E5 Electrical Principles

A. Resonance

At resonance in a series RLC circuit, the inductive and capacitive reactances cancel, so only the resistance limits the current and the current is large. That current flowing through each reactance develops I x XL across the inductor and I x XC across the capacitor, which can be Q times the applied voltage; the two voltages are opposite in phase, so they cancel in the total and the source never sees them.

• B. Capacitance alone does not raise the voltage; a capacitor's voltage exceeds the source voltage only near resonance, where the inductor cancels its reactance and the current becomes large.
• C. A low Q means resistance dominates the circuit, which limits the current and keeps the reactive voltages small; voltage magnification grows with high Q, not low Q.
• D. Resistance limits the current at resonance, so more resistance lowers the voltage across the reactances rather than raising it.

C. 3.56 MHz

Resonant frequency depends only on L and C: f = 1 / (2 x π x √(L x C)). With L in microhenries and C in picofarads, the result in megahertz is 1000 / (2 x π x √(50 x 40)) = 1000 / (2 x π x 44.72) = 1000 / 281.0 = 3.56 MHz. The 22-ohm resistance affects Q, not the resonant frequency.

• A. 44.72 is √(50 x 40), an intermediate step in the calculation, not the resonant frequency.
• B. 22.36 million is 1 / √(L x C) without the 2 x π; that is the angular frequency in radians per second, not the frequency in hertz.
• D. 1.78 MHz is half the correct value, the result of dividing by 4 x π instead of 2 x π.

D. Approximately equal to circuit resistance

At resonance, the inductive reactance and capacitive reactance of a series circuit are equal and opposite, so they cancel. The only opposition left is the circuit resistance, so the impedance is at its minimum and approximately equals R.

• A. High impedance at resonance is a property of parallel resonant circuits; in a series circuit the reactances cancel and leave only the resistance, the minimum impedance.
• B. The capacitive reactance is cancelled by an equal inductive reactance at resonance, so it does not set the impedance.
• C. The inductive reactance is cancelled by an equal capacitive reactance at resonance, so it does not set the impedance.

A. Approximately equal to circuit resistance

In a parallel RLC circuit at resonance, the inductor and capacitor branch currents are equal and opposite, so they cancel at the input and the LC pair acts like an open circuit. The source sees only the resistance in parallel with it, so the impedance rises to its maximum, which is approximately R.

• B. The two reactances cancel each other at resonance, so neither one sets the impedance; the resistance does.
• C. The impedance of a parallel circuit falls below R away from resonance, where one reactive branch draws current; at resonance the reactive branches draw no net current, so the impedance is R, not less.
• D. At resonance the reactive branches drop out and only the parallel resistance remains, so the impedance rises to R but no higher.

A. Matching bandwidth is decreased

Bandwidth and Q are inversely related: bandwidth = f / Q. Raising the Q of a matching network makes it more frequency selective, so it holds a good match over a narrower range of frequencies and needs retuning more often as the operating frequency changes.

• B. Higher Q narrows the bandwidth; a wider matching bandwidth comes from lower Q.
• C. Higher Q raises circulating currents and voltages, which can add loss in imperfect components, but that depends on component quality; the result that always follows from higher Q is a narrower bandwidth.
• D. A higher Q network is more selective, so it attenuates harmonics more, not less.

B. It is at a maximum

At resonance, energy swings back and forth between the inductor's magnetic field and the capacitor's electric field. The current circulating around the LC loop is about Q times the current drawn from the source and is at its maximum at resonance, even though the line current is at its minimum.

• A. The current drawn from the source (the line current) is at a minimum at resonance; the current circulating inside the tank is at a maximum.
• C. 1 / (2 x π x √(L x C)) is the formula for the resonant frequency, not for a current.
• D. 2 x π x √(L x C) is the period of one cycle at resonance, in seconds (the reciprocal of the resonant frequency), not a current.

A. Minimum

At resonance, the inductor and capacitor branch currents in a parallel circuit are equal and opposite, so they cancel at the input. The source supplies only the current for the resistance; the impedance is at its maximum, so the input current is at its minimum.

• B. Maximum current describes the current circulating inside the tank, or the input current of a series circuit at resonance, not the input current of a parallel circuit.
• C. R / L is the reciprocal of an RL time constant, with units of 1 per second; it is not a current.
• D. L / R is the time constant of an RL circuit, in seconds; it is not a current.

