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Circuit Surgery Regular clinic by Ian Bell Frequency shifting and superheterodyne receivers – Part 1 T he topic for this month was suggested by PE’s editor, Matt Pulzer – we will be looking at the principles behind superheterodyne radio receivers (often shortened to ‘superhets’). Heterodyning is the process of shifting the frequency of a signal. This month, we will discuss the mixer circuits which are used to perform frequency shifting and next month we will look at radio systems in more detail. The word heterodyne is from the widely used Greek root hetero, meaning ‘other’, and another ancient Greek root dyne, which is from dunamis, meaning ‘power’, although in the context of superhets it really means ‘frequency’. The ‘super’ part of superheterodyne is short for ‘supersonic’, to indicate that use is made of frequencies beyond the audio range. Today, the term supersonic would usually refer to something moving faster than the speed of sound, such as a supersonic jet plane, and we would use the term ultrasonic for frequencies beyond the audio range. Heterodyning Heterodynes are frequency shifts used in a variety of electronic signal processing systems (not just in radio systems), for example, chopper-stabilised amplifiers for high-precision amplification of verylow-frequency signals. Heterodyning is often referred to as ‘upshifting’ / ‘up conversion’ or downshifting’ / ‘down conversion’ (depending on the direction of frequency shift). Heterodyning occurs when signals of different frequencies are combined in a nonlinear way (more details on this later). Heterodyning is not restricted to electronic signal processing; in physics, heterodyning is used in a variety of measurement and detection techniques. For example, heterodyning of light waves in some types of microscopy and spectroscopy. The term ‘superheterodyne’ was coined in 1918 by the inventor of the superheterodyne radio receiver, American electrical engineer Edwin Howard Armstrong. He developed and demonstrated the idea while serving in 48 the US Army Signal Corps in France. However, the general principle of heterodyning predates this by almost 20 years and was invented by another pioneering radio engineer, Reginald Fessenden, working in the US, although he was born in Canada. Despite being involved in various long-running patent disputes over various developments in radio technology, Armstrong was able to work with the Radio Corporation of America (RCA) on developing relatively easy-to-use and low-cost implementations of superheterodyne receivers. RCA produced the first commercial superheterodyne radios in the mid 1920s, and since the 1930s superheterodyne designs have dominated the commercial radio receiver market. Radio fundamentals Radio transmission systems are fundamentally based on heterodyning. The signal to be transmitted, referred to as the ‘message signal’ (for example, speech) is upshifted from its original frequency range (called the baseband) to the much higher frequencies (radio frequencies – RF) required for practical wireless transmission of electromagnetic signals. This is achieved by modulating (varying) properties (amplitude, frequency or phase) of a high frequency signal (the carrier wave) in sympathy with the message signal. Thus, we have the familiar and long-established AM (amplitude modulation) and FM (frequency modulation) and more modern techniques such as digital 64QAM and 256QAM (64 and 256-state quadrature amplitude modulation), which varies both phase and amplitude. QAM is used in systems such as Wi-Fi and 4G. A radio receiver has to downshift the signal from the RF carrier frequency to the original baseband to recover the message. Recovering the original signal is referred to as ‘demodulation’ or ‘detection’. For many radio receivers, such as domestic radios for speech and music AM and FM stations, there is a requirement to handle a wide range of carrier frequencies. Direct demodulation of wide-range RF signals makes the design of the demodulation circuits very challenging, potentially leading to high costs and/or implementations which require complex and skilled manual adjustments, not suited to use by the general public. A key issue is that