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Electronics II Tutorials
   Overview Analog Wireless Communcation
   Resistors, Capacitors, RC Networks
   Diodes, Amplitute Modulation, Diode Detection
   RL Circuits, Inductive Kicks, Diode Snubbers
   RC filters. Series resonance and quality factor, Matching, Soldering
   Ladder filters Butterworth and Chebyshev filters Filter tables ADS
   Bandpass ladder filters Quartz crystals
   Impedance inverter
   Ideal Transformers
   Transformer shunt inductance
   BJT-Large signal models
   Transistor switches. Voltage regulators
   Transistor switches. Voltage regulators
   Common emitter amplifier. Max. efficiency of class A amps. Transformer coupled loads
   Available power. Distortion. Emitter degeneration. Miller effect
   Emitter follower and differential amplifiers
   JFET Source follower amplifier
   Oscillators. Clapp oscillator. VFO startup
   Variable frequency oscillator. Gain limiting
   Receiver incremental tuning. Crystal oscillators
   Mixers. Gilbert cell
   Superheterodyne receivers. Spurious responses of mixers
   Decreasing channel bandwidth by using CW
   Audio amplifiers
   JFETs as variable resistors
   Automatic gain control
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   Nyquist noise formula. Cascading noisy components. Noise figure
   Receiver intermodulation and dynamic range
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Ladder Filters. Butterworth and Chebyshev Filters

Ladder Filters. Butterworth and Chebyshev Filters. Filter Tables. ADS.
Ladder filters are networks that are composed of alternating series and shunt elements.

Notice that the same source and load resistances are assumed. This is called “doubly terminated” filters. All of our filters will be doubly terminated.
Ladder filters are actually one of the oldest types of filters. They have been around since the mid-1800’s. A circuit designer can achieve a sharper (or steeper) frequency roll off with ladder filters than with simple RC or RL circuits. Consequently, one can obtain more ideal low, high or band pass filter responses and with little resistive loss. Additionally, doubly terminated ladder filters have a low sensitivity to component variation. That is a good characteristic.
There are four basic types of ladder filters:
1. Maximally flat, also called Butterworth filters,
2. Equal ripple, also called Chebyshev filters,
3. Elliptic, also called Cauer filters,
4. Linear phase filters.
We will consider the first two in this course. The circuits in Figs. 1 and 2 can be either Butterworth or Chebyshev filters. The topology is the same for both. Only the values for L and C vary between the two types of filters. We will characterize these two filter types by the response of the loss factor L( f ) magnitude versus frequency. [The loss factor is
sometimes referred to as the insertion loss = IL = 10 log10(L).]
Maximum Available Power
Before further discussion of ladder filters, we must first define maximum available power, P+. This is the maximum time average power that can be provided by a source, or by the previous stage in the circuit, to a matched load. Consider that a source or previous circuit stage has been modeled by this Thévenin equivalent circuit:

As you determined in homework prob. 1, a dc source delivers maximum power when a resistive load Rs is connected to the output, similar to that shown above. For the ac circuit shown here, the maximum power delivered to the load Rs is

In summary, P+ is the maximum available power from an ac source (or a Thévenin equivalent) with internal resistance Rs. It is the maximum time-average power that can be delivered to a matched source. Very important formula. (Note that Vs is the amplitude, not p-to-p.)
1. Maximally Flat, or Butterworth, Low Pass Filter
For this filter, the values of the inductors and capacitors are somehow chosen so that

Where LB is the loss factor as a function of f. In this expression:
• Pi = maximum available power from the source (see Lecture 10),
• P = delivered power to the load,
• fc = cutoff frequency of the filter,
• n = order of the filter (number of L’s and C’s in high and low pass filter; number of L-C pairs in bandpass filters).
For the Butterworth (maximally flat) low pass filter

2. Equal Ripple, or Chebyshev Low Pass Filter
The values of the inductors and capacitors in this type of filter are somehow chosen so that