C. The voltage and current are in phase

At resonance, a series circuit's inductive and capacitive reactances cancel, leaving a purely resistive impedance. In a resistance, the voltage and current rise and fall together, so they are in phase (a phase angle of zero).

• A. Voltage leading current by 90 degrees describes a pure inductance, not a resonant circuit where the reactances cancel.
• B. Current leading voltage by 90 degrees describes a pure capacitance, not a resonant circuit where the reactances cancel.
• D. At series resonance, the voltages across the inductor and the capacitor are 180 degrees apart from each other, but the source voltage and current are in phase.

C. Resistance divided by the reactance of either the inductance or capacitance

In a parallel resonant circuit, the resistance is in parallel with the reactances, so a larger R loads the circuit less and gives a higher Q: Q = R / X, where X is the reactance of either the inductor or the capacitor (they are equal at resonance). This is the inverse of the series formula.

• A. Reactance divided by resistance (X / R) is the Q of a series resonant circuit, where resistance is in series with the reactances.
• B. Multiplying reactance by resistance gives a quantity in ohms squared, not the unitless ratio that Q is.
• D. XL x XC equals L / C, which is used to find the characteristic impedance of an LC circuit (the square root of L / C); it is not Q.

A. 7.12 MHz

Resonant frequency depends only on L and C: f = 1 / (2 x π x √(L x C)). With L in microhenries and C in picofarads, the result in megahertz is 1000 / (2 x π x √(50 x 10)) = 1000 / (2 x π x 22.36) = 1000 / 140.5 = 7.12 MHz. The 33-ohm resistance affects Q, not the resonant frequency.

• B. 23.5 kHz is far too low; 50 µH and 10 pF resonate at 7.12 MHz.
• C. 7.12 kHz has the right digits but the wrong prefix; the result is 7.12 MHz, 1000 times higher.
• D. 23.5 MHz is about 3.3 times too high; f = 1 / (2 x π x √(L x C)) with 50 µH and 10 pF gives 7.12 MHz.

C. 47.3 kHz

Half-power bandwidth equals the resonant frequency divided by Q: BW = f / Q = 7,100,000 Hz / 150 = 47,333 Hz, or 47.3 kHz. This is the full width between the two half-power (-3 dB) points.

• A. 157.8 Hz is half of 315.6 Hz, which comes from dividing by Q twice; the bandwidth divides the frequency by Q only once.
• B. 315.6 Hz is 7.1 MHz divided by Q twice (by 150²); the bandwidth divides the frequency by Q only once.
• D. 23.67 kHz is half the bandwidth, the distance from the center frequency to one half-power point, not the full width between the two points.

C. 31.4 kHz

Half-power bandwidth equals the resonant frequency divided by Q: BW = f / Q = 3,700,000 Hz / 118 = 31,356 Hz, or 31.4 kHz. This is the full width between the two half-power (-3 dB) points.

• A. 436.6 comes from multiplying 3.7 by 118; the bandwidth divides the frequency by Q instead.
• B. 218.3 kHz is half of 3.7 x 118; the bandwidth divides the frequency by Q rather than multiplying.
• D. 15.7 kHz is half the bandwidth, the distance from the center frequency to one half-power point, not the full width between the two points.

C. Internal voltages increase

At resonance, the voltage across the inductor and across the capacitor of a series circuit is Q times the applied voltage. Raising Q raises these internal voltages, so components in a high-Q circuit must be rated for voltages well above the source voltage.

• A. Q describes selectivity and loss, not part count; raising Q does not reduce the number of components needed.
• B. Parasitic effects, such as lead inductance and stray capacitance, come from component construction and layout; raising Q does not reduce them.
• D. Phase shift through a resonant circuit follows frequency in a predictable way; higher Q makes the phase change more steeply near resonance, but it does not become uncontrolled.

B. One time constant

One time constant (τ = R x C for an RC circuit) is the time a capacitor takes to charge to 63.2% of the applied voltage, or to discharge to 36.8% of its starting voltage. These figures come from the exponential curve: 1 - 1/e = 0.632 and 1/e = 0.368, where e is about 2.718.