the performance of a demodulator circuit that is very good at a particular carrier frequency will degrade as the frequency moves away from this optimum value. The superheterodyne receiver solved this problem. The principle of the superheterodyne receiver is conversion (heterodyning) to a fixed intermediate frequency (IF) at which the demodulation is performed. This allows much of the radio’s circuitry to operate at the fixed intermediate frequency, making it easier to design, providing good performance at relatively low cost, and not requiring adjustments by the user other than tuning into the desired station. The ‘super’ part of superheterodyne refers to the fact that the IF is much higher than the baseband frequencies (such as audio, hence ‘supersonic’, as noted earlier). IF stages occur in many radio systems, including modern, largely digital, receivers where the IF signal(s) are fed to ADCs and then on to digital signal processing (DSP) for demodulation using custom integrated circuits (ICs) or FPGAs (field-programmable gate arrays). However, there are modern receivers that do not use IF and downconvert straight to the baseband. These systems, called direct-conversion receivers, are challenging to design due to a number of non-ideal behaviours, but with most of the circuitry implementable on an IC they have the potential for low cost, flexibility and low power consumption. Mixers A frequency mixer (or simply mixer) is a nonlinear circuit that combines signals at two frequencies to produce new frequencies (heterodynes). The Practical Electronics | December | 2023 Frequency shifting and superheterodyne receivers Frequency shifting and superheterodyne receivers VCC R1 S1 L1 C3 Vout 𝑆𝑆! = 𝐴𝐴! cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) 𝑆𝑆! = 𝐴𝐴! cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) 𝑆𝑆" = 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) 𝑆𝑆" = 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) 𝑆𝑆! 𝑆𝑆" = 𝐴𝐴! 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) C1 Q1 𝑆𝑆! 𝑆𝑆" = 𝐴𝐴! 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) C2 1 1 cos 𝛼𝛼 cos 𝛽𝛽 = cos(𝛼𝛼 − 𝛽𝛽) + cos(𝛼𝛼 + 𝛽𝛽) 2 mixer. 2 Fig.3. LTspice schematic for behavioural simulation of multiplying R2 R3 1 1 cos 𝛼𝛼 cos 𝛽𝛽 = cos(𝛼𝛼 − 𝛽𝛽) + cos(𝛼𝛼 + 𝛽𝛽) 2 2 considering two sinusoidal signals 𝐴𝐴! 𝐴𝐴" 𝐴𝐴! 𝐴𝐴" Frequency shifting and superheterodyne receivers Frequency shifting and superheterodyne receivers 𝑆𝑆! 𝑆𝑆" = cos(2𝜋𝜋(𝑓𝑓! − 𝑓𝑓" )𝑡𝑡) + cos(2𝜋𝜋(𝑓𝑓! + 𝑓𝑓" (S1 and S2) as inputs to the mixer. 2 2 We write these signals𝐴𝐴 as: 𝐴𝐴! 𝐴𝐴" ! 𝐴𝐴" Fig.1. Basic single-transistor RF mixer circuit. 𝑆𝑆! 𝑆𝑆" = cos(2𝜋𝜋(𝑓𝑓! − 𝑓𝑓" )𝑡𝑡) + cos(2𝜋𝜋(𝑓𝑓! + 𝑓𝑓" )𝑡𝑡) = 𝐴𝐴 𝐴𝐴!! cos(2𝜋𝜋𝑓𝑓 cos(2𝜋𝜋𝑓𝑓!!𝑡𝑡) 𝑡𝑡) 2 2 𝑆𝑆𝑆𝑆!! = 𝑘𝑘# + 𝑘𝑘! 𝑥𝑥 + 𝑘𝑘" 𝑥𝑥 " + 𝑘𝑘$ 𝑥𝑥 $ + 𝑘𝑘𝑥𝑥 % … and From this we can see that the mixed term ‘mixer’ canFrequency also be used when signal consists " $ % of two new sinusoids shifting and superheterodyne receivers 𝑘𝑘 + 𝑘𝑘 𝑥𝑥 + 𝑘𝑘 𝑆𝑆 = 𝐴𝐴 cos(2𝜋𝜋𝑓𝑓 𝑡𝑡) # ! " 𝑥𝑥 + 𝑘𝑘$ 𝑥𝑥 + 𝑘𝑘𝑥𝑥 … 𝑆𝑆"" = 𝐴𝐴"" cos(2𝜋𝜋𝑓𝑓""𝑡𝑡) signals are scaled and added (this is a at the sum (f1 + f2𝑆𝑆) and difference (f1 – ! = 𝐴𝐴! cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) linear operation that does not produce Here, A1 and A2 are the signal amplitudes f2), of the input frequencies. Note that new frequencies). A key example is an and f1 and their frequencies. The input frequencies are not present =f2𝐴𝐴are ! ! cos(2𝜋𝜋𝑓𝑓 ! 𝑡𝑡) "𝑡𝑡) 𝑆𝑆! = 𝐴𝐴the 𝑆𝑆 𝑆𝑆 = = 𝐴𝐴 𝐴𝐴!!𝐴𝐴 𝐴𝐴""𝑆𝑆cos(2𝜋𝜋𝑓𝑓 cos(2𝜋𝜋𝑓𝑓 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓 cos(2𝜋𝜋𝑓𝑓 𝑡𝑡) ! cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) !!𝑡𝑡) " audio mixing desk. The meaning of𝑆𝑆!!𝑆𝑆""2π factor converts the ordinary frequency in the output. 𝑆𝑆" = 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) the term should be clear from context. of the signal (f) in hertz to an angular An i d e a l mi xe r m u l ti p l i e s two frequency (ω) measured𝑡𝑡)in radians per Multiplying mixer simulation 𝑆𝑆" = 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) 11𝑆𝑆" = 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓 11 " signals. The circuitry in a basic radio second. Multiplying We can use" LTspice to simulate idealised cos 𝛼𝛼𝛼𝛼 cos cos = cos(𝛼𝛼 cos(𝛼𝛼 − 𝛽𝛽) 𝛽𝛽) + + these cos(𝛼𝛼 two + 𝛽𝛽) 𝛽𝛽)signals cos 𝛽𝛽𝛽𝛽 = − cos(𝛼𝛼 + 𝑘𝑘" 𝑥𝑥 = 𝑘𝑘" (𝐴𝐴! cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) + 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡))" 22 gives: 22 receiver – particularly when built together radio subsystems to help understand their from discrete devices – is often not a principles of operation. 