Chapter 20: Frequency Translation

The Colpitts oscillator

Frequency translation circuits make a new frequency from signals at other frequencies: multipliers, dividers, synthesisers and mixers. An oscillator such as the Colpitts only generates a signal; the divider is a translation circuit too.

A perfect sine wave

Only a perfect sine wave is free of harmonics; every other shape contains some. A triangle wave looks smooth, but it still contains odd harmonics of its fundamental.

21,15 MHz

The nth harmonic is n times the fundamental: 3 × 7,05 = 21,15 MHz. Counting three steps above the fundamental gives 4 × 7,05 = 28,2 MHz, which is the fourth harmonic.

18 MHz

The fifth harmonic is 5 × 3,65 = 18,25 MHz, so 18 MHz is nearest. Taking it as the fifth step above the fundamental gives 6 × 3,65 = 21,9 MHz, nearest the 22 MHz option.

The fourth harmonic

Since 14,4 ÷ 3,6 = 4, it is the fourth harmonic; the fundamental itself counts as the first. Calling it the third counts only the steps above the fundamental.

It distorts the input, then a bandpass filter selects one harmonic

The input is distorted on purpose so that it is rich in harmonics, and a bandpass filter tuned to the wanted harmonic passes it alone. A lowpass filter would also pass the fundamental and the lower harmonics.

Non-linearly, so that it creates harmonics

A multiplier must distort the signal to create harmonics, so its stage is run non-linearly: in the manual’s doubler the diodes do this, and a tripler may use class C. A linear stage adds no harmonics to select.

144 MHz

Each stage multiplies the frequency it is given: 12 × 3 = 36 MHz, then 36 × 4 = 144 MHz, an overall factor of 12. Adding the factors, 3 + 4 = 7, gives the wrong 84 MHz.

16 MHz

The two triplers multiply by 3 × 3 = 9, so the crystal runs at 145,8 ÷ 9 = 16,2 MHz, nearest 16 MHz. Adding the factors to 6 gives 24,3 MHz, nearest the 24 MHz option.

A doubler, as its output repeats twice per input cycle

A full-wave rectified sine repeats twice in every input cycle, so it is rich in the second harmonic and suits a doubler. It has almost no third harmonic; the manual uses class C for its tripler.

Its output has a strong third harmonic

A class C stage conducts in short pulses, for less than half of each cycle, so its output has a high third-harmonic content that suits a tripler. Conducting for more than half of each cycle is class AB, or A.

It passes it straight through to ground

For DC a coil is almost a short circuit, so L1 conducts the DC part to ground while L1-C1 resonates at the output frequency. A coil passes DC; blocking DC is what a capacitor does.

They block DC but pass the RF signal

C2 stops L1 shorting Q1’s base bias, set by R1 and R2, to ground, and C4 keeps Q1’s collector DC out of the next stage; both pass the RF. Neither is wired as a supply decoupler.

To weaken the unwanted harmonics further

Tuned to the output frequency, L2-C3 further weakens the harmonics left after L1-C1, such as three and four times the input. It is a filter; a stage that oscillated would be useless as a multiplier.

A sine wave at twice the input frequency

The tuned circuits remove the DC and the other harmonics, leaving a sine wave at twice the input frequency. A square wave would be full of harmonics, which those tuned circuits are there to remove.

As a common-emitter amplifier

Q1 takes its input at the base and gives its output at the collector, through L2-C3 and C4, with R1, R2 and R3 setting its bias: a common-emitter amplifier. An emitter follower takes its output from the emitter.

A varactor multiplier

At VHF and UHF a varicap (varactor) diode is often the non-linear part of a multiplier, and tuned circuits select the wanted harmonic. A varicap-tuned oscillator, such as a VCO, makes its own signal; it takes no RF input.

A phase-modulated signal

The manual allows multipliers for unmodulated carriers, CW, FM and phase modulation. An AM signal, even with its full carrier, holds a range of frequencies that the distortion would intermodulate.

A double-sideband AM signal

An AM signal holds several frequencies at once, which the multiplier’s distortion would intermodulate. FM, CW and a plain carrier are safe, though multiplying an FM signal multiplies its deviation too.

