 47.3 kHz.
E5A12 What is the half-power bandwidth of a resonant circuit that has a resonant frequency of 3.7 MHz and a Q of 118?
Answer 31.4 kHz
Tip Again bandwidth = f / Q. 3.7 MHz divided by 118 is 31.4 kHz.
E5A13 What is an effect of increasing Q in a series resonant circuit?
Answer Internal voltages increase
Tip Higher Q rings harder, so the internal reactive voltages climb.
3.2Time Constants and the Phase Dance
Two ideas govern circuits in motion: how fast they settle, and who leads whom.
The time constant measures settling: one time constant is the time for a capacitor in an RC circuit to charge to 63.2 percent of the applied voltage, or discharge to 36.8 percent of its initial voltage. The pool's computation: two 220 µF capacitors and two 1-megohm resistors, all in parallel. Module 3 of the General course does the combining (parallel capacitors add: 440 µF; parallel resistors halve: 0.5 megohm), and τ = RC = 220 seconds. That nearly-four-minute answer is also a safety lesson: big filter capacitors bleed slowly, which is why the bleeder resistor exists and why "off" is not "safe."
The phase dance starts with two partners you can memorize with one phrase, ELI the ICE man:
- In an inductor (L), voltage (E) leads current (I): ELI.
- In a capacitor (C), current (I) leads voltage (E): ICE.
- Both by 90 degrees in the pure component.
Mix R with net reactance and the angle lands between 0 and 90: the tangent of the phase angle is net reactance over resistance. The pool's three, each with the same recipe (net the reactances, divide by R, arctangent):
- XC 500, R 1000, XL 250: net 250 capacitive, angle 14 degrees, voltage lagging (capacitive circuits lag, per ICE).
- XC 300, R 100, XL 100: net 200 capacitive, angle 63 degrees, voltage lagging.
- XC 25, R 100, XL 75: net 50 inductive, angle 27 degrees, voltage leading (ELI).
The reciprocal world completes your vocabulary: admittance is the inverse of impedance (from General), and its imaginary part is susceptance, represented by the letter B. Converting impedance in polar form to admittance: take the reciprocal of the magnitude and change the sign of the angle. Converting a pure reactance to susceptance: replace it with its reciprocal. Parallel-circuit analysis runs on admittances the way series analysis runs on impedances, which is the entire reason this mirror-image vocabulary exists.
Knowledge Check 3.2
Tap an answer. In your own words first, then check yourself.
1. After one time constant, an RC circuit's capacitor has charged to what fraction of the applied voltage?
2. In a capacitor, what is the current-voltage phase relationship?
3. What is susceptance?
E5B the easy way: all 12 questions, with the right answer
Every pool question in this group. Read the answer, let the tip make it stick.
E5B01 What is the term for the time required for the capacitor in an RC circuit to be charged to 63.2% of the applied voltage or to discharge to 36.8% of its initial voltage?
Answer One time constant
Tip One time constant is the 63.2 percent charge or 36.8 percent discharge mark. Five of them is essentially done.
E5B02 What letter is commonly used to represent susceptance?
Answer B
Tip Susceptance gets the letter B. It is the reactive partner to conductance G, both measured in siemens.
E5B03 How is impedance in polar form converted to an equivalent admittance?
Answer Take the reciprocal of the magnitude and change the sign of the angle
Tip Admittance is one over impedance, so invert the magnitude and flip the sign of the angle.
E5B04 What is the time constant of a circuit having two 220-microfarad capacitors and two 1-megohm resistors, all in parallel?
Answer 220 seconds
Tip tau = RC. Two 220 uF in parallel is 440 uF, two 1 Mohm in parallel is 0.5 Mohm, and 0.5 M times 440 uF is 220 seconds.
E5B05 What is the effect on the magnitude of pure reactance when it is converted to susceptance?
Answer It is replaced by its reciprocal
Tip Susceptance is one over reactance, so converting a pure reactance just takes its reciprocal.
E5B06 What is susceptance?
Answer The imaginary part of admittance
Tip Admittance is G + jB, so susceptance B is its imaginary part, the reactive piece.
E5B07 What is the phase angle between the voltage across and the current through a series RLC circuit if XC is 500 ohms, R is 1 kilohm, and XL is 250 ohms?
Answer 14.0 degrees with the voltage lagging the current
Tip Net X is XL minus XC, so 250 minus 500 is negative, meaning capacitive. atan(250/1000) gives 14.0 degrees with voltage lagging.
E5B08 What is the phase angle between the voltage across and the current through a series RLC circuit if XC is 300 ohms, R is 100 ohms, and XL is 100 ohms?