𝑆𝑆! 𝑆𝑆" = 𝐴𝐴! 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓 𝑡𝑡)𝑘𝑘" (𝐴𝐴! cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) + 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡))" A multiplying 𝑘𝑘" 𝑥𝑥 " "= direct implementation of the multiply mixer can be implemented using a Frequency shifting and superheterodyne receivers 𝐴𝐴!𝐴𝐴 𝐴𝐴" 𝐴𝐴!𝐴𝐴 𝐴𝐴" 𝐴𝐴 𝐴𝐴 𝑥𝑥 " = 𝑘𝑘" (𝐴𝐴!" cos " (2𝜋𝜋𝑓𝑓! 𝑡𝑡) + 𝐴𝐴! 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) + 𝐴𝐴"" )𝑡𝑡) + )𝑡𝑡) = ! " cos(2𝜋𝜋(𝑓𝑓 cos(2𝜋𝜋(𝑓𝑓 −implications 𝑓𝑓"")𝑡𝑡) + ! " cos(2𝜋𝜋(𝑓𝑓 cos(2𝜋𝜋(𝑓𝑓 + 𝑓𝑓𝑓𝑓𝑘𝑘""")𝑡𝑡) function – typically the nonlinearity To see the of this clearly in behavioural voltage source, as shown in 𝑆𝑆𝑆𝑆!!𝑆𝑆𝑆𝑆"" = 𝑓𝑓 !− !+ ! ! 22 terms of different frequencies 22 inherent in diodes and transistors is we need Fig.3. Here we generate two sinusoids 1 𝑘𝑘 𝑥𝑥 " = 𝑘𝑘 (𝐴𝐴" 1cos " (2𝜋𝜋𝑓𝑓 𝑡𝑡) +to 𝐴𝐴! 𝐴𝐴"(signal1 cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓 𝐴𝐴"" cos(2𝜋𝜋𝑓𝑓 " product " + !𝛽𝛽) " 𝑡𝑡) + " 𝑡𝑡)) at ! two 𝛼𝛼 cos 𝛽𝛽 = the cos(𝛼𝛼 − 𝛽𝛽) cos(𝛼𝛼 + used (in the early days it was vacuum costo convert of cosines at 20kHz and signal2 2𝑆𝑆! = 𝐴𝐴! cos(2𝜋𝜋𝑓𝑓 2 ! 𝑡𝑡) tubes/values). Fig.1 shows an example individual sine or cosine functions. This 2kHz) using standard voltages sources + 𝑘𝑘𝑘𝑘!!𝑥𝑥𝑥𝑥 + + 𝑘𝑘𝑘𝑘""𝑥𝑥𝑥𝑥"" + + 𝑘𝑘𝑘𝑘$$𝑥𝑥𝑥𝑥$$ + + 𝑘𝑘𝑥𝑥 𝑘𝑘𝑥𝑥%% … … 𝑘𝑘𝑘𝑘# + of a basic mixer based on a single # problem was solved in in the sixteenth (V1 and V2 respectively). These signals bipolar transistor. by people are multiplied together using the 𝐴𝐴! 𝐴𝐴century 𝐴𝐴! 𝐴𝐴""in 𝑆𝑆" = 𝐴𝐴interested 𝑡𝑡) astronomy" " cos(2𝜋𝜋𝑓𝑓 cos(2𝜋𝜋(𝑓𝑓 − 𝑓𝑓" )𝑡𝑡) + This cos(2𝜋𝜋(𝑓𝑓 𝑓𝑓" )𝑡𝑡) behavioural voltage source B1 (with However, it is possible to 𝑆𝑆build based ship led to the ! 𝑆𝑆" = ! navigation. !+ 2 = 𝐴𝐴 𝐴𝐴!! cos(2𝜋𝜋𝑓𝑓 cos(2𝜋𝜋𝑓𝑓!!𝑡𝑡) 𝑡𝑡) 2 𝑆𝑆𝑆𝑆!! = analogue circuits that multiply signals. Prosthaphaeresis formulas, also known equation V=v(signal1)*v(signal2)). In particular, the Gilbert multiplier as Simpson’s a set of𝑡𝑡)four The frequencies are not representative of 𝑆𝑆! 𝑆𝑆" = 𝐴𝐴! 𝐴𝐴"formulas cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡)–cos(2𝜋𝜋𝑓𝑓 " " $ circuit (also called the Gilbert cell), trigonometric identities, of %which we typical radio systems, but are convenient 𝑘𝑘# + 𝑘𝑘 𝑥𝑥 + 𝑘𝑘 𝑥𝑥 + 𝑘𝑘 𝑥𝑥 + 𝑘𝑘𝑥𝑥 … ! " $ 𝑆𝑆 = = 𝐴𝐴 𝐴𝐴"" cos(2𝜋𝜋𝑓𝑓 cos(2𝜋𝜋𝑓𝑓 𝑡𝑡) which was developed by Barrie Gilbert need𝑆𝑆""the following:""𝑡𝑡) for displaying the results. in the late 1960s, is often used. The core When considering the operation 1 1 − 𝛽𝛽) + cos(𝛼𝛼 + 𝛽𝛽) of the Gilbert cell uses six transistors," cos 𝛼𝛼 cos𝑆𝑆𝛽𝛽 ==𝐴𝐴 cos(𝛼𝛼 of radio systems, we are often more " cos(2𝜋𝜋𝑓𝑓 𝑡𝑡) 2!𝑡𝑡) + 𝐴𝐴 cos(2𝜋𝜋𝑓𝑓 2 𝑡𝑡))" ! ! 𝑥𝑥" = 𝑘𝑘 (𝐴𝐴 cos(2𝜋𝜋𝑓𝑓 ""𝑥𝑥 = 𝑘𝑘""(𝐴𝐴!! cos(2𝜋𝜋𝑓𝑓!!𝑡𝑡) + 𝐴𝐴""cos(2𝜋𝜋𝑓𝑓!!𝑡𝑡)) but more are required to implement𝑘𝑘𝑘𝑘the interested in the signal spectra (what biasing (for example, a current source). Applying this to the above signal frequencies are present) rather than This Gilbert multiplier is suited to expression gives: the waveforms in the time domain. 𝐴𝐴 𝐴𝐴 𝐴𝐴 𝐴𝐴 " " (2𝜋𝜋𝑓𝑓 ! " 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓 " 𝑡𝑡) ! ""cos(2𝜋𝜋(𝑓𝑓 (𝐴𝐴!"!" cos = 𝑘𝑘𝑘𝑘""(𝐴𝐴 cos 𝑡𝑡) + cos(2𝜋𝜋(𝑓𝑓 𝐴𝐴!!𝑆𝑆𝐴𝐴 𝐴𝐴"""= cos(2𝜋𝜋𝑓𝑓 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓 𝑡𝑡) + 𝐴𝐴"" cos(2𝜋𝜋𝑓𝑓 𝑡𝑡)) )𝑡𝑡)cos(2𝜋𝜋𝑓𝑓 𝑘𝑘𝑘𝑘""𝑥𝑥𝑥𝑥" = cos(2𝜋𝜋𝑓𝑓 𝑆𝑆!"𝑆𝑆(2𝜋𝜋𝑓𝑓 + integrated circuit implementations, " 𝑡𝑡) + 𝐴𝐴""cos(2𝜋𝜋𝑓𝑓 ""𝑡𝑡)) " = !!𝑡𝑡) + 𝐴𝐴 ! − 𝑓𝑓!! "𝑡𝑡) ! + 𝑓𝑓" )𝑡𝑡) 2 2 where the high transistor count is not an issue, and it is common in IC-based 𝑘𝑘" 𝑥𝑥 " = 𝑘𝑘" (𝐴𝐴! cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) + 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡))" radio circuits. 𝑘𝑘# + 𝑘𝑘! 𝑥𝑥 + 𝑘𝑘" 𝑥𝑥 " + 𝑘𝑘$ 𝑥𝑥 $ + 𝑘𝑘𝑥𝑥 % … When considering the general principles of radio systems, "we usually 𝑘𝑘" 𝑥𝑥 = 𝑘𝑘" (𝐴𝐴!" cos " (2𝜋𝜋𝑓𝑓! 𝑡𝑡) + 𝐴𝐴! 