It would create IMD too close to filter out

The distortion makes the many frequencies in an SSB signal intermodulate, and the IMD products fall too near the wanted ones to be filtered out. A multiplier works on whatever frequencies it is fed, carrier or not.

180 Hz

A multiplier multiplies every frequency it is fed, so a 60 Hz error at 7 MHz becomes 3 × 60 = 180 Hz at 21 MHz. The error is not left at 60 Hz; it grows with the multiplication factor.

±2,1 kHz

Multiplying the frequency multiplies every swing in it too: 6 × 350 Hz = 2,1 kHz either side, which is how an FM signal’s deviation grows in a multiplier chain. The swing does not stay at ±350 Hz.

No filter could pick out the wanted harmonic from its neighbours

The 99th, 100th and 101st harmonics lie only 1% apart, so no practical filter could separate them, and a multiplier makes most harmonics. Transistor speed is not the limit the manual gives.

A square wave

A digital divider switches between two levels, so its output is typically a square wave, rich in harmonics. A sawtooth ramps steadily up and drops back, which a switching circuit does not do.

75 kHz, 125 kHz and 175 kHz

The divider’s square wave is strongest in the odd harmonics, at three, five and seven times its frequency: 75, 125 and 175 kHz. A symmetrical square wave has almost no even harmonics.

A lowpass filter, or a bandpass one tuned to it

The harmonics all lie above the wanted frequency, so a lowpass filter, or a bandpass filter tuned to the wanted frequency, leaves a clean sine. A bandstop filter there would remove the very signal that is wanted.

12 kHz

Each divider divides the frequency it is fed: 6 MHz ÷ 10 = 600 kHz, then 600 kHz ÷ 50 = 12 kHz, an overall ÷500. Leaving out the first divider gives 6 MHz ÷ 50 = 120 kHz.

60 kHz

The overall division is 10 × 12 = 120, and 7 MHz ÷ 120 = 58,3 kHz, so 60 kHz is nearest. Leaving out the ÷10 gives 7 MHz ÷ 12 = 583 kHz, nearest the 580 kHz option.

0,05 Hz

A divider divides every frequency it is fed, errors included: 50 Hz ÷ 1 000 = 0,05 Hz. The output is wrong by the same fraction, five parts in a million, not by the same 50 Hz.

It drifts with heat and jumps if knocked

A VFO is hard to make really stable: its frequency drifts as the temperature changes and can jump if it is knocked. A tiny tuning range is the crystal oscillator’s drawback, not the VFO’s.

Its frequency can be pulled only slightly

A crystal oscillator is very stable against heat and knocks, but its frequency can be moved only a little, far less than a band. Drifting with temperature is the VFO’s weakness, which the crystal avoids.

A phase locked loop

Modern transceivers are mostly built around phase locked loop synthesisers, which hold a VCO at N times a crystal reference. A phase-shift oscillator simply oscillates; nothing ties it to a reference.

The VCO’s output, divided by N

The VCO’s output passes through the ÷N divider, and the comparator matches that against the reference. With the reference divided by N instead, the output would settle at a fraction of the reference.

It gives a voltage set by its inputs’ phase difference

The comparator compares the phases of the reference and the divided VCO signal and gives a voltage that depends on their difference. Dividing by N is the divider’s job, and it divides the VCO’s output, not the reference.

To keep the comparator’s fast ripple away from the VCO

Most phase comparators also give out ripple at their input frequencies; the lowpass filter removes it, leaving the slow phase-error voltage to steer the VCO. It filters the control voltage, not the output signal.

Negative feedback

Any drift of the VCO produces a comparator voltage that pushes it back the other way, which is negative feedback. Positive feedback is what makes an oscillator start and keep oscillating.

It is brought back to its set frequency

The drift changes the phase difference at the comparator, and the new voltage brings the VCO back down to N times the reference. Pushing it further away would be positive feedback, and it would never settle.

Nearly as stable as the reference itself

The VCO is held at N times the reference, so its stability nearly matches the crystal’s, though it adds some phase noise. The circuit cannot make it more stable than the crystal it follows.