In this expression:
• ? = ripple size,
• Cnn)) Chebyshev polynomial of order n
Chebyshev filters might be more susceptible to variations in component values than Butterworth filters. This is due to the large coefficients of the polynomials listed

• Whether to use Butterworth, Chebyshev or another filter type depends on the specifications/requirements of the circuit (required rejection, roll off, phase variation, etc.), the available components, component value variations and so on. • Once you have the specifications, then you can synthesize the filter. The required filter specifications are:

• With these specifications, you can calculate the specific inductor and capacitor values needed to realize the filter (i.e., “synthesize” it). It is a complicated procedure to derive the formulas for these component values. There are entire books devoted to this topic. (See the attachment at the end of this lecture for a simple example.)
• Instead of deriving these formulas, designers often simply use filter tables. These are tabulated values for normalized susceptance and reactance (collectively called immittance, a). To un-normalize values from filter tables for low pass filters, Use

RN and ?N are the normalization values used in the tables (often both = 1), while R and ?c are the actual circuit values. An example will help explain this procedure.
Design a fifth-order, Butterworth, low-pass filter (see Fig. 1 above) with a cutoff frequency of 8 MHz, a rejection of at least 23 dB at 14 MHz and an impedance level of 50 ?. With a fifth order filter, n = 5. From (5.1) and f/fc = 14/8 then

which meets the 23 dB spec. (Note that there is also loss in the passband. At 7 MHz, for example, IL = 10 log10[1+(7/8)10]=1.0 dB. Where does this “lost” energy go?) Now, for this fifth-order Butterworth filter we read the immittance coefficients from Table 5.1 to be a1 = 0.618, a2 =1.618, a3 = 2, a 4=1.618 and a5 = 0.618.
For a low pass filter, these immittance coefficients are the normalized susceptances of the shunt elements at fc and the normalized reactances of the series elements at fc.

For R = 50 ? and W3=2 ? Fc=5.027 10 ×107 rad/s (at 8 MHz), then

All of these values are “in the ballpark” for the Harmonic Filter. Of course, one generally needs to use standard values of components for the filter, unless you build your own inductors and/or capacitors. Consequently, the circuit may need to be “tweaked” after completing this synthesis step.
Advanced Design System (ADS)
This tweaking process can be performed using analysis software such as SPICE, Puff or Advanced Design System (ADS). Your text uses the passive microwave circuit simulator called Puff, which comes with your text. It is DOS-based and requires the use of “scattering parameters” to characterize the behavior of circuits, including filters. (S parameters are discussed extensively in EE 481 Microwave Engineering.) For these, and other, reasons we will NOT be using Puff in this course. Instead, we will be using Advanced Design System (ADS) from Agilent Technologies. Consequently, all of the text problems that refer to Puff have been rewritten to use ADS. These can be found on the course web site. The manual “Getting Started with ADS” has been written to help you get going with ADS. It can also be found on the course web site. ADS has just a couple of nuances. Other than that, it is very straightforward to use. To illustrate the use of ADS, we will verify the proper operation of the low-pass filter designed previously.
ADS Simulation of a Low-Pass Ladder Filter
ADS Startup Window:

To get going with ADS, you must first create a “project”:

ADS example with Rs = 50 ?:

Here is a plot of Pout/Pin in dB:

This doesn’t look like the response of a maximally flat low pass filter. What’s wrong? Here’s a plot of |Vout| Vin| in dB:

This plot has the general shape of a maximally flat filter, but there is an extra 6 dB of attenuation at the design frequency of 7 MHz. What’s going on here? Lastly, here’s a plot of Pout/P+ in dB where P+ is the maximum available power from the source:

Alas, this is the plot we’ve been looking for. Why? Because by definition, insertion loss is the ratio of the output power to the maximum avaliable source power. See (5.1) as an example. From this last plot, we can see that ADS predicts an insertion loss of –1.017 dB at 7.000 MHz. This is very close to our design prediction of –1.0 dB at 7 MHz.
ADS example with Rs = 100 ?:

Changing the impedance “level” (from 50 ? to 100 ?) has a dramatic effect on the performance of the filter. Can you explain why?

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