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) + 𝐴𝐴"" cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡)) model mixers as multipliers and 𝑆𝑆! = 𝐴𝐴! cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) draw block diagrams of radio system architectures with mixers represented by the multiplier symbol, as shown 𝑆𝑆" = 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) in Fig.2. The operation of a multiplying mixer can be explained mathematically by 𝑘𝑘" 𝑥𝑥 " = 𝑘𝑘" (𝐴𝐴! cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) + 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡))" S2 S2 f2 S1×S 𝑘𝑘2" 𝑥𝑥 " = 𝑘𝑘" (𝐴𝐴!" cos " (2𝜋𝜋𝑓𝑓! 𝑡𝑡) + 𝐴𝐴! 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) + 𝐴𝐴"" cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡)) f1 + f2 f1 – f2 S1 f1 Fig.2. Multiplier/mixer symbol. Practical Electronics | December | 2023 Fig.4. Waveform results from the circuit in Fig.3. 49 Beats and heterodynes Fig.6. Spectra of signals from the waveforms in Fig.4. Heterodyne (mixer) circuits are sometimes referred to using the term ‘beat’; for example, a ‘beat frequency oscillator’ used as an input to a mixer in a radio receiver. However, this could be confusing because the term ‘beat’ or ‘beat frequency’ in acoustics refers to the interference pattern which occurs between tones at two slightly different frequencies. Interference is the addition of the two waves at each point in space, so it is a linear process. Similar interference patterns can occur in other physics domains, such as with light waves (see Thomas Young’s famous double-slit experiment). Unlike multiplying, adding two sinusoidal signals does not produce any new frequencies in the spectrum of the output signal, but the amplitude of the combined signal varies at a rate proportional to the difference in the two frequencies. In acoustics, if we produce two pure tones (sinusoidal) the human listener may perceive this amplitude variation as either a pulsing in volume or a separate audible tone, depending on the frequency of amplitude variation in relation to the human hearing range and the degree of separation of the tones. Fig.7 shows an LTspice simulation set up in a similar way to that in Fig.3. The key difference is that the behaviour source (B1) is adding rather than multiplying the signals (signal1 and signal2). The resulting signal is labelled beat in reference to the audio beat frequencies just discussed. The signals being added are at 18kHz and 22kHz (the same as the output frequencies in the previous example). The waveforms are shown in Fig.8. The waveform shape of the output (beat signal) is the same as the output from the multiplier in Fig.7 – this is to be expected as the same two frequencies are present at equal amplitudes in both cases. The spectra in Fig.9 show the key difference between the additive and multiplicative circuits. For the circuit in Fig.7 the frequencies in the output (beat signal) are the same as the frequencies in the input (signal1 and signal2). Unlike for the multiplying mixer, none of the signals match the envelope of the output waveform. This is a sinewave at a frequency which is half the difference between frequencies of the added sinusoids (since 50 Practical Electronics | December | 2023 Fig.5. Waveform results from the circuit in Fig.3. Therefore, the simulation is configured to facilitate viewing of the spectrum with LTspice’s FFT (Fast Fourier Transform) function. This requires a relatively long simulation with a relatively small timestep in comparison with what is required just to view the waveforms. We also use the directives to prevent LTspice from compressing the waveform data (.option plotwinsize=0) and use double-precision math (.options numdgt=7) to improve FFT accuracy. The waveforms from the simulation in Fig.3 are shown in Fig.4. We see that the mixed waveform looks like the amplitude of the higher frequency signal is being varied by the low frequency signal. Indeed, this is the case – an amplitude modulated (AM) radio signal can be produced by a multiplying mixer. Fig.5 shows the low-frequency waveform (signal2) and its inverse (-signal2) match the amplitude envelope of the mixer output. Although the setup for AM is not exactly as shown in figure, in