• A. "Exponential rate of one" is not a standard term; the interval described is one time constant.
• C. "Exponential period" is not a standard term; the interval described is one time constant.
• D. "Time factor" is not a standard term; the interval described is one time constant.

D. B

Susceptance is represented by B. It is the reciprocal of reactance and the imaginary part of admittance, which is written Y = G + jB and measured in siemens.

• A. G represents conductance, the real part of admittance and the reciprocal of resistance.
• B. X represents reactance, the opposition of inductors and capacitors to AC, measured in ohms.
• C. Y represents admittance, the reciprocal of impedance, which combines conductance and susceptance.

B. Take the reciprocal of the magnitude and change the sign of the angle

Admittance is the reciprocal of impedance (Y = 1 / Z). In polar form, the reciprocal of a complex number has the reciprocal magnitude and the opposite angle, so an impedance of 50 ohms at 30 degrees becomes an admittance of 0.02 siemens at -30 degrees.

• A. Taking the reciprocal of an angle has no meaning here; the reciprocal applies to the magnitude, and the angle changes sign.
• C. Neither a square root nor a 180-degree shift is part of the conversion; Y = 1 / Z inverts the magnitude and reverses the sign of the angle.
• D. Squaring the magnitude and subtracting 90 degrees does not invert a complex number; Y = 1 / Z inverts the magnitude and reverses the sign of the angle.

D. 220 seconds

The time constant is τ = R x C. Two 220 µF capacitors in parallel add to 440 µF, and two 1 MΩ resistors in parallel give 0.5 MΩ, so τ = 500,000 ohms x 0.000440 farads = 220 seconds.

• A. 55 seconds treats the capacitors as if they were in series (110 µF) while using the parallel resistance (0.5 MΩ); capacitors in parallel add.
• B. 110 seconds halves the resistance without doubling the capacitance; in parallel, the capacitance doubles to 440 µF while the resistance halves to 0.5 MΩ.
• C. 440 seconds doubles the capacitance but leaves the resistance at 1 MΩ; two 1 MΩ resistors in parallel give 0.5 MΩ.

D. It is replaced by its reciprocal

Susceptance is the reciprocal of reactance, B = 1 / X, so converting a pure reactance to susceptance replaces its magnitude with its reciprocal. For example, 100 ohms of reactance becomes 0.01 siemens of susceptance.

• A. The magnitude does change: 100 ohms of reactance becomes 0.01 siemens of susceptance.
• B. The sign of the imaginary part does flip when 1 / (jX) is worked out, but a sign is not part of a magnitude; the magnitude becomes its reciprocal.
• C. A 90-degree shift describes the phase between voltage and current in a reactance; it is not what happens to the magnitude when reactance becomes susceptance.

C. The imaginary part of admittance

Admittance (Y), the reciprocal of impedance, is written in rectangular form as Y = G + jB. The real part G is conductance and the imaginary part B is susceptance, both measured in siemens.

• A. "Magnetic impedance" is not what susceptance means; the opposition to magnetic flux in a magnetic circuit is called reluctance.
• B. Ratios of electric and magnetic field strength describe a radio wave in space (the wave impedance, about 377 ohms in free space); susceptance is a circuit quantity.
• D. Transformer efficiency is output power divided by input power; it is unrelated to susceptance.

C. 14.0 degrees with the voltage lagging the current

The net reactance is X = XL - XC = 250 - 500 = -250 ohms, and the phase angle is arctan(X / R) = arctan(-250 / 1000) = -14.0 degrees. A negative angle means the circuit is net capacitive, so the voltage lags the current by 14.0 degrees.

• A. Both parts are wrong: a 68.2-degree angle would need a net reactance 2.5 times the resistance, and a leading voltage would need XL larger than XC.
• B. The angle is right but the direction is wrong; XC is larger than XL, so the circuit is capacitive and the voltage lags the current.
• D. The direction is right but the angle is wrong; a 68.2-degree angle would need a net reactance 2.5 times the resistance, and here it is only 250 ohms against 1000 ohms.

A. 63 degrees with the voltage lagging the current

The net reactance is X = XL - XC = 100 - 300 = -200 ohms, and the phase angle is arctan(X / R) = arctan(-200 / 100) = -63.4 degrees. A negative angle means the circuit is net capacitive, so the voltage lags the current by about 63 degrees.