Answer 63 degrees with the voltage lagging the current
Tip Net X is 100 minus 300, negative so capacitive. atan(200/100) is about 63 degrees with voltage lagging current.
E5B09 What is the relationship between the AC current through a capacitor and the voltage across a capacitor?
Answer Current leads voltage by 90 degrees
Tip In a capacitor, current leads voltage by 90 degrees. Remember ICE, the I comes before the E.
E5B10 What is the relationship between the AC current through an inductor and the voltage across an inductor?
Answer Voltage leads current by 90 degrees
Tip In an inductor, voltage leads current by 90 degrees. Remember ELI, the E comes before the I.
E5B11 What is the phase angle between the voltage across and the current through a series RLC circuit if XC is 25 ohms, R is 100 ohms, and XL is 75 ohms?
Answer 27 degrees with the voltage leading the current
Tip Net X is 75 minus 25, positive so inductive. atan(50/100) is about 27 degrees with voltage leading current.
E5B12 What is admittance?
Answer The inverse of impedance
Tip Admittance is simply the inverse of impedance, the measure of how easily AC flows.
3.3Complex Impedance on Paper
Impedance has two dimensions, so plotting it takes two axes, and two coordinate systems share the job.
Rectangular coordinates split impedance into parts: the X axis carries the resistive component, the Y axis the reactive component, written as R ± jX. The conventions:
- Pure resistance sits on the horizontal axis.
- Inductive reactance is +j (up); capacitive is -j (down). A pure capacitive reactance of 100 ohms is 0 - j100.
- 50 - j25 reads as 50 ohms of resistance in series with 25 ohms of capacitive reactance. The minus sign is the whole diagnosis.
Polar coordinates describe the same point by magnitude and phase angle, which is how your antenna analyzer thinks. Pure inductive reactance in polar form is a positive 90-degree angle; pure capacitive, negative 90. A phasor diagram is the drawing that shows phase relationships between impedances at a given frequency, Section 3.2's dance rendered as arrows.
The pool exercises this on Figure E5-1, a rectangular grid of numbered points where you compute a reactance (X = 1/2πfC or 2πfL) and pick the point at (R, ±jX). The three worked cases: 400 ohms with 38 pF at 14 MHz lands near 400 - j300; 300 ohms with 18 µH at 3.505 MHz near 300 + j400; 300 ohms with 19 pF at 21.2 MHz near 300 - j400. Compute the reactance, mind the sign, find the point.
Website note: this page will include a redrawn E5-1 grid with worked solutions traced onto it, plus a blank version for practice.
One graphing convention finishes the section: frequency-response curves use a logarithmic Y-axis scale, because filter responses spanning 60 dB are unreadable on linear paper. Every filter plot you have seen since the General course was quietly logarithmic; now it is official.
Quiz yourself: plot the impedances yourself
Knowledge Check 3.3
Tap an answer. In your own words first, then check yourself.
1. What does 50 - j25 ohms represent?
2. How do polar coordinates describe an impedance?
3. Which Y-axis scale do frequency-response graphs normally use?
E5C the easy way: all 12 questions, with the right answer
Every pool question in this group. Read the answer, let the tip make it stick.
E5C01 Which of the following represents pure capacitive reactance of 100 ohms in rectangular notation?
Answer 0 - j100
Tip A pure capacitor has no resistance and a negative reactance, so it is written 0 - j100.
E5C02 How are impedances described in polar coordinates?
Answer By magnitude and phase angle
Tip Polar form gives a vector as magnitude and phase angle, like a length and a direction.
E5C03 Which of the following represents a pure inductive reactance in polar coordinates?
Answer A positive 90 degree phase angle
Tip A pure inductor has reactance pointing straight up, so its polar angle is positive 90 degrees.
E5C04 What type of Y-axis scale is most often used for graphs of circuit frequency response?
Answer Logarithmic
Tip Frequency response spans huge ranges, so a logarithmic Y axis keeps both small and large values readable.
E5C05 What kind of diagram is used to show the phase relationship between impedances at a given frequency?
Answer Phasor diagram
Tip A phasor diagram draws each impedance as an arrow, showing their phase relationships at one frequency.
E5C06 What does the impedance 50 - j25 ohms represent?
Answer 50 ohms resistance in series with 25 ohms capacitive reactance
Tip 50 - j25 reads as 50 ohms of resistance with 25 ohms of capacitive reactance in series, since minus j means capacitive.
E5C07 Where is the impedance of a pure resistance plotted on rectangular coordinates?