radio terms signal1 in Fig.4 can be thought of as acting like a carrier wave and signal2 represents the message. The spectra of the signals in Fig.4 are shown in Fig.6. These are obtained in LTspice by right-clicking the waveform of interest and selecting View > FFT. The default configuration for the FFT was used here. The initial display shows a lot of detail at lower amplitudes, but the minimum dB axis value was changed to –100 dB to simplify the plots. The two input signals have a single peak at their respective frequencies of 20kHz and 2kHz, as expected for ideal sinewaves. The sub –100 dB amplitudes at other frequencies (less the 1/10000 of the peak value) are due to the fact that the FFT calculations are not perfectly accurate because of practical limits on the number of data points and the numerical precision of the simulation and FFT. The spectrum of the mixed signal in Fig.6 has two peaks at 18kHz and 22kHz. These are at the sum (20kHz + 2kHz = 22kHz) and difference (20kHz – 2kHz = 18kHz) frequencies. This is as expected from the formula obtained above. The output spectrum of the mixer does not contain the two input frequencies, just their sum and difference. Again, this follows from the equations above. using discrete components. If we apply the sum of two signals to any circuit with a nonlinear response heterodyning will occur. Looking at Fig.1 we see that the voltage across the base-emitter junction of the transistor is the difference between the two input signals. This is effectively the sum of receivers the signals (with a sign Frequency shifting and superheterodyne change). The voltage-current relationship Fig.7. LTspice schematic for behavioural simulation of signal addition. of the junctionreceivers is nonlinear (thanks to Frequency shifting shifting and and superheterodyne superheterodyne receivers Frequency Nonlinear mixers the exponential diode𝑡𝑡)voltage-current (22 – 18)/2 = 2 in kHz). In Fig.7 there is an 𝑆𝑆! = 𝐴𝐴! cos(2𝜋𝜋𝑓𝑓 ! As noted above, not all mixer circuits are shifting relationship), so the base and hence additional sinusoidal voltage source (V3) Frequency and superheterodyne receivers multiplier implementations, particularly collector 𝑆𝑆 current will contain which produces a signal that matches = 𝐴𝐴 cos(2𝜋𝜋𝑓𝑓 𝑡𝑡) 𝑆𝑆!! = 𝐴𝐴!! cos(2𝜋𝜋𝑓𝑓!! 𝑡𝑡) additional shifting and superheterodyne receivers those in basicFrequency or early radios and those frequencies the amplitude envelope (see Fig.10). 𝑆𝑆" = 𝐴𝐴(heterodyning). " cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) We can show the general behaviour of receivers Frequency shifting superheterodyne 𝑆𝑆and ! = 𝐴𝐴! cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) a nonlinear mathematically by 𝑆𝑆"" = =circuit 𝐴𝐴"" cos(2𝜋𝜋𝑓𝑓 cos(2𝜋𝜋𝑓𝑓 𝑡𝑡) " 𝑆𝑆 𝐴𝐴 𝑡𝑡) " 𝑆𝑆! =superheterodyne 𝐴𝐴!the cos(2𝜋𝜋𝑓𝑓 Frequency shifting and assuming relationship between input ! 𝑡𝑡) receivers 𝑆𝑆! 𝑆𝑆" = 𝐴𝐴! 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) and output is a polynomial. Mathematical 𝑆𝑆" = 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓𝑆𝑆"!𝑡𝑡)= 𝐴𝐴! cos(2𝜋𝜋𝑓𝑓! Frequency shifting and superheterodyne receivers theory shows that many functions can 𝑆𝑆 𝑆𝑆!! 𝑆𝑆 𝑆𝑆"" = = 𝐴𝐴 𝐴𝐴!! 𝐴𝐴 𝐴𝐴"" cos(2𝜋𝜋𝑓𝑓 cos(2𝜋𝜋𝑓𝑓!! 𝑡𝑡) 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓 cos(2𝜋𝜋𝑓𝑓"" 𝑡𝑡) 𝑡𝑡) be approximated – you 𝑆𝑆" = 𝐴𝐴1" cos(2𝜋𝜋𝑓𝑓 𝑆𝑆by =polynomials 𝐴𝐴1! cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) " 𝑡𝑡) ! ‘polynomial fit’+to cos 𝛼𝛼may cos have 𝛽𝛽 = used cos(𝛼𝛼a − 𝛽𝛽) + cos(𝛼𝛼 𝛽𝛽)draw 𝑆𝑆" = 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓 2 𝑆𝑆! 𝑆𝑆" = 𝐴𝐴! 𝐴𝐴2 cos(2𝜋𝜋𝑓𝑓 " ! 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓 𝑆𝑆! = 𝐴𝐴! cos(2𝜋𝜋𝑓𝑓 a line data "on a graph. A " 𝑡𝑡) ! 𝑡𝑡) through 1 1 1 1 cos 𝛼𝛼 cos 𝛽𝛽 = cos(𝛼𝛼 − 𝛽𝛽) + cos(𝛼𝛼 + 𝛽𝛽) cos 𝛼𝛼 cos 𝛽𝛽 = cos(𝛼𝛼 − 𝛽𝛽) + cos(𝛼𝛼 + 𝛽𝛽) of a variable is the weighted 𝑆𝑆! 𝑆𝑆" polynomial = 𝐴𝐴! 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓 𝑡𝑡)𝑆𝑆cos(2𝜋𝜋𝑓𝑓 𝑡𝑡) = 𝐴𝐴x " cos(2𝜋𝜋𝑓𝑓 " 𝑡𝑡) 2 2 2 ! " 2" x-cubed 𝐴𝐴! 𝐴𝐴"sum of its powers (x,𝐴𝐴x-squared, 𝐴𝐴 !1 " 𝑆𝑆! 𝑆𝑆" = 𝐴𝐴! 𝐴𝐴 1" cos(2𝜋𝜋𝑓𝑓 ! 𝑡𝑡) c )𝑡𝑡) 𝛽𝛽 )𝑡𝑡) + 𝛽𝛽) cos(2𝜋𝜋(𝑓𝑓 +each cos(2𝜋𝜋(𝑓𝑓 + and power of x 𝑓𝑓is 𝑆𝑆𝑆𝑆"! 𝑆𝑆=" 𝐴𝐴=" cos(2𝜋𝜋𝑓𝑓 𝑡𝑡) on, ! −where !