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431/531 Class Notes 4
3.5 Frequency Domain Analysis (contd.)
Before we lo ok at some more examples using our technique of complex imp edance, let's lo ok at some related general concepts.
3.5.1 Reactance
First, just a rede ntion of what we already have learned. The term reactance is often used in place of imp edance for capacitors and inductors. Reviewing our de nitions of imp edances from Section 3.2 we de ne the reactance of a capacitor X~C to just b e equal to its imp edance:
X~C {=(! C ). Similarly, for an inductor X~L {! L. This is the notation used in the text.
However, an alternative but common useage is to de ne the reactances as real quantities. This is done simply by dropping the { from the de nitions ab ove. The various reactances present in a circuit can by combined to form a single quantity X , which is then equal to the imaginary part of the imp edance. So, for example a circuit with R, L, and C in series would have total imp edance
Z~ = R + {X = R + {(XL + XC ) = R + {(! L 1
! C
A circuit which is \reactive" is one for which X is non-negligible compared with R.
3.5.2 General Solution
As stated b efore, our technique involves solving for a single Fourier frequency comp onent such as V~ = V e{(!^ t+ )^. You may wonder how our results generalize to other frequencies and to input waveforms other than pure sine waves. The answer in words is that we Fourier decomp ose the input and then use these decomp osition amplitudes to weight the output we found for a single frequency, Vout. We can formalize this within the context of the Fourier transform, whch will also allow us to see how our time-domain di erential equation b ecame transformed to an algebraic equation in frequency domain. Consider the example of the RC low-pass lter, or integrator, circuit of Fig. 7. We obtained the di erential equation given by Eq. 2. We wish to take the Fourier transform of this equation. De ne the Fourier transform of V (t) as
v (! ) F fV (t)g =
p
Z + 1
dte {!^ t^ V (t) (10)
Recall that F fdV =dtg = {! F fV g. Therefore our di erential equation b ecomes
{! v (! ) + v (! )=(RC ) = F fVin (t)g=(RC ) (11)
Solving for v (! ) gives
v (! ) =
F fVin (t)g
1 + {! RC
The general solution is then the real part of the inverse Fourier transform:
V~ (t) = F ^1 fv (! )g = p^1
Z + 1
d! 0 e{!^
(^0) t v (! 0 ) (13)
In the sp eci c case we have considered so far of a single Fourier comp onent of frequency
! , i.e. V~in = Vi e{!^ t^ , then F f V~in (t)g =
p
2 (! ! 0 ), and we recover our previous result for
the transfer function:
T~ = V~ = V~in = 1 1 + {! RC
For an arbitrary functional form for Vin (t), one could use Eqns. 12 and 13. Note that one would go through the same steps if Vin (t) were written as a Fourier series rather than a Fourier integral. Note also that the pro cedure carried out to give Eqn. 11 is formally equivalent to our use of the complex imp edances: In b oth cases the di erential equation is converted to an algebraic equation.
3.6 Phase Shift
We now need to discuss nding the phase of our solution. To do this, we pro ceed as previ- ously, for example like the high-pass lter, but this time we preserve the phase information by not taking the mo dulus of V~out. The input to a circuit has the form V~in = Vin e{(!^ t+ ^1 )^ , and the output V~out = Vout e{(!^ t+ ^2 )^. We are usually only interested in the phase di erence
2 1 b etween input and output, so, for convenience, we can cho ose 1 = 0 and set the
phase shift to b e 2 . Physically, we must cho ose the real or imaginary part of these
expressions. Conventionally, the real part is used. So we have:
Vin (t) = <( V~in ) = Vin (! ) cos(! t)
and
Vout (t) = <( V~out ) = Vout (! ) cos (! t + )
Let's return to our example of the high-pass lter to see how to calculate the phase shift. We rewrite the expression from Section 3.3 and then multiply numerator and denominator by the complex conjugate of the denominator:
V~out = V~in
R
R {=(! C )
= Vin e{!^ t^
1 + {=(! RC )
1 + 1 =(! RC )^2
By recalling the general form a + {b =
p
a^2 + b^2 e{ ^ , where = tan ^1 (b=a), we can write
1 + {=(! RC ) =
! RC
2 #^1 = 2
e{
allowing us to read o the phase shift:
= tan ^1 (1=(! RC )) (15)
Our solution for V~out is then
V~out = Vin^ e
{! t+
h
1 + ( ! RC^1 )^2
i 1 = 2
Vin R Vout
L C
Figure 12: A RLC circuit. Several lter typ es are p ossible dep ending up on how Vout is chosen. In the case shown, the circuit gives a resonant output.
