Lab 4: Diode Circuits
Reading Assignment
1. HH Lab Manual Class 3 and Worked Examples (pg 61-74 of lab manual)
2. HH Text pg 44-53
3. Hands-On, Ch. 3-3.4
Additional background:
A. Transmission lines and proper termination
Source: Eyler, lab 3
In an infinitely long sequence (transmission line) of LC components, the
amplitude should be the same at each stage, while the phase is shifted by each
stage by an amount φ given by
4sin 2
! (" ) " 2
1
= 2 , where "0 2 =
.
2
"0
LC
The cutoff frequency ωc occurs when the sine function becomes equal to 1, at
ωc=2ω0. Above this frequency the phase shift is imaginary, corresponding to an
attenuation of the amplitude at each stage.
Although this applies only to an infinite line, we can terminate a finite line with an
impedance that makes it indistinguishable from an infinite line. To find this
characteristic impedance, we need to ask what impedance is the same that we
would see if the continuation of the line were still present. A good trick for
determining this is to realize that if we add one more stage to the front of an
infinite line, the impedance is unchanged. Thus:
This tells us that
1
Z = i!L +
i!C +
1 .
Z
Solving for Z we find,
i! L
L
Z=
+
2
C
2
"!L #
$%
& .
' 2 (
Re-writing this in terms of the cutoff frequency gives an alternative form,
2
"
#
"! #
L $
! %
Z=
1& $ % + i
.
C $
!c (
!c %
'
'
(
This makes it obvious that the impedance has a magnitude L C independent of
frequency, multiplied by a frequency-dependent phase term with magnitude one.
In any case, we can physically realize this impedance by terminating our
transmission line with an inductance L′ in series with a resistor R′, with values
L
L$ = L / 2 and R $ =
C
2
"!#
1% & ' .
( !c )
Thus for every frequency ω a different value of the resistor R is required. For
frequencies above cutoff a pure inductor will do the job, but it’s really of little
importance, since the signal is attenuated exponentially and has a very small
amplitude at the termination.
To carry over what we’ve learned to a continuous transmission line such as a
coaxial cable is a surprisingly easy task. Consider the effects of subdividing the
line further and further, in such a way that the individual L’s and C’s become
smaller and smaller, but their ratio L/C is preserved. That’s all there is to it! The
cutoff frequency approaches infinity and the impedance becomes Z = L C ,
independent of frequency. This is why 50W coaxial cables should always be
terminated with 50W resistors if they are to be used at high frequencies or with
fast pulses.
B. Diode Mixers
Source: Eyler, lab 4
The nonlinearity of a diode can be put to explicit use when you need to create
new frequencies that are sums, multiples, or differences of other frequencies
present in a circuit. In this lab we will construct a diode mixer, whose input is two
sine waves at frequencies ω1 and ω2 and whose output is a sinusoid at the
difference frequency ω1 - ω2.
As we have already seen (Lab 2), the current through a diode is an exponential
function of the voltage drop. For present purposes we can represent the forwardbiased current adequately by a simple exponential,
I = A1 (eV / A2 ! 1),
where the constants A1 and A2 are determined by the device physics and
temperature.
If we take a Taylor expansion, the response is displayed explicitly as a sum of
linear and nonlinear terms,
'! V
1- V
I = A1 &
+ ++
!% A2 2 , A2
2
$!
*
(( + L#
)
!"
(1)
Thus if the input is a sum of two sinusoids with different frequencies,
V = V1 cos !1t + V2 cos ! 2t ,
the output will have components not only at frequencies ω1 and ω2, but also at dc
(zero frequency) and at frequencies 2ω1, 2ω2, ω1- ω2, and ω1+ ω2. If the cubic and
higher order terms are included in Eq. 2, we will obtain even more new
frequencies. The dc term corresponds to using the diode as a rectifier, as in the
first part of this laboratory. For the mixer we will be looking specifically for the
difference frequency, the “mixture” of the two inputs.
Mixers are used widely in RF technology, where an incoming signal at ω1
containing information is commonly combined with a stable “local oscillator” at ω2
to produce a difference frequency much lower than the input frequency, but still
containing all of the information. This process is called heterodyning, and is used
in almost every radio receiver. It is also seen commonly in the physics
laboratory, where the procedure is used not only at RF frequencies, but even for
coherent optical radiation from lasers.
(2)