+ "cos(𝛼𝛼 cos 𝛼𝛼𝑓𝑓"cos = 2 cos(𝛼𝛼 − 𝛽𝛽) " so 2 2 2 (coefficient) 1 𝑆𝑆!by 𝐴𝐴 𝐴𝐴 𝐴𝐴 𝑆𝑆" a=fixed 𝐴𝐴! 𝐴𝐴1" value cos(2𝜋𝜋𝑓𝑓 𝐴𝐴!!multiplied 𝐴𝐴𝛽𝛽""= 𝐴𝐴!! 𝐴𝐴 𝐴𝐴""+ ! 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) cos(𝛼𝛼 − + + cos(𝛼𝛼 𝛽𝛽) )𝑡𝑡) )𝑡𝑡) 𝑆𝑆cos 𝑆𝑆" 𝛼𝛼= =cos cos(2𝜋𝜋(𝑓𝑓 −𝛽𝛽) 𝑓𝑓"this +as: cos(2𝜋𝜋(𝑓𝑓 + 𝑓𝑓 )𝑡𝑡) ! ! ! )𝑡𝑡) 𝑆𝑆 cos(2𝜋𝜋(𝑓𝑓 cos(2𝜋𝜋(𝑓𝑓 k). We can write ! 𝑆𝑆" ! − 𝑓𝑓" ! + 𝑓𝑓" 2 2 2 2 1 " 2 2 cos 𝛼𝛼 cos 𝛽𝛽 = cos(𝛼𝛼 − 𝛽𝛽) + 𝑆𝑆! 𝑆𝑆" = 𝐴𝐴! 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓𝑘𝑘! 𝑡𝑡)+cos(2𝜋𝜋𝑓𝑓 𝑡𝑡)𝑥𝑥 " + 𝑘𝑘 𝑥𝑥 $ + 𝑘𝑘𝑥𝑥 % … 𝐴𝐴 𝐴𝐴 𝑘𝑘! 𝑥𝑥 +𝐴𝐴!𝑘𝑘"𝐴𝐴 # "" $ ! 2" 𝑆𝑆! 𝑆𝑆" = cos(2𝜋𝜋(𝑓𝑓 cos(2𝜋𝜋(𝑓𝑓! + 1 ! − 𝑓𝑓" )𝑡𝑡) + 1 2cos 𝛽𝛽to 2 cos 𝛼𝛼+ =our cos(𝛼𝛼 + 𝛽𝛽) " $ − 𝛽𝛽) % +circuit 𝐴𝐴! 𝐴𝐴" If x𝑘𝑘 𝐴𝐴"𝑘𝑘cos(𝛼𝛼 input nonlinear " !+ 𝑘𝑘 𝑘𝑘 𝑥𝑥𝐴𝐴 𝑥𝑥 2 𝑘𝑘##is+ +the 𝑘𝑘−!! 𝑥𝑥 𝑥𝑥𝑓𝑓 + 𝑥𝑥 $ + + 𝑘𝑘𝑥𝑥 𝑘𝑘𝑥𝑥 !% … … )𝑡𝑡)𝑘𝑘""+𝑥𝑥 +2𝑘𝑘$$cos(2𝜋𝜋(𝑓𝑓 )𝑡𝑡) 𝑆𝑆! 𝑆𝑆" = cos(2𝜋𝜋(𝑓𝑓 + 𝑓𝑓 ! " " and it1 happens to2 be the sum of two 1 2 𝐴𝐴! cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡)𝐴𝐴! 𝐴𝐴" ! = 𝐴𝐴+ cos 𝛼𝛼 cos 𝛽𝛽 = cos(𝛼𝛼 −sinusoids 𝛽𝛽) + 𝑆𝑆cos(𝛼𝛼 (S1!+𝛽𝛽) S2𝑆𝑆),! 𝑆𝑆where: cos(2𝜋𝜋(𝑓𝑓 − 𝑓𝑓 )𝑡𝑡) + " = 2 2 𝑘𝑘# + 𝑘𝑘! 𝑥𝑥 + 𝑘𝑘"2𝑥𝑥 " + 𝑘𝑘$ 𝑥𝑥 $ +! 𝑘𝑘𝑥𝑥 %"… 𝐴𝐴!𝑆𝑆𝐴𝐴" = 𝐴𝐴 𝐴𝐴! 𝐴𝐴" 𝐴𝐴!! cos(2𝜋𝜋𝑓𝑓 cos(2𝜋𝜋𝑓𝑓!! 𝑡𝑡) 𝑡𝑡) Fig.8. Waveform results from the circuit in Fig.3. 𝑆𝑆!𝑘𝑘𝑆𝑆"𝑥𝑥=+ 𝑘𝑘 𝑆𝑆𝑥𝑥!! "= cos(2𝜋𝜋(𝑓𝑓 !− 𝑘𝑘# + + 𝑘𝑘$ 𝑥𝑥 $ + 𝑘𝑘𝑥𝑥𝑓𝑓%" )𝑡𝑡) … + 2 cos(2𝜋𝜋(𝑓𝑓! + 𝑓𝑓" ) ! " 2 𝑆𝑆 = 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) 𝐴𝐴! 𝐴𝐴" 𝐴𝐴! 𝐴𝐴" " And 𝑘𝑘# + 𝑘𝑘! !𝑥𝑥𝑡𝑡)+ 𝑘𝑘" 𝑥𝑥 " + 𝑘𝑘$ 𝑥𝑥 $ 𝑆𝑆! 𝑆𝑆" = cos(2𝜋𝜋(𝑓𝑓! − 𝑓𝑓" )𝑡𝑡) + cos(2𝜋𝜋(𝑓𝑓! + 𝑓𝑓𝑆𝑆"!)𝑡𝑡) = 𝐴𝐴! cos(2𝜋𝜋𝑓𝑓 2 2 𝑆𝑆 = 𝐴𝐴 cos(2𝜋𝜋𝑓𝑓 𝑡𝑡) 𝑆𝑆"" = 𝐴𝐴"" cos(2𝜋𝜋𝑓𝑓"" 𝑡𝑡) 𝑆𝑆 = 𝐴𝐴! 𝑘𝑘 cos(2𝜋𝜋𝑓𝑓 + 𝑘𝑘" 𝑥𝑥 " + 𝑘𝑘$ 𝑥𝑥 $ +" 𝑘𝑘𝑥𝑥 % … # + 𝑘𝑘! 𝑥𝑥! 𝑡𝑡) 𝑘𝑘" 𝑥𝑥 " = 𝑘𝑘"!(𝐴𝐴! cos(2𝜋𝜋𝑓𝑓 ! 𝑡𝑡) + 𝐴𝐴discussed " cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡)) as previously for the 𝑆𝑆"!𝑡𝑡)= 𝐴𝐴! cos(2𝜋𝜋𝑓𝑓! 𝑆𝑆 = 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓 " $ % " multiplier, then, if we just 𝑘𝑘# + 𝑘𝑘! 𝑥𝑥 + 𝑘𝑘" 𝑥𝑥 𝑘𝑘+ 𝑥𝑥𝑘𝑘""$ 𝑥𝑥= 𝑘𝑘+ (𝐴𝐴 𝑘𝑘𝑥𝑥 … " " cos(2𝜋𝜋𝑓𝑓!! 𝑡𝑡) 𝑡𝑡) + + 𝐴𝐴 𝐴𝐴"" cos(2𝜋𝜋𝑓𝑓 cos(2𝜋𝜋𝑓𝑓!! 𝑡𝑡)) 𝑡𝑡)) 𝑘𝑘"" 𝑥𝑥 = 𝑘𝑘"" (𝐴𝐴!! cos(2𝜋𝜋𝑓𝑓 2 term, we have: 𝑆𝑆" = 𝐴𝐴consider 2x 𝑆𝑆the =k𝐴𝐴 " cos(2𝜋𝜋𝑓𝑓 " 𝑡𝑡) ! ! cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) " " " 𝑘𝑘" 𝑥𝑥 = 𝑘𝑘" (𝐴𝐴! cos (2𝜋𝜋𝑓𝑓! 𝑡𝑡) + 𝐴𝐴! 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) + 𝐴𝐴"" cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡)) " = 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓 𝑘𝑘" 𝑥𝑥 " = 𝑘𝑘" (𝐴𝐴! cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) + 𝑆𝑆𝐴𝐴"" cos(2𝜋𝜋𝑓𝑓 ! 𝑡𝑡)) " 𝑆𝑆!𝑘𝑘=(𝐴𝐴 𝐴𝐴!"" cos(2𝜋𝜋𝑓𝑓 " " ! 𝑡𝑡)𝑡𝑡) + 𝐴𝐴 𝐴𝐴 cos(2𝜋𝜋𝑓𝑓 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓 𝑡𝑡) + 𝐴𝐴" " " = " (2𝜋𝜋𝑓𝑓 𝑘𝑘 𝑥𝑥 cos cos(2𝜋𝜋𝑓𝑓 " " ! ! " ! " " 𝑡𝑡 " 𝑘𝑘" 𝑥𝑥 = 𝑘𝑘" (𝐴𝐴!!" cos (2𝜋𝜋𝑓𝑓! 𝑡𝑡) + 𝐴𝐴! 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓 " " 𝑡𝑡) + 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡 𝑘𝑘" 𝑥𝑥 = 𝑘𝑘" (𝐴𝐴! cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) +𝑆𝑆𝐴𝐴" "= cos(2𝜋𝜋𝑓𝑓 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓 ! 𝑡𝑡)) " 𝑡𝑡) " " " (2𝜋𝜋𝑓𝑓 𝑘𝑘""𝑥𝑥cos(2𝜋𝜋𝑓𝑓 = 𝑘𝑘squared " (𝐴𝐴 ! cos(2𝜋𝜋𝑓𝑓 !"𝑡𝑡) 𝑥𝑥 " = 𝑘𝑘" (𝐴𝐴 cos(2𝜋𝜋𝑓𝑓 𝑡𝑡)++𝐴𝐴 Multiplying the ! 𝑡𝑡) + 𝐴𝐴out ! 𝐴𝐴 ! 𝑡𝑡) ! cos 𝑆𝑆" = 𝐴𝐴𝑘𝑘""cos(2𝜋𝜋𝑓𝑓 " 𝑡𝑡) terms get: " " " (𝐴𝐴!!𝑡𝑡) 𝑘𝑘" 𝑥𝑥 " = 𝑘𝑘" (𝐴𝐴!" cos " (2𝜋𝜋𝑓𝑓! 𝑡𝑡) + 𝐴𝐴𝑘𝑘! 𝐴𝐴 cos(2𝜋𝜋𝑓𝑓 = 𝑘𝑘"we cos(2𝜋𝜋𝑓𝑓 ++𝐴𝐴𝐴𝐴 " cos(2𝜋𝜋𝑓𝑓 " 𝑡𝑡) " 𝑡𝑡)) " 𝑥𝑥 ! 𝑡𝑡) " cos(2𝜋𝜋𝑓𝑓 ! 𝑡𝑡)) " cos(2𝜋𝜋𝑓𝑓 𝑘𝑘" 𝑥𝑥 " = 𝑘𝑘"" (𝐴𝐴!" cos " (2𝜋𝜋𝑓𝑓! 𝑡𝑡) + 𝐴𝐴! 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡 𝑘𝑘" 𝑥𝑥 " = 𝑘𝑘" (𝐴𝐴! cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) + 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓 ! 𝑡𝑡)) 𝑘𝑘" 𝑥𝑥 " = 𝑘𝑘" (𝐴𝐴!" cos " (2𝜋𝜋𝑓𝑓! 𝑡𝑡) + 𝐴𝐴! 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) + 𝐴𝐴"" c 𝑘𝑘" 𝑥𝑥 " = 𝑘𝑘" (𝐴𝐴!" cos " (2𝜋𝜋𝑓𝑓! 𝑡𝑡) + 𝐴𝐴! 𝐴𝐴" cos(2𝜋𝜋𝑓𝑓! 