Amateur Extra · correct answers only, in question-pool order.
At series resonance, large circulating current can develop. The voltage across the inductor or capacitor can therefore exceed the voltage applied to the entire RLC circuit.
Resonance occurs where XL equals XC, with f = 1/(2π√LC). Using 50 microhenries and 40 picofarads gives about 3.56 MHz.
In a series RLC circuit at resonance, inductive and capacitive reactance cancel. The remaining impedance is approximately the circuit resistance.
The official Extra pool states that the magnitude of impedance of a parallel RLC circuit at resonance is approximately equal to circuit resistance.
Increasing Q makes an impedance-matching network more selective, so the range of frequencies over which it provides a good match becomes narrower.
At parallel resonance, energy circulates strongly between the inductor and capacitor, so current within those reactive components reaches a maximum.
At parallel resonance, reactive branch currents largely cancel at the input, so the current drawn from the source is at a minimum.
At series resonance, XL and XC cancel and the circuit is purely resistive, so voltage and current are in phase.
For the parallel RLC relationship used in this pool, Q is resistance divided by the magnitude of either inductive or capacitive reactance at resonance.
Using f = 1/(2π√LC) with 50 microhenries and 10 picofarads gives approximately 7.12 MHz.
Half-power bandwidth is resonant frequency divided by Q. 7.1 MHz / 150 is about 47.3 kHz.
Half-power bandwidth equals resonant frequency divided by Q. 3.7 MHz / 118 is about 31.4 kHz.
Increasing Q in a series resonant circuit increases circulating current and therefore increases the internal voltages across the reactive components.
One RC time constant is the time required to charge a capacitor to 63.2% of the applied voltage or discharge it to 36.8% of its initial voltage.
Susceptance is conventionally represented by the letter B.
Admittance is the reciprocal of impedance. In polar form, taking a reciprocal means invert the magnitude and reverse the sign of the phase angle.
Two 220-microfarad capacitors in parallel total 440 microfarads. Two 1-megohm resistors in parallel total 500 kilohms. Their RC time constant is 0.000440 × 500,000 = 220 seconds.
Susceptance is the reciprocal counterpart of reactance, so converting pure reactance to susceptance replaces the magnitude with its reciprocal.
Susceptance is the imaginary component of admittance, analogous to reactance being the imaginary component of impedance.
Net reactance is XL - XC = 250 - 500 = -250 ohms. arctan(-250/1000) is about -14 degrees, so voltage lags current by 14 degrees.
Net reactance is 100 - 300 = -200 ohms. arctan(-200/100) is about -63 degrees, so the voltage lags the current by about 63 degrees.
For an ideal capacitor, current reaches its peak one-quarter cycle before voltage does, so current leads voltage by 90 degrees.
For an ideal inductor, voltage reaches its peak one-quarter cycle before current does, so voltage leads current by 90 degrees.
Net reactance is 75 - 25 = +50 ohms. arctan(50/100) is about +27 degrees, so voltage leads current by about 27 degrees.
Admittance is the inverse, or reciprocal, of impedance.
Pure capacitive reactance lies on the negative imaginary axis, so 100 ohms of capacitive reactance is written 0 - j100.
Polar impedance is described by a magnitude and a phase angle.
A pure inductor has a +90-degree impedance phase angle because voltage leads current by 90 degrees.
Circuit frequency-response graphs commonly use a logarithmic vertical scale, often expressed in decibels, so large amplitude ranges are easier to display.
A phasor diagram shows the magnitudes and phase relationships of AC quantities such as resistance, inductive reactance, and capacitive reactance.
The real term, 50 ohms, is resistance. The negative imaginary term, -j25, represents 25 ohms of capacitive reactance.
Pure resistance has no reactive component, so it lies entirely on the horizontal real axis of a rectangular impedance plot.
Polar coordinates directly express both impedance magnitude and phase angle, so they are well suited for circuits containing resistance and reactance.
On a rectangular impedance plot, the horizontal X axis represents resistance and the vertical Y axis represents reactance.
At 14 MHz, a 38-picofarad capacitor has about 299 ohms of capacitive reactance. The impedance is therefore about 400 - j299 ohms, which corresponds to Point 4 on Figure E5-1.
At 3.505 MHz, an 18-microhenry inductor has about 396 ohms of inductive reactance. The impedance is about 300 + j396 ohms, corresponding to Point 3 on Figure E5-1.
At 21.200 MHz, a 19-picofarad capacitor has about 395 ohms of capacitive reactance. The impedance is about 300 - j395 ohms, corresponding to Point 1 on Figure E5-1.
At RF, current crowds toward the conductor surface. As frequency increases, the effective cross-sectional area carrying current decreases, so resistance increases.
Long component leads act as unwanted inductors at VHF and above. Keeping them short minimizes inductive reactance.
Reactive power involves energy moving into and out of electric or magnetic fields. Current and voltage are 90 degrees out of phase.
At microwave frequencies, even short conductors represent significant fractions of a wavelength. Short connections reduce unwanted phase shift along the path.
Electrolytic capacitors have significant parasitic inductance, which makes them unsuitable as ideal capacitors at RF.
An inductor's turns have capacitance between them. That inter-turn capacitance resonates with the intended inductance and produces the inductor's self-resonant frequency.
A component self-resonates when its intended reactance interacts with an opposite-sign parasitic reactance.
The official Extra pool identifies skin effect as the primary cause of RF loss in film capacitors.
Ideal inductors and capacitors alternately store energy in magnetic or electric fields and return it to the circuit. They do not dissipate reactive power as heat.
The official Extra pool states that increasing a conductor's diameter increases its electrical length.
Only the resistor consumes real power. With 1 ampere through 100 ohms, P = I²R = 100 watts; the ideal inductive reactance consumes no real power.
Reactive power is wattless, nonproductive power associated with energy alternately stored and returned by inductors and capacitors.