Admittance (Y), measured in siemens (S), is the reciprocal of impedance (Z). While impedance quantifies the opposition to the flow of AC, admittance can be said to quantify the ease of flow.

```Y = 1 / Z
```

The exams ask you to convert between impedance and admittance. This sounds easy until you remember that impedance and admittance are complex numbers. Still, when you use polar coordinates, it's not so bad. Just take the reciprocal of the magnitude and change the sign of the angle.

`1 / (r ∠ θ) = (1 / r) ∠ -θ`

You also need to remember how to convert from polar to rectangular coordinates:

```x = r cos(θ)
y = r sin(θ)
```

## Bandwidth

Bandwidth is the amount of frequency spectrum occupied by a signal. A simple signal like Morse code has a lot smaller bandwidth than a complex signal like television. When choosing a transmitting frequency, you should allow sufficient frequency separation to avoid interfering with other signals. Of course, the required frequency separation depends on the bandwidth of the signals.

Bandwidth varies considerably based on factors such as information rate, frequency shift, etc., but some some typical values are:

```Signal type                      Typical bandwidth
PSK31                           30-70 Hz
Morse code (CW)                 10-150 Hz
MFSK16                          316 Hz
HF Packet (300 baud)            500 Hz
Single sideband voice (SSB)     2-3 kHz
Amplitude modulated voice (AM)  6 kHz
VHF/UHF Packet (1200-9600 baud) 5-20 kHz
Frequency modulated voice (FM)  5-20 kHz
Fast-scan television            6 MHz
```

## Component arithmetic

Resistors

```Series
R = R1 + R2 + R3

Parallel
1/R = 1/R1 + 1/R2 + 1/R3
```

Capacitors

```Series
C = C1 + C2 + C3

Parallel
1/C = 1/C1 + 1/C2 + 1/C3
```

When shopping for a receiver, three key factors to consider are:

```    * Selectivity — filtering out adjacent signals.
* Sensitivity — the signal-plus-noise to noise ratio
* Stability   — staying exactly on frequency.
```

## Frequency Modulation Formulas

```                   Dmax
deviation-ratio = ------
Mmax

Dmax is the maximum deviation of the carrier
Mmax is the maximum frequency of the modulating signal

Dmax
modulation-index = ------
m

m is the instantaneous frequency of the modulating signal

bandwidth = 2 x ( Dmax  + Mmax )
```

## Impedance of a Parallel RLC Circuit

We've calculated the impedance of series RLC circuits. Now parallel. Read the question carefully, as it's easy to miss that one little word.

The impedance of parallel RLC circuits is harder, as it uses the reciprocals. There are only two of these in the question pool, and you have only a 9% chance of seeing one on your actual exam, so many students either memorize the answers or just skip them altogether.

```r = 1 / ( sqrt ( (1/R)^2 + (1/X)^2 ) )
θ = arctan(R/X)
```

## ITU Emission Designator

The ITU emission designator classifies radio emissions. The designators mentioned on the exams consist of three symbols:

```   * 1st symbol — type of modulation of the main carrier
* 2nd symbol — nature of signals modulating the main carrier
* 3rd symbol — type of information to be transmitted
```

There are lots of choices for each symbol, but you only need to know the highlighted ones.

```1st symbol — Type of Modulation of the Main Carrier

Symbol	Description
N 	Emission of an unmodulated carrier
Emission in which the main carrier is amplitude-modulated (including cases where sub-carriers are angle-modulated):
A 	Double-sideband
H 	Single-sideband, full carrier
R 	Single-sideband, reduced or variable level carrier
J 	Single-sideband, suppressed carrier
B 	Independent sidebands
C 	Vestigial sideband
Emission in which the main carrier is angle-modulated:
F 	Frequency modulation
G 	Phase modulation
D 	Emission in which the main carrier is amplitude and angle-modulated either simultaneously or in a pre-established sequence
Emission of pulses:
P 	Sequence of unmodulated pulses
A sequence of pulses:
K 	Modulated in amplitude
L 	Modulated in width/duration
M 	Modulated in position/phase
Q 	In which the carrier is angle-modulated during the period of the pulse
V 	Which is a combination of the foregoing or is produced by other means
W 	Cases not covered above, in which an emission consists of the main carrier modulated, either simultaneously or in pre-established sequence, in a combination of two or more of the following modes: amplitude, angle, pulse
X 	Cases not otherwise covered

2nd symbol — Nature of Signals Modulating the Main Carrier

You don't need to know much about the 2nd symbol, but here are the values anyway:
Symbol	Description
0 	No modulating signal
1 	A single channel containing quantized or digital information without the use of a modulating sub-carrier, excluding time-division multiplex
2 	A single channel containing quantized or digital information with the use of a modulating sub-carrier, excluding time-division multiplex
7 	Two or more channels containing quantized or digital information
9 	Composite system with one or more channels containing quantized or digital information, together with one or more channels containing analog information
X 	Cases not otherwise covered

3rd symbol — Type of Information to be Transmitted

Symbol	Description
N 	No information transmitted
A 	Telegraphy - for aural reception
B 	Telegraphy - for automatic reception
C 	Facsimile
D 	Data transmission, telemetry, telecommand
F 	Television (video)
W 	Combination of the above
X 	Cases not otherwise covered.
```