7,035 MHz

The VCO settles at N times the reference: 70 350 × 100 Hz = 7 035 000 Hz = 7,035 MHz. Dividing N by the reference instead gives the wrong 703,5 Hz.

11 642

N = output ÷ reference = 145 525 000 Hz ÷ 12 500 Hz = 11 642. Taking the reference as 1,25 kHz, one decimal place out, gives the wrong 116 420.

140 000 to 144 000

N = output ÷ reference: 7 000 000 Hz ÷ 50 Hz = 140 000, and 7 200 000 ÷ 50 = 144 000. Leaving the output in kilohertz gives the wrong 140 to 144.

40

Each step of N moves the output by the reference, 25 Hz, so 1 kHz needs 1 000 ÷ 25 = 40 steps. Taking each step of N as 1 Hz gives 1 000, the step-size slip.

1,5 kHz

Each unit of N is worth one reference step of 10 Hz, so 150 × 10 Hz = 1 500 Hz = 1,5 kHz. Taking each unit of N as 1 Hz gives only 150 Hz.

A microprocessor, from the user’s choice

The dial is read by a microprocessor, which sets N and with it the output frequency. The phase comparator only gives a phase-error voltage to steer the VCO; it never chooses N.

A quartz crystal for 10 Hz would be impractically large

The lower a crystal’s frequency, the bigger it must be, and 10 Hz is far too low for a crystal of sensible size. Thin, fragile crystals are the high-frequency ones, so the crystal runs higher and is divided down.

3 kHz

12 800 000 Hz ÷ 4 096 = 3 125 Hz, so 3 kHz is nearest. Slipping one decimal place gives about 31 kHz, nearest the 30 kHz option.

Small steps need a low filter cutoff, which responds slowly

Small steps need a low reference frequency, which needs a low cutoff in the lowpass filter, and that limits how fast the synthesiser can follow a change. Nothing makes the VCO visit every step on its way.

200 Hz

The filter must remove the comparator’s outputs at the 2 kHz reference and above, so its cutoff lies below the reference: 200 Hz suits. A 4 kHz cutoff would let the 2 kHz components through to the VCO.

A multiple-loop synthesiser

The manual calls this a multiple-loop synthesiser: a coarse loop with a high reference makes the big jumps, and a fine loop with a small step fills in. No multiplier chain is involved.

It spoilt both received and transmitted signals

Phase noise spreads the synthesiser’s signal, and since a transceiver uses it on receive and on transmit, it degraded both. Slow drift with temperature is a VFO fault, not phase noise.

Tiny rapid wobbles in the output’s phase and frequency

As the circuit keeps adjusting the VCO to hold it to the reference, its phase and frequency wobble very slightly: that is phase noise. Harmonics are separate signals at multiples of the frequency, not noise.

As fuzz on both sides of the signal line

Phase noise widens the spectrum: instead of a single clean line there is fuzz above and below it. Clean, separate extra lines would be spurious signals, a different problem.

2 sin A cos B = sin(A + B) + sin(A − B)

With A = 2πf1t and B = 2πf2t, multiplying two waves gives waves at f1 + f2 and f1 − f2 added together. The minus form is the identity for 2 cos A sin B instead.

9 MHz and 11 MHz

The inputs satisfy f1 + f2 = 20 MHz and f1 − f2 = 2 MHz, so f1 = 11 MHz and f2 = 9 MHz. The 2 MHz and 20 MHz signals are the products, not the inputs.

5,25 MHz

The sum product is 9 MHz + fLO = 14,25 MHz, so fLO = 14,25 − 9 = 5,25 MHz. An LO of 23,25 MHz would also give 14,25 MHz, but as the difference, not the sum.

144,5 MHz

The sum product is 116 + 28,5 = 144,5 MHz, which lands in the 2 m band. The difference, 116 − 28,5 = 87,5 MHz, is also produced, but it is not the sum that was asked for.

10,7 MHz

The difference product is 17,85 − 7,15 = 10,7 MHz. Adding them gives 25 MHz, the sum product, which is also present but is not the difference asked for.