• B. The angle is right but the direction is wrong; XC is larger than XL, so the circuit is capacitive and the voltage lags the current.
• C. Both parts are wrong: 27 degrees comes from inverting the ratio (arctan of 100 / 200), and the circuit is capacitive, so the voltage lags.
• D. The direction is right but the angle is wrong; 27 degrees comes from inverting the ratio (arctan of R / X instead of X / R).

D. Current leads voltage by 90 degrees

A capacitor's voltage builds up only as charge flows into it, so the current peaks first, a quarter cycle (90 degrees) before the voltage. The memory aid "ELI the ICE man" puts I (current) before E (voltage) in ICE, where C stands for capacitor.

• A. Voltage and current in phase describes a pure resistance, not a capacitor.
• B. A 180-degree relationship does not occur between voltage and current in an ideal capacitor; the shift is 90 degrees, with current leading.
• C. Voltage leading current by 90 degrees describes an inductor, the opposite of a capacitor.

A. Voltage leads current by 90 degrees

An inductor opposes changes in current, so the current builds up only after voltage is applied, and the voltage peaks a quarter cycle (90 degrees) before the current. The memory aid "ELI the ICE man" puts E (voltage) before I (current) in ELI, where L stands for inductor.

• B. Current leading voltage by 90 degrees describes a capacitor, the opposite of an inductor.
• C. A 180-degree relationship does not occur between voltage and current in an ideal inductor; the shift is 90 degrees, with voltage leading.
• D. Voltage and current in phase describes a pure resistance, not an inductor.

B. 27 degrees with the voltage leading the current

The net reactance is X = XL - XC = 75 - 25 = +50 ohms, and the phase angle is arctan(X / R) = arctan(50 / 100) = +26.6 degrees. A positive angle means the circuit is net inductive, so the voltage leads the current by about 27 degrees.

• A. The angle is right but the direction is wrong; XL is larger than XC, so the circuit is inductive and the voltage leads the current.
• C. Both parts are wrong: 63 degrees comes from inverting the ratio (arctan of 100 / 50), and the circuit is inductive, so the voltage leads.
• D. The direction is right but the angle is wrong; 63 degrees comes from inverting the ratio (arctan of R / X instead of X / R).

A. The inverse of impedance

Admittance (Y) is the inverse of impedance: Y = 1 / Z, measured in siemens. Impedance describes how strongly a circuit opposes AC; admittance describes how readily it passes AC.

• B. The gain of a field effect transistor is described by its transconductance, the change in drain current per change in gate voltage; it is not admittance.
• C. The inverse of reactance is susceptance, which is only the imaginary part of admittance.
• D. A conducting FET's resistance is its on-resistance, RDS(on), the resistance from drain to source; it is not admittance.

A. 0 - j100

Rectangular notation writes impedance as R ± jX: the real part is resistance and the imaginary part is reactance, positive for inductive and negative for capacitive. A pure capacitive reactance of 100 ohms has no resistance, so it is 0 - j100.

• B. 0 + j100 has a positive imaginary part, which is a pure inductive reactance of 100 ohms.
• C. 100 - j0 has a real part of 100 and no reactance, which is a pure resistance of 100 ohms.
• D. 100 + j0 has a real part of 100 and no reactance, which is a pure resistance of 100 ohms.

C. By magnitude and phase angle

Polar coordinates describe an impedance by its magnitude (the total opposition in ohms) and its phase angle (how far current and voltage are shifted from each other), written for example as 50 ohms at 30 degrees.

• A. R and X values (resistance and reactance) describe an impedance in rectangular form, not polar form.
• B. Real and imaginary parts describe an impedance in rectangular form, not polar form.
• D. Y and G are the symbols for admittance and conductance; they describe admittance, not the polar form of an impedance.

C. A positive 90 degree phase angle

A pure inductive reactance has no resistance and a positive imaginary part (+jX), so it points straight up from the origin: in polar form, its magnitude is X at a phase angle of +90 degrees. For example, 0 + j100 ohms is 100 ohms at +90 degrees.

• A. A +45-degree angle describes equal resistance and inductive reactance, such as 100 + j100 ohms, not a pure reactance.
• B. A -45-degree angle describes equal resistance and capacitive reactance, such as 100 - j100 ohms.
• D. A -90-degree angle describes a pure capacitive reactance.

