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**EKT 441 MICROWAVE COMMUNICATIONS**

CHAPTER 4: MICROWAVE FILTERS

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**INTRODUCTION What is a Microwave filter ? linear 2-port network**

controls the frequency response at a certain point in a microwave system provides perfect transmission of signal for frequencies in a certain passband region infinite attenuation for frequencies in the stopband region a linear phase response in the passband (to reduce signal distortion). f2

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**Commonly used block Diagram of a Filter**

INTRODUCTION The goal of filter design is to approximate the ideal requirements within acceptable tolerance with circuits or systems consisting of real components. f1 f3 f2 Commonly used block Diagram of a Filter

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**INTRODUCTION Why Use Filters? RF signals consist of:**

Desired signals – at desired frequencies Unwanted Signals (Noise) – at unwanted frequencies That is why filters have two very important bands/regions: Pass Band – frequency range of filter where it passes all signals Stop Band – frequency range of filter where it rejects all signals

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**INTRODUCTION Categorization of Filters**

Low-pass filter (LPF), High-pass filter (HPF), Bandpass filter (BPF), Bandstop filter (BSF), arbitrary type etc. In each category, the filter can be further divided into active and passive types. In active filter, there can be amplification of the of the signal power in the passband region, passive filter do not provide power amplification in the passband. Filter used in electronics can be constructed from resistors, inductors, capacitors, transmission line sections and resonating structures (e.g. piezoelectric crystal, Surface Acoustic Wave (SAW) devices, and also mechanical resonators etc.). Active filter may contain transistor, FET and Op-amp. Filter LPF BPF HPF Active Passive

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**INTRODUCTION Types of Filters Low-pass Filter Passes low freq**

Rejects high freq High-pass Filter Passes high freq Rejects low freq f1 f1 f1 f2 f2 f2

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**INTRODUCTION Band-pass Filter Band-stop Filter**

Passes a small range of freq Rejects all other freq Band-stop Filter Rejects a small range of freq Passes all other freq f1 f1 f1 f2 f2 f2 f3 f3 f3

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**INTRODUCTION Filter Parameters**

Pass bandwidth; BW(3dB) = fu(3dB) – fl(3dB) Stop band attenuation and frequencies, Ripple difference between max and min of amplitude response in passband Input and output impedances Return loss Insertion loss Group Delay, quality factor

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**INTRODUCTION (1.1a) (1.1b) Low-pass filter (passive). c 1 A Filter**

|H()| 1 Transfer function Arg(H()) Low-pass filter (passive). A Filter H() V1() V2() ZL A()/dB c 3 10 20 30 40 50 (1.1b)

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INTRODUCTION For impedance matched system, using s21 to observe the filter response is more convenient, as this can be easily measured using Vector Network Analyzer (VNA). Filter Zc Vs a1 b2 Transmission line is optional Zc c 20log|s21()| 0dB Arg(s21()) Complex value

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**INTRODUCTION Low pass filter response (cont) Transition band Passband**

A()/dB c 3 10 20 30 40 50 Passband Stopband Transition band Cut-off frequency (3dB) A Filter H() V1() V2() ZL

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**INTRODUCTION High Pass filter Passband Stopband Transfer function**

A()/dB c 3 10 20 30 40 50 c |H()| 1 Transfer function Stopband Passband

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**INTRODUCTION Band-pass filter (passive). Band-stop filter. A()/dB**

40 1 3 30 20 10 2 o A()/dB 40 1 3 30 20 10 2 o 1 |H()| 1 Transfer function 2 o 1 |H()| 1 Transfer function 2 o

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**INTRODUCTION Figure 4.1: A 10 GHz Parallel Coupled Filter Response**

Insertion Loss Pass BW (3dB) Q factor Figure 4.1: A 10 GHz Parallel Coupled Filter Response Stop band frequencies and attenuation

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**FILTER DESIGN METHODS Filter Design Methods**

Two types of commonly used design methods: - Image Parameter Method - Insertion Loss Method Image parameter method yields a usable filter However, no clear-cut way to improve the design i.e to control the filter response

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**FILTER DESIGN METHODS Filter Design Methods**

The insertion loss method (ILM) allows a systematic way to design and synthesize a filter with various frequency response. ILM method also allows filter performance to be improved in a straightforward manner, at the expense of a ‘higher order’ filter. A rational polynomial function is used to approximate the ideal |H()|, A() or |s21()|. Phase information is totally ignored.Ignoring phase simplified the actual synthesis method. An LC network is then derived that will produce this approximated response. Here we will use A() following [2]. The attenuation A() can be cast into power attenuation ratio, called the Power Loss Ratio, PLR, which is related to A()2.