𝑡𝑡) cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡) + 𝐴𝐴"" cos(2𝜋𝜋𝑓𝑓" 𝑡𝑡)) Fig.10. Envelope of beat signal from Fig.8. The A1A2 term in the middle is equivalent to the output for the multiplier discussed above – so this will produce the sum and difference frequencies: (f1 + f2) and (f1 – f2). The cos2 terms imply multiplying two signals of the same frequency – again we get the sum (f + f =2f) and difference (f – f = 0), so the output will also contain frequencies of twice the two input frequencies and some DC (zero frequency). The k 1 x polynomial term will add the original input frequencies to the output and Practical Electronics | December | 2023 51 Fig.9. Spectra of signals from the circuit in Fig.7. Note that there are no new frequencies in the output (beat signal). Introduction to LTspice Want to learn the basics of LTspice? Ian Bell wrote an excellent series of Circuit Surgery articles to get you up and running, see PE October 2018 to January 2019, and July/August 2020. All issues are available in print and PDF from the PE website: https://bit.ly/pe-backissues Fig.11. LTspice schematic for behavioural simulation of nonlinear mixer using squaring. the mixer circuit in Fig.1, the LC circuit (L1 and C3) implements the filtering of the output signal. The LC circuits passes a narrow range of frequencies around it’s resonant frequency to the next stage via the transformer secondary. Nonlinear mixer simulation Fig.12. LTspice schematic for obtaining the frequency response of the filter in Fig.11. the k0 indicates the possibility of DC in the output. The higher-order terms produce many other frequencies, including at integer multiples of the two input frequencies and various additive combinations of these. The fact that a nonlinear mixer produces a wide range of output frequencies means that we usually need a filter circuit on the output to remove the unwanted frequencies. Even with the multiplier mixer we may only want one of the outputs (either the sum or difference frequencies), so a filter may be needed here too. In Fig.11 shows another simulation schematic similar to the previous examples. This is a mixer configured to produce the square of the sum of two input sinusoids – signal1 at 20kHz and signal2 at 2kHz – the same inputs as in Fig.3. From the mathematics discussed above we expect the circuit to produce four frequencies f1 – f2 (18kHz), f1 + f2 (22kHz), 2f1 (40kHz) and 2f2 (4kHz), plus some DC. The schematic also includes a bandpass filter (U 1 ) to attenuate unwanted frequencies. The ‘wanted’ signal here is selected to be the 18kHz (f 1 – f 2 ) output. The filter is implemented using the LTspice behavioural second-order bandpass filter, which is available in the SpecialFunctions folder of the component selector with name 2ndOrderBandpass. The filter is for illustrative purposes only – it was not set up to meet any specific requirements. The LTspice schematic in Fig.12 can be used to plot the frequency response of the filter. The output waveforms for the circuit in Fig.11 are shown in Fig.13. Despite having the same inputs as the circuit in Fig.3 the mixer output waveform has a significantly different shape due to the additional frequencies present. It also has a DC offset, as predicted above. The filtered waveform is closer to a pure 18kHz sinewave in shape, but not perfectly so as the filter does not complexly remove the other frequencies. The spectrum of the mixed signal is shown in the top trace in Fig.14. The four frequencies mentioned above can be seen. The spectra in this figure were obtained using the Blackman windowing function in the LTspice FFT configuration. This is because the full waveforms are not necessarily integer number of cycles of repeating wave shape. Windowing functions ‘fade out’ the ends of the waveform to avoid abrupt signal changes or other differences at the ends from causing errors in the FFT. This nonlinear mixer has a relatively simple behaviour (just a squaring function) but produces a more complex output than the ideal multiplier mixer discussed earlier. It is often the case that we need to remove some of the output from a mixer using a filter. The middle trace shows the filter frequency response used in this example. This is an 18kHz bandpass filter, which reduces the amplitude of frequencies away from 18kHz. This can be seen in the lower spectrum, which is for the filter output. The attenuation of the unwanted signals is not perfect here, but the filter was not set up to provide a specific performance. In general, unwanted mixer output frequencies may be close to wanted frequencies, which leads to demanding filter specifications. Simulation files Fig.13. Waveform results from the circuit in Fig.11. Input signals are the same as in Fig.4. 