14 MHz

The difference is 21,1 − 7,45 = 13,65 MHz, so 14 MHz is nearest. Adding the two instead gives the sum, 28,55 MHz, nearest the 29 MHz option.

45 kHz

The difference is 145,000 − 144,955 = 0,045 MHz, and 0,045 MHz × 1 000 = 45 kHz. Moving the decimal comma one place too far gives 450 kHz, ten times too much.

7 MHz and 13 MHz

13 − 10 = 3 and 10 − 7 = 3: both give a 3 MHz product. The second sits on the LO’s other side (receivers call it the image), not at 13 + 2 × 3 = 19 MHz.

It is hard to multiply accurately without distortion

Multiplying two sine waves accurately without adding distortion is hard, so switching mixers chop one signal with the other instead. It is the switching mixer, not true multiplication, that needs a strong LO.

It uses one input to flip the other’s polarity

The LO switches the diodes so that, on alternate half cycles, the RF reaches the IF port with its polarity reversed. Accurate multiplication of sine waves is exactly what switching mixers avoid.

It changes to the opposite polarity

The other pair of diodes now conducts and earths the opposite end of T1’s winding, so the same RF voltage drives the IF port the other way. The LO thus flips the RF’s polarity every half cycle.

The centre tap of the RF transformer’s winding

The IF port is the centre tap of T1’s secondary, the RF transformer. The centre tap of T2, the LO transformer, is earthed, which is what the conducting diode pair is balanced against.

Intermediate frequency

IF stands for intermediate frequency, a name that makes sense in receivers, where the mixer’s output lies between the antenna and audio stages. The IF port is the output, not an input.

So the conducting pair’s midpoint sits at earth

With identical forward voltages, the junction of the conducting pair sits at the potential of the LO transformer’s centre tap, which is earth, and the mixer stays balanced. The diodes always conduct in pairs.

Multiplying it by a ±1 square wave

Multiplying by +1 leaves the signal as it is and multiplying by −1 reverses it, so the switching is multiplication by a ±1 square wave. Adding a square wave would not mix the two signals at all.

43 MHz

The harmonic and the RF give 24 + 19 = 43 MHz (and 24 − 19 = 5 MHz). 57 MHz is three times the RF, while 27 MHz and 11 MHz come from the LO’s fundamental, 8 MHz.

By filters placed after the mixer

Unwanted mixing products, including those from the LO’s harmonics in a switching mixer, are removed by suitable filters after the mixer. A stronger LO does not remove them.

5 mW

A level of 0 dBm is 1 mW, and +7 dBm is 7 dB above it, a factor of about 5, so about 5 mW. Reading dBm as milliwatts gives the wrong 7 mW.

Each product comes out 6 to 7 dB below the RF input

Conversion loss compares each output product with the RF input: here each comes out 6 to 7 dB weaker. A mixer with this loss gives no gain, and the LO level is a separate matter.

8 µW

A 6 dB loss divides the power by about 4, so 30 µW ÷ 3,98 = 7,5 µW and 8 µW is nearest. Treating the 6 dB as a gain gives 119 µW, nearest the 120 µW option.

Its output holds neither the RF nor the LO input

Double balanced means that neither input signal, RF or LO, appears at the IF output; single balanced keeps out only one. Every mixer gives the sum and difference, so that is not what the name means.

10 MHz and 32 MHz

The products are 21 − 11 = 10 MHz and 21 + 11 = 32 MHz, and a double-balanced mixer keeps both inputs out of its output. With the 11 and 21 MHz inputs present it would be unbalanced.

One input, usually the RF signal

A single-balanced mixer keeps one input out of its output but lets the other through, typically the weaker RF signal. Both inputs get through only in an unbalanced mixer.

Both the RF and the LO inputs

An unbalanced mixer does nothing to keep its inputs out, so the RF and LO inputs both appear at its output. Letting only one of them through is the single-balanced case.

A frequency mixer

A properly designed mixer adds little distortion, so it can shift AM, SSB and even audio signals. A synthesiser generates its own signal from a reference; it cannot carry an existing AM signal to a new frequency.