Answer On the horizontal axis
Tip Pure resistance has zero reactance, so it sits right on the horizontal axis.
E5C08 What coordinate system is often used to display the phase angle of a circuit containing resistance, inductive, and/or capacitive reactance?
Answer Polar coordinates
Tip Polar coordinates state a magnitude and a phase angle, which is handy for showing a circuit's phase angle directly.
E5C09 When using rectangular coordinates to graph the impedance of a circuit, what do the axes represent?
Answer The X axis represents the resistive component, and the Y axis represents the reactive component
Tip On rectangular impedance plots the X axis is resistance and the Y axis is reactance.
E5C10 Which point on Figure E5-1 best represents the impedance of a series circuit consisting of a 400-ohm resistor and a 38-picofarad capacitor at 14 MHz?
Answer Point 4
Tip XC = 1 / (2 pi f C) gives about 299 ohms at 14 MHz, so plot 400 ohms right with 299 ohms down, which is Point 4.
E5C11 Which point in Figure E5-1 best represents the impedance of a series circuit consisting of a 300-ohm resistor and an 18-microhenry inductor at 3.505 MHz?
Answer Point 3
Tip XL = 2 pi f L gives about 396 ohms at 3.505 MHz, so plot 300 ohms right with inductance up, which is Point 3.
E5C12 Which point on Figure E5-1 best represents the impedance of a series circuit consisting of a 300-ohm resistor and a 19-picofarad capacitor at 21.200 MHz?
Answer Point 1
Tip XC = 1 / (2 pi f C) gives about 395 ohms at 21.2 MHz, so plot 300 ohms right with capacitance down, which is Point 1.
3.4What RF Does to Real Parts
The General course warned that at RF every part is all three of R, L, and C. This section names the mechanisms.
Skin effect is the big one: as frequency increases, RF current flows closer to the conductor's surface, so resistance increases with frequency because the current uses less of the copper. Consequences ripple everywhere: it is the primary loss mechanism in film capacitors at RF, it is why RF conductors are silver-plated and why the General course preferred wide strap, and it adds a subtlety to antennas: as a conductor's diameter increases, its electrical length increases, which is why fat elements resonate slightly lower than the wire formula predicts (and, from General Module 7, why they are broader banded).
Parasitics are the components you did not install:
- Every component's self-resonance comes from its nominal reactance combining with its parasitic reactance. In an inductor specifically, the parasite is inter-turn capacitance, the General course's self-resonant coil now explained.
- Electrolytic capacitors are unsuitable at RF because of their parasitic inductance, that rolled-foil construction being effectively a coil.
- The cures are geometric: keep component leads short at VHF and above to minimize inductive reactance, and at microwave, short connections reduce phase shift along the connection, because at 10 GHz a centimeter of wire is an appreciable fraction of a wavelength.
Real and reactive power close the module. In reactive power, voltage and current are 90 degrees out of phase, and the result is genuinely strange: ideal inductors and capacitors store energy in their fields but dissipate no power. Reactive power is wattless, nonproductive power, volt-amperes circulating without heating anything. Only the resistive component consumes: a circuit of 100 ohms resistance in series with 100 ohms inductive reactance drawing 1 ampere consumes I²R = 100 watts of real power, not the 141 the total impedance suggests. Your wattmeter, your power bill, and your finals all agree: only R gets paid.
Knowledge Check 3.4
Tap an answer. In your own words first, then check yourself.
1. Why does resistance rise with frequency in a conductor?
2. What makes electrolytic capacitors unsuitable at RF?
3. How much real power is consumed by 100 ohms of resistance in series with 100 ohms of inductive reactance drawing 1 ampere?
E5D the easy way: all 12 questions, with the right answer
Every pool question in this group. Read the answer, let the tip make it stick.
E5D01 What is the result of conductor skin effect?
Answer Resistance increases as frequency increases because RF current flows closer to the surface
Tip Skin effect pushes RF current toward the conductor surface, shrinking the usable cross section, so resistance climbs with frequency.
E5D02 Why is it important to keep lead lengths short for components used in circuits for VHF and above?
Answer To minimize inductive reactance
Tip Every bit of wire adds inductive reactance, so at VHF and up you keep leads short to hold that reactance down.
E5D03 What is the phase relationship between current and voltage for reactive power?
Answer They are 90 degrees out of phase
Tip Reactive power flows when current and voltage are 90 degrees out of phase, so no net energy is delivered.
E5D04 Why are short connections used at microwave frequencies?
Answer To reduce phase shift along the connection
Tip At microwave frequencies a wavelength is tiny, so even short leads shift phase, and short connections keep that shift small.