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**FILTER DESIGN METHODS (2.1a) ZL Vs Lossless 2-port network 1 Zs PA**

Pin PL PLR large, high attenuation PLR close to 1, low attenuation For example, a low-pass filter response is shown below: PLR(f) (2.1a) f 1 Low attenuation High fc Low-Pass filter PLR

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PLR and s21 In terms of incident and reflected waves, assuming ZL=Zs = ZC. Zc Vs Lossless 2-port network PA Pin PL a1 b1 b2 (2.1b)

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**FILTER RESPONSES Filter Responses Several types filter responses:**

- Maximally flat (Butterworth) - Equal Ripple (Chebyshev) - Elliptic Function - Linear Phase

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**THE INSERTION LOSS METHOD**

Practical filter response: Maximally flat: - also called the binomial or Butterworth response, - is optimum in the sense that it provides the flattest possible passband response for a given filter complexity. - no ripple is permitted in its attenuation profile [8.10] – frequency of filter c – cutoff frequency of filter N – order of filter

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**THE INSERTION LOSS METHOD**

Equal ripple - also known as Chebyshev. - sharper cutoff - the passband response will have ripples of amplitude 1 +k2 [8.11] – frequency of filter c – cutoff frequency of filter N – order of filter

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**THE INSERTION LOSS METHOD**

Figure 5.3: Maximally flat and equal-ripple low pass filter response.

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**THE INSERTION LOSS METHOD**

Elliptic function: - have equal ripple responses in the passband and stopband. - maximum attenuation in the passband. - minimum attenuation in the stopband. Linear phase: - linear phase characteristic in the passband - to avoid signal distortion - maximally flat function for the group delay.

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**THE INSERTION LOSS METHOD**

Figure 5.4: Elliptic function low-pass filter response

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**THE INSERTION LOSS METHOD**

Low-pass Prototype Design Filter Specification Scaling & Conversion Normally done using simulators Optimization & Tuning Filter Implementation Figure 5.5: The process of the filter design by the insertion loss method.

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**THE INSERTION LOSS METHOD**

Low Pass Filter Prototype Figure 5.6: Low pass filter prototype, N = 2

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**THE INSERTION LOSS METHOD**

Low Pass Filter Prototype – Ladder Circuit Figure 5.7: Ladder circuit for low pass filter prototypes and their element definitions. (a) begin with shunt element. (b) begin with series element.

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**THE INSERTION LOSS METHOD**

g0 = generator resistance, generator conductance. gk = inductance for series inductors, capacitance for shunt capacitors. (k=1 to N) gN+1 = load resistance if gN is a shunt capacitor, load conductance if gN is a series inductor. As a matter of practical design procedure, it will be necessary to determine the size, or order of the filter. This is usually dictated by a specification on the insertion loss at some frequency in the stopband of the filter.

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**THE INSERTION LOSS METHOD**

Low Pass Filter Prototype – Maximally Flat Figure 4.8: Attenuation versus normalized frequency for maximally flat filter prototypes.

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**THE INSERTION LOSS METHOD**

Figure 4.9: Element values for maximally flat LPF prototypes

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**THE INSERTION LOSS METHOD**

Low Pass Filter Prototype – Equal Ripple For an equal ripple low pass filter with a cutoff frequency ωc = 1, The power loss ratio is: [5.12] Where 1 + k2 is the ripple level in the passband. Since the Chebyshev polynomials have the property that [5.12] shows that the filter will have a unity power loss ratio at ω = 0 for N odd, but the power loss ratio of 1 + k2 at ω = 0 for N even.

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**THE INSERTION LOSS METHOD**

Figure 4.10: Attenuation versus normalized frequency for equal-ripple filter prototypes. (0.5 dB ripple level)

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**THE INSERTION LOSS METHOD**

Figure 4.11: Element values for equal ripple LPF prototypes (0.5 dB ripple level)

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**THE INSERTION LOSS METHOD**

Figure 4.12: Attenuation versus normalized frequency for equal-ripple filter prototypes (3.0 dB ripple level)

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**THE INSERTION LOSS METHOD**

Figure 4.13: Element values for equal ripple LPF prototypes (3.0 dB ripple level).