3.8 An RLC Circuit Example
We can apply our technique of imp edance to increasingly more intricate examples, with no more e ort than a commensurate increase in the amount of algebra. The RLC circuit of Fig. 12 exempli es some new qualitative b ehavior. We can again calculate the output using our generalized voltage divider result of Eqn. 5. In this case, the Z~ 1 consists of the inductor and capacitor in series, and Z~ 2 is simply R. So,
Z~ 1 = {! L {=(! C ) = {L
where we have de ned the LC resonant frequency! 0 1 =
p
LC. We then obtain for the transfer function:
T (! )
j V~out j
j V~in j
R
jR + Z~ 1 j
[! 2 2 + (! 2 ! 20 )^2 ]^1 =^2
where R=L is the \R-L frequency".
T (! ) indeed exhibits a resonance at! =! 0. The quality factor Q, de ned as the ratio
of! 0 to the width of the resonance is given by Q ! 0 =(2 ) for ! 0. Such circuits have
many applications. For example, a high-Q circuit, where Vin (t) is the signal on an antenna, can b e used as a receiver. As was shown in class, we achieve di erent b ehavior if we cho ose to place the output across the capacitor or inductor, rather than across the resistor, as ab ove. Rather than a resonant circuit, cho osing Vout = VC yields a low-pass lter of the form
T (! ) =
j {=(! C )j
jR + Z~ 1 j
[! 2 2 + (! 2 ! 20 )^2 ]^1 =^2
The cuto frequency is! 0 , and for! ! 0 then T ! ^2 (\12 db p er o ctave"), which more
closely approaches ideal step function-like b ehavior than the RC low pass lter, for which
T ! ^1 for! ! 0 (\6 db p er o ctave"). As you might susp ect, cho osing Vout = VL provides
a high-pass lter with cuto at! 0 and T ! ^2 for! ! 0.
3.9 More Filters
3.9.1 Combining Filter Sections
Filter circuits can b e combined to pro duce new lters with mo di ed functionality. An ex- ample is the homework problem (6) of page 59 of the text, where a high-pass and a low-pass lter are combined to form a \band-pass" lter. As discussed at length in Section 1.5, it is imp ortant to design a \sti " circuit, in which the next circuit element do es not load the previous one, by requiring that the output imp edance of the rst b e much smaller than the input imp edance of the second. We can standardize this inequality by using a factor of 10
for the ratio j Z~in j=j Z~out j.
3.9.2 More Powerful Filters
This technique of cascading lter elements to pro duce a b etter lter is discussed in detail in Chapter 5 of the text. In general, the transfer functions of such lters take the form (for the low-pass case):
T (! ) =
h
1 + (^) n (f =fc )^2 n^
i 1 = 2
where fc is the 3 db frequency, (^) n is a co ecient dep ending up on the typ e of lter, and n is the lter \order," often equal to the numb er of ltering capacitors.
3.9.3 Active Filters
Filters involving LC circuits are very go o d, b etter than the simple RC lters, as discussed ab ove. Unfortunately, inductors are, in practice, not ideal lump ed circuit elements and are dicult to fabricate. In addition, lters made entirely from passive elements tend to have a lot of attenuation. For these reasons active lters are most commonly used where go o d ltering is required. These typically use op erational ampli ers (which we will discuss later), which can b e con gured to b ehave like inductors, and can have provide arbitrary voltage gain. Again, this is discussed in some detail in Chapter 5. When we discuss op amps later, we will lo ok at some examples of very simple active lters. At high frequencies (for example RF), op amps fail, and one most fall back on inductors.
4 Dio de Circuits
The gure b elow is from Lab 2, which gives the circuit symb ol for a dio de and a drawing of a dio de from the lab. Dio des are quite common and useful devices. One can think of a dio de as a device which allows current to ow in only one direction. This is an over-simpli cation, but a go o d approximation.
IF
Figure 13: Symb ol and drawing for dio des.
V
I
1 μΑ
10 mA
0.7 V
Forward Biased
Reverse Biased
Figure 15: The V - I b ehavior of a dio de.