E5D05 What parasitic characteristic causes electrolytic capacitors to be unsuitable for use at RF?
Answer Inductance
Tip Electrolytics have rolled construction that adds series inductance, which spoils them for RF use.
E5D06 What parasitic characteristic creates an inductor’s self-resonance?
Answer Inter-turn capacitance
Tip The gaps between an inductor's turns act like small capacitors, and that inter-turn capacitance sets its self-resonance.
E5D07 What combines to create the self-resonance of a component?
Answer The component’s nominal and parasitic reactance
Tip Self-resonance appears when a part's intended reactance teams up with its parasitic reactance.
E5D08 What is the primary cause of loss in film capacitors at RF?
Answer Skin effect
Tip Film capacitor leads and plates carry RF current near their surface, so skin effect is the main loss source.
E5D09 What happens to reactive power in ideal inductors and capacitors?
Answer Energy is stored in magnetic or electric fields, but power is not dissipated
Tip Ideal L and C only store and return energy in their fields, so reactive power is never actually dissipated.
E5D10 As a conductor’s diameter increases, what is the effect on its electrical length?
Answer It increases
Tip A fatter conductor shifts its resonance, making it act electrically longer, so it needs trimming shorter to tune.
E5D11 How much real power is consumed in a circuit consisting of a 100-ohm resistor in series with a 100-ohm inductive reactance drawing 1 ampere?
Answer 100 watts
Tip Real power is I squared times R only. 1 amp squared times 100 ohms is 100 watts, and the reactance adds nothing.
E5D12 What is reactive power?
Answer Wattless, nonproductive power
Tip Reactive power is wattless, nonproductive power that swings in and out without doing real work.
Review: every Exam Anchor from this module
E5A f = 1/(2π√LC): 50 µH + 40 pF = 3.56 MHz; 50 µH + 10 pF = 7.12 MHz. Series resonance: impedance ≈ R, voltage and current in phase; resonance can raise reactance voltages above the applied voltage, and higher Q raises them further. Parallel resonance: impedance ≈ R, circulating current maximum, input current minimum. Parallel Q = R / X. Half-power bandwidth = f/Q: 47.3 kHz at 7.1 MHz with Q 150; 31.4 kHz at 3.7 MHz with Q 118. Higher matching-circuit Q: narrower matching bandwidth.
E5B One time constant: charge to 63.2 percent or discharge to 36.8 percent. Two 220 µF caps and two 1 MΩ resistors all in parallel: 220 seconds. Capacitor: current leads voltage by 90 degrees; inductor: voltage leads current by 90 degrees. Phase angles: 14 degrees lagging (XC 500, R 1000, XL 250); 63 degrees lagging (XC 300, R 100, XL 100); 27 degrees leading (XC 25, R 100, XL 75). Admittance: inverse of impedance; susceptance: its imaginary part, letter B; polar conversion: reciprocal magnitude, negated angle; reactance to susceptance: reciprocal.
E5C Rectangular: X axis resistive, Y axis reactive; pure resistance on the horizontal axis; pure capacitive 100 ohms = 0 - j100; 50 - j25 = 50 ohms resistance in series with 25 ohms capacitive reactance. Polar: magnitude and phase angle; pure inductive = +90 degrees; polar coordinates display circuit phase angle. Phasor diagram: phase relationships at a given frequency. Frequency response graphs: logarithmic Y axis. Figure E5-1: compute reactance, sign it, locate the point.
E5D Skin effect: RF current crowds the surface, resistance rises with frequency; it is the main film-capacitor RF loss; larger conductor diameter increases electrical length. Self-resonance: nominal plus parasitic reactance; in inductors, inter-turn capacitance. Electrolytics fail at RF from parasitic inductance. Short leads at VHF: minimize inductive reactance; short microwave connections: reduce phase shift. Reactive power: voltage and current 90 degrees apart; ideal L and C store energy without dissipation; reactive power is wattless and nonproductive. 100 ohms R in series with 100 ohms XL at 1 A: 100 watts real power.
Before you move on
The boxed formula earned its box: you can now compute a resonant frequency, its bandwidth from Q, the phase angle of any RLC combination, and the real power hiding inside a complex impedance, and you can write any of it in rectangular or polar form as the job demands. ELI the ICE man rides along for free.
Module 4 spends this vocabulary in the parts drawers: semiconductor physics, the full diode family, logic families, ferrites with their physics attached, RF packaging, and the electro-optics that switch your amplifier without touching it.
© 2026 Anthony Pelfanio, K4DIA. All rights reserved.