D. Logarithmic

Frequency response graphs usually show gain or attenuation in decibels, a logarithmic scale, so large and small ratios fit on the same graph and equal steps represent equal ratios. The frequency axis is often logarithmic as well.

• A. A linear scale crowds small values near zero, so responses spanning many orders of magnitude, such as a filter's stopband, become hard to read.
• B. A scatter plot is a kind of graph made of individual data points, not a type of axis scale.
• C. There is no "random" axis scale; axis divisions follow a defined rule.

C. Phasor diagram

A phasor diagram draws AC quantities, such as impedances, voltages and currents, as arrows whose length is the magnitude and whose angle is the phase at a given frequency. It shows at a glance how resistance and reactance combine.

• A. A Venn diagram shows overlapping sets in logic and mathematics, not phase relationships.
• B. The near field is the region close to an antenna; it is not a diagram of impedance phase relationships.
• D. The far field is the region well away from an antenna where its radiation pattern is measured; it is not a diagram of impedance phase relationships.

B. 50 ohms resistance in series with 25 ohms capacitive reactance

In rectangular form, the real part of an impedance is resistance and the imaginary part is reactance, with a negative sign for capacitive reactance. So 50 - j25 ohms is 50 ohms of resistance in series with 25 ohms of capacitive reactance.

• A. Inductive reactance would carry a positive sign, 50 + j25; the minus sign marks capacitive reactance.
• C. 25 ohms of resistance with 50 ohms of inductive reactance would be written 25 + j50; the resistance is the real part, 50, and the sign is negative.
• D. 25 ohms of resistance with 50 ohms of capacitive reactance would be written 25 - j50; the real part, here 50, is the resistance.

D. On the horizontal axis

On rectangular impedance coordinates, the horizontal axis is resistance and the vertical axis is reactance. A pure resistance has zero reactance, so it lies on the horizontal axis at its resistance value.

• A. The vertical axis is where pure reactances lie, with no resistance.
• B. A 45-degree line through the origin holds impedances with equal resistance and reactance, such as 100 + j100 ohms.
• C. A horizontal line above the axis holds impedances with the same positive (inductive) reactance and differing resistance, not a pure resistance.

D. Polar coordinates

Polar coordinates express an impedance as a magnitude and a phase angle, so they show the phase angle of a circuit with resistance and reactance directly, as the angle of a line from the origin.

• A. The Maidenhead grid is the geographic locator system amateurs use to describe station locations, not a way to show phase angle.
• B. A "Faraday grid" is not a coordinate system; Faraday's name is attached to electromagnetic induction and to the shielding enclosure called a Faraday cage.
• C. Elliptical coordinates are a mathematical system built on ellipses and hyperbolas; they are not used to display circuit phase angles.

A. The X axis represents the resistive component, and the Y axis represents the reactive component

On a rectangular impedance graph, the X (horizontal) axis is resistance and the Y (vertical) axis is reactance, with inductive reactance above the axis and capacitive reactance below it. An impedance R + jX is plotted at the point (R, X).

• B. The axes are reversed; resistance goes on the horizontal axis and reactance on the vertical axis.
• C. Phase angle and magnitude are polar quantities; they are not the axes of a rectangular graph.
• D. Magnitude and phase angle are polar quantities; they are not the axes of a rectangular graph.

B. Point 4

With f in megahertz and C in picofarads, the capacitor's reactance is XC = 1,000,000 / (2 x π x f x C) = 1,000,000 / (2 x π x 14 x 38) = 299 ohms, about 300 ohms. Capacitive reactance is negative, so the impedance is 400 - j300 ohms, the point at 400 on the horizontal axis and -300 on the vertical axis: Point 4.

• A. Point 2 is at 400 + j300, a resistance with inductive reactance; a capacitor puts the point below the horizontal axis.
• C. Point 5 is at -400 - j300, which has negative resistance; a resistor puts the point to the right of the vertical axis.
• D. Point 6 lies on the horizontal axis at 400 ohms, a pure resistance with no reactance; the capacitor adds about 300 ohms of capacitive reactance.