## Half-Power Bandwidth

Take another look at the graphs demonstrating the Q of a circuit. A series R-L-C circuit stops, and a parallel passes, signals at or near a particular frequency. The Q describes the "sharpness" of the frequency response of a resonant circuit — in other words, how near.

Half-power bandwidth describes the same thing, but in slightly different terms. As the name suggests, half-power bandwidth is the width of the frequency range in which the series circuit stops, or parallel passes, at least half the power as at the resonant frequency. Half-power bandwidth is closely related to Q:

```    Half-power bandwidth = fR / Q
```

```Total reactance: X = Xl - Xc

/ X \
angle = arctan |---|
\ R /

negative angle: voltage lagging current
```

## Power Factor

In an AC circuit with a non-zero phase angle, the true power, the power actually consumed by the circuit, is less than the apparent power, the product of the voltage and current. This is because some of the power into the circuit, rather than being consumed, is temporarily stored by inductors or capacitors and later returned to the source. To distinguish it from true power, apparent power is often specified in volt-amperes (VA) rather than watts.

The power factor is the ratio of true power to apparent power. It is always a number between 0 and 1, and can be calculated from the phase angle:

```    power-factor = true-power / apparent-power = cos( θ )
```

## Quality Factor (Q)

Resonant circuits are not perfect. They resonate to some extent at frequencies slightly off their resonant frequency. The quality factor, or Q factor, or simply the Q, is the measure of the "quality" of a resonant system. A higher Q circuit shows a narrower peak or dip in the frequency response function.

```Series R-L-C Circuit
Q = reactance / resistance
= (2 pi fR L) / R

pi = 3.14
fR = resonant frequency

Parallel R-L-C Circuit
Q = resistance / reactance
= R / (2 pi fR L)
```

 antenna wave length spaced apart feed phase pattern 1/4 1/8 in phase 1/4 1/4 in phase elliptical 1/4 1/2 in phase figure 8 broadside 1/4 1/8 90 degrees 1/4 1/4 90 degrees unidirectional cardioid 1/4 1/2 90 degrees 1/4 1/8 180 degrees figure 8 end-fire in line 1/4 1/4 180 degrees figure 8 end-fire in line 1/4 1/2 180 degrees figure 8 end-fire in line

## Reactance

```Xc = 1 / 2 π f C
Xl = 2 π f C
```

## Relationship between voltage and current through inductors and capacitors

Current leads voltage by 90 degrees through a capacitor.

Voltage leads current by 90 degrees through an inductor.

## Resonant Frequency of an R-L-C circuit

```           1
fR = -------------
2 pi sqrt(LC)

fR = resonant frequency (in hertz)
L = inductance (in henrys)
```

## Time Constant

When charging a capacitor through a resistor, it takes time for the charge to build up. One time constant is the time required to charge to 63.2% of the supply voltage.

During each additional time constant, the capacitor charges by 63.2% of the remaining difference in charge.

When discharging a capacitor through a resistor, it takes time for the charge to dissipate. One time constant is the time required to discharge to 36.8% of the initial voltage.

During each additional time constant, the capacitor discharges to 36.8% of the remaining charge.

```T = R x C

T = time in seconds
R = resistance in ohms
```

For inductors one time constant is the amount of time required for the current to build up to 63.2% of the full current.

For inductors one time constant is the amount of time required for the current to be reduced to 36.8% of the initial current.

```     L
T = ---
R

T = time in seconds
L = inductance in henrys
R = resistance in ohms
```

## Toroidal Inductors

```For a powdered-iron toroidal core inductor:
N = 100 * sqrt ( L / AL )

N = the required number of turns
L = the desired inductance (in μH)
AL = the inductance index of the core (in μH/100 turns)

For a ferrite toroidal core inductor:
N = 1000 * sqrt ( L / AL )

N = the required number of turns
L = the desired inductance (in mH)
AL = the inductance index of the core (in mH/1000 turns)
```