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**FILTER TRANSFORMATIONS**

Low Pass Filter Prototype – Impedance Scaling [8.13a] [8.13b] [8.13c] [8.13d]

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**FILTER TRANSFORMATIONS**

Frequency scaling for the low pass filter: [8.14] The new element values of the prototype filter: [8.15a] [8.15b]

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**FILTER TRANSFORMATIONS**

The new element values are given by: [8.16a] [8.16b]

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**FILTER TRANSFORMATIONS**

Low pass to high pass transformation The frequency substitution: [8.17] The new component values are given by: [8.18a] [8.18b]

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**BANDPASS & BANDSTOP TRANSFORMATIONS**

Low pass to Bandpass transformation [8.19] Where, [8.20] The center frequency is: [8.21]

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**BANDPASS & BANDSTOP TRANSFORMATIONS**

The series inductor, Lk, is transformed to a series LC circuit with element values: [8.22a] [8.22b] The shunt capacitor, Ck, is transformed to a shunt LC circuit with element values: [8.23a] [8.23b]

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**BANDPASS & BANDSTOP TRANSFORMATIONS**

Low pass to Bandstop transformation [8.24] Where, The center frequency is:

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**BANDPASS & BANDSTOP TRANSFORMATIONS**

The series inductor, Lk, is transformed to a parallel LC circuit with element values: [8.25a] [8.25b] The shunt capacitor, Ck, is transformed to a series LC circuit with element values: [8.26a] [8.26b]

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**BANDPASS & BANDSTOP TRANSFORMATIONS**

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EXAMPLE 5.1 Design a maximally flat low pass filter with a cutoff freq of 2 GHz, impedance of 50 Ω, and at least 15 dB insertion loss at 3 GHz. Compute and compare with an equal-ripple (3.0 dB ripple) having the same order.

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**EXAMPLE 5.1 (Cont) Solution:**

First find the order of the maximally flat filter to satisfy the insertion loss specification at 3 GHz. We can find the normalized freq by using:

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EXAMPLE 5.1 (Cont) The ladder diagram of the LPF prototype to be used is as follow:

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**EXAMPLE 5.1 (Cont) pF nH pF nH pF**

LPF prototype for maximally flat filter pF nH pF nH pF

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**EXAMPLE 5.1 (Cont) pF nH pF nH pF**

LPF prototype for equal ripple filter: pF nH pF nH pF

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**THE INSERTION LOSS METHOD**

Low-pass Prototype Design Filter Specification Scaling & Conversion Normally done using simulators Optimization & Tuning Filter Implementation

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**SUMMARY OF STEPS IN FILTER DESIGN**

Filter Specification Max Flat/Equal Ripple, If equal ripple, how much pass band ripple allowed? If max flat filter is to be designed, cont to next step Low Pass/High Pass/Band Pass/Band Stop Desired freq of operation Pass band & stop band range Max allowed attenuation (for Equal Ripple)

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**SUMMARY OF STEPS IN FILTER DESIGN (cont)**

Low Pass Prototype Design Min Insertion Loss level, No of Filter Order/Elements by using IL values Determine whether shunt cap model or series inductance model to use Draw the low-pass prototype ladder diagram Determine elements’ values from Prototype Table

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**SUMMARY OF STEPS IN FILTER DESIGN (cont)**

Scaling and Conversion Determine whether if any modification to the prototype table is required (for high pass, band pass and band stop) Scale elements to obtain the real element values

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**SUMMARY OF STEPS IN FILTER DESIGN (cont)**

Filter Implementation Put in the elements and values calculated from the previous step Implement the lumped element filter onto a simulator to get the attenuation vs frequency response

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EXAMPLE 5.2 Design a band pass filter having a 0.5 dB equal-ripple response, with N = 3. The center frequency is 1 GHz, the bandwidth is 10%, and the impedance is 50 Ω.

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**EXAMPLE 5.2 (Cont) RS L1 L3 C2 RL**

Solution: The low pass filter (LPF) prototype ladder diagram is shown as follow: = 0.1 N = 3 = 1 GHz RS L1 L3 C2 RL

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EXAMPLE 5.2 (Cont) From the equal ripple filter table (with 0.5 dB ripple), the filter elements are as follow;

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**EXAMPLE 5.2 (Cont) RS RL C1 C2 C3 L1 L2 L3**

Transforming the LPF prototype to the BPF prototype RS RL C1 C2 C3 L1 L2 L3

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EXAMPLE 5.2 (Cont)

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EXAMPLE 5.2 (Cont)

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EXAMPLE 5.3 Design a five-section high pass lumped element filter with 3 dB equal-ripple response, a cutoff frequency of 1 GHz, and an impedance of 50 Ω. What is the resulting attenuation at 0.6