B. Point 3

With f in megahertz and L in microhenries, the inductor's reactance is XL = 2 x π x f x L = 2 x π x 3.505 x 18 = 396 ohms, about 400 ohms. Inductive reactance is positive, so the impedance is 300 + j400 ohms, the point at 300 on the horizontal axis and +400 on the vertical axis: Point 3.

• A. Point 1 is at 300 - j400, a resistance with capacitive reactance; an inductor puts the point above the horizontal axis.
• C. Point 7 is at -300 - j400, which has negative resistance and capacitive reactance; neither fits a resistor and an inductor.
• D. Point 8 sits close to the horizontal axis at 300 ohms, showing very little reactance; the inductor adds about 396 ohms of inductive reactance.

A. Point 1

With f in megahertz and C in picofarads, the capacitor's reactance is XC = 1,000,000 / (2 x π x f x C) = 1,000,000 / (2 x π x 21.2 x 19) = 395 ohms, about 400 ohms. Capacitive reactance is negative, so the impedance is 300 - j400 ohms, the point at 300 on the horizontal axis and -400 on the vertical axis: Point 1.

• B. Point 3 is at 300 + j400, a resistance with inductive reactance; a capacitor puts the point below the horizontal axis.
• C. Point 7 is at -300 - j400, which has negative resistance; a resistor puts the point to the right of the vertical axis.
• D. Point 8 sits close to the horizontal axis at 300 ohms, showing very little reactance; the capacitor adds about 395 ohms of capacitive reactance.

A. Resistance increases as frequency increases because RF current flows closer to the surface

As frequency increases, RF current crowds toward the surface of a conductor and flows in a thinner and thinner layer (the skin depth). Less of the conductor's cross section carries current, so its effective resistance rises with frequency.

• B. Electron mobility does not increase with frequency, and resistance rises rather than falls; skin effect confines the current to a thin surface layer.
• C. The resistance of most metals does rise with temperature, but that is the temperature coefficient of resistance, a separate effect; skin effect depends on frequency.
• D. Resistance that falls as temperature rises (a negative temperature coefficient) describes semiconductors and some special materials, not skin effect.

B. To minimize inductive reactance

Every length of wire has inductance, and inductive reactance rises with frequency (XL = 2 x π x f x L). At VHF and above even a few nanohenries matter: 10 nH of lead inductance has about 9.4 ohms of reactance at 150 MHz, which can detune circuits and create unwanted coupling or resonances.

• A. Lead length has little to do with how fast a component heats or cools, and a longer thermal time constant is not a goal at VHF.
• C. Short leads are used to reduce unwanted reactance, not to extend component life.
• D. Only minimizing inductive reactance applies; the thermal and lifetime reasons are not why leads are kept short at VHF.

C. They are 90 degrees out of phase

Reactive power flows in inductors and capacitors, where the current and voltage are 90 degrees out of phase. Energy flows into the reactance during one quarter cycle and back out during the next, so the average power is zero.

• A. Out of phase is true but incomplete; reactive power comes from current and voltage that are exactly 90 degrees apart.
• B. Current and voltage in phase describes real power in a resistance, which is dissipated as heat.
• D. A 45-degree angle describes a load with equal resistance and reactance, which has equal amounts of real and reactive power, not purely reactive power.

B. To reduce phase shift along the connection

At microwave frequencies the wavelength is very short, so even a short connection is a large fraction of a wavelength and shifts the phase of the signal. At 10 GHz the free-space wavelength is 3 cm, so 1 cm of connection is a third of a wavelength, 120 degrees, and more in a slower circuit-board medium.

• A. Short connections do not add resistance; neutralization cancels unwanted feedback in amplifiers and is unrelated to connection length.
• C. Short connections reduce stray capacitance and inductance; they do not increase compensating capacitance.
• D. Noise figure depends mainly on the first amplifier stage and any loss ahead of it; short connections are needed because even short lengths are a large fraction of a wavelength at microwaves.

C. Inductance

Electrolytic capacitors are built from long strips of foil rolled into a cylinder, which gives them substantial internal inductance. At RF the inductive reactance outweighs the capacitive reactance, so the part no longer behaves like a capacitor.

• A. Skin effect raises the resistance of all conductors at RF; it is not what makes electrolytic capacitors unsuitable.
• B. Capacitance is what a capacitor is meant to provide; extra shunt capacitance would not stop it working as a capacitor at RF.
• D. Electrolytic capacitors do have relatively high leakage current, but that is a DC effect; it is not what prevents their use at RF.