52 Most, but not every month, LTSpice is used to support descriptions and analysis in Circuit Surgery. The examples and files are available for download from the PE website: https://bit.ly/pe-downloads Practical Electronics | December | 2023 Fig.14. Spectra of signals in Fig.13 together with frequency response of the bandpass filter. STEWART OF READING 17A King Street, Mortimer, near Reading, RG7 3RS Telephone: 0118 933 1111 Fax: 0118 933 2375 USED ELECTRONIC TEST EQUIPMENT Check website www.stewart-of-reading.co.uk Fluke/Philips PM3092 Oscilloscope 2+2 Channel 200MHz Delay TB, Autoset etc – £250 LAMBDA GENESYS LAMBDA GENESYS IFR 2025 IFR 2948B IFR 6843 R&S APN62 Agilent 8712ET HP8903A/B HP8757D HP3325A HP3561A HP6032A HP6622A HP6624A HP6632B HP6644A HP6654A HP8341A HP83630A HP83624A HP8484A HP8560E HP8563A HP8566B HP8662A Marconi 2022E Marconi 2024 Marconi 2030 Marconi 2023A PSU GEN100-15 100V 15A Boxed As New £400 PSU GEN50-30 50V 30A £400 Signal Generator 9kHz – 2.51GHz Opt 04/11 £900 Communication Service Monitor Opts 03/25 Avionics POA Microwave Systems Analyser 10MHz – 20GHz POA Syn Function Generator 1Hz – 260kHz £295 RF Network Analyser 300kHz – 1300MHz POA Audio Analyser £750 – £950 Scaler Network Analyser POA Synthesised Function Generator £195 Dynamic Signal Analyser £650 PSU 0-60V 0-50A 1000W £750 PSU 0-20V 4A Twice or 0-50V 2A Twice £350 PSU 4 Outputs £400 PSU 0-20V 0-5A £195 PSU 0-60V 3.5A £400 PSU 0-60V 0-9A £500 Synthesised Sweep Generator 10MHz – 20GHz £2,000 Synthesised Sweeper 10MHz – 26.5 GHz POA Synthesised Sweeper 2 – 20GHz POA Power Sensor 0.01-18GHz 3nW-10µW £75 Spectrum Analyser Synthesised 30Hz – 2.9GHz £1,750 Spectrum Analyser Synthesised 9kHz – 22GHz £2,250 Spectrum Analsyer 100Hz – 22GHz £1,200 RF Generator 10kHz – 1280MHz £750 Synthesised AM/FM Signal Generator 10kHz – 1.01GHz £325 Synthesised Signal Generator 9kHz – 2.4GHz £800 Synthesised Signal Generator 10kHz – 1.35GHz £750 Signal Generator 9kHz – 1.2GHz £700 HP/Agilent HP 34401A Digital Multimeter 6½ Digit £325 – £375 HP 54600B Oscilloscope Analogue/Digital Dual Trace 100MHz Only £75, with accessories £125 (ALL PRICES PLUS CARRIAGE & VAT) Please check availability before ordering or calling in HP33120A HP53131A HP53131A Audio Precision Datron 4708 Druck DPI 515 Datron 1081 ENI 325LA Keithley 228 Time 9818 Practical Electronics | December | 2023 Marconi 2305 Marconi 2440 Marconi 2945/A/B Marconi 2955 Marconi 2955A Marconi 2955B Marconi 6200 Marconi 6200A Marconi 6200B Marconi 6960B Tektronix TDS3052B Tektronix TDS3032 Tektronix TDS3012 Tektronix 2430A Tektronix 2465B Farnell AP60/50 Farnell XA35/2T Farnell AP100-90 Farnell LF1 Racal 1991 Racal 2101 Racal 9300 Racal 9300B Solartron 7150/PLUS Solatron 1253 Solartron SI 1255 Tasakago TM035-2 Thurlby PL320QMD Thurlby TG210 Modulation Meter £250 Counter 20GHz £295 Communications Test Set Various Options POA Radio Communications Test Set £595 Radio Communications Test Set £725 Radio Communications Test Set £800 Microwave Test Set £1,500 Microwave Test Set 10MHz – 20GHz £1,950 Microwave Test Set £2,300 Power Meter with 6910 sensor £295 Oscilloscope 500MHz 2.5GS/s £1,250 Oscilloscope 300MHz 2.5GS/s £995 Oscilloscope 2 Channel 100MHz 1.25GS/s £450 Oscilloscope Dual Trace 150MHz 100MS/s £350 Oscilloscope 4 Channel 400MHz £600 PSU 0-60V 0-50A 1kW Switch Mode £300 PSU 0-35V 0-2A Twice Digital £75 Power Supply 100V 90A £900 Sine/Sq Oscillator 10Hz – 1MHz £45 Counter/Timer 160MHz 9 Digit £150 Counter 20GHz LED £295 True RMS Millivoltmeter 5Hz – 20MHz etc £45 As 9300 £75 6½ Digit DMM True RMS IEEE £65/£75 Gain Phase Analyser 1mHz – 20kHz £600 HF Frequency Response Analyser POA PSU 0-35V 0-2A 2 Meters £30 PSU 0-30V 0-2A Twice £160 – £200 Function Generator 0.002-2MHz TTL etc Kenwood Badged £65 Function Generator 100 microHz – 15MHz Universal Counter 3GHz Boxed unused Universal Counter 225MHz SYS2712 Audio Analyser – in original box Autocal Multifunction Standard Pressure Calibrator/Controller Autocal Standards Multimeter RF Power Amplifier 250kHz – 150MHz 25W 50dB Voltage/Current Source DC Current & Voltage Calibrator £350 £600 £350 POA POA £400 POA POA POA POA Marconi 2955B Radio Communications Test Set – £800 53