D. Inter-turn capacitance

Adjacent turns of a coil act as small capacitors. Together this inter-turn (distributed) capacitance resonates with the coil's inductance at the self-resonant frequency; above that frequency the inductor behaves like a capacitor.

• A. Skin effect raises the resistance of the wire at RF, which lowers Q; it does not create a resonance.
• B. Dielectric loss in coil forms or insulation wastes energy and lowers Q; resonance requires a capacitance, not a loss.
• C. Coupling transfers energy between a coil and nearby conductors; self-resonance comes from the coil's own inter-turn capacitance.

B. The component’s nominal and parasitic reactance

Every real component has the reactance it is designed to have (its nominal reactance) plus unintended parasitic reactance of the opposite kind: a capacitor has lead inductance, and an inductor has inter-turn capacitance. At the frequency where the two are equal and opposite, the component is self-resonant.

• A. Resistance cannot resonate with reactance; resonance needs an inductive and a capacitive reactance that cancel.
• C. An inductance and a capacitance are involved, but self-resonance specifically pairs the component's intended reactance with its unintended parasitic reactance, which this choice does not describe.
• D. Electrical length and impedance describe transmission lines and antennas, not the cause of a component's self-resonance.

D. Skin effect

Film capacitors use plastic dielectrics, such as polypropylene, with very little dielectric loss. At RF, skin effect confines the current to a thin surface layer of the metal plates and leads, raising their resistance, and this resistive loss becomes the main loss.

• A. Inductance in the leads and plates sets a film capacitor's self-resonant frequency, but an ideal inductance stores and returns energy rather than dissipating it.
• B. Film dielectrics are chosen because their dielectric loss is low, so at RF the conductor loss from skin effect is the larger loss.
• C. Self-discharge is the slow loss of stored charge through leakage over seconds to hours, a DC effect rather than an RF loss.

B. Energy is stored in magnetic or electric fields, but power is not dissipated

An ideal inductor stores energy in its magnetic field and an ideal capacitor stores energy in its electric field. Each stores energy during one part of the AC cycle and returns it to the circuit during the next, so no power is dissipated.

• A. Only resistance turns energy into heat, and ideal inductors and capacitors have no resistance.
• C. Coulomb forces act between electric charges; they do not cancel reactive power, which is energy stored and returned each cycle.
• D. Energy does go into building the fields, but it is returned to the circuit when the fields collapse, so none is dissipated.

D. It increases

As a conductor's diameter increases relative to its length, it resonates at a lower frequency than a thinner conductor of the same physical length, so its electrical length increases. This is why a half-wave dipole made of thick tubing is cut somewhat shorter than one made of thin wire for the same frequency.

• A. Thickness does affect electrical length; a thick element resonates at a lower frequency than a thin element of the same length.
• B. The effect is predictable, not random: electrical length increases steadily as diameter increases.
• C. The effect is the opposite; a thicker conductor is electrically longer, which is why thick elements are cut shorter for a given frequency.

B. 100 watts

Only resistance consumes real power; reactance stores and returns energy. The real power is P = I² x R = (1 A)² x 100 ohms = 100 watts.

• A. 70.7 watts multiplies the 100 watts by the 0.707 power factor a second time; the power factor converts apparent power to real power, and 141.4 VA x 0.707 = 100 W.
• C. 141.4 is the apparent power, I² x Z = (1 A)² x 141.4 ohms, in volt-amperes; it includes reactive power, which is not consumed.
• D. 200 watts adds the resistance and reactance as if both consumed power; the 100 ohms of inductive reactance consumes none.

D. Wattless, nonproductive power

Reactive power is the energy that flows back and forth between the source and the inductors or capacitors each cycle. It is stored and returned, not consumed, so it is called wattless or nonproductive power and is measured in volt-amperes reactive (VAR) rather than watts.

• A. Q, the quality factor, is a ratio of reactance to resistance; it does not consume power, and reactive power is not consumed at all.
• B. Power lost in an inductor's wire resistance is real power, dissipated as heat.
• C. Ideal inductors and capacitors store and return energy each cycle rather than consuming it, which is why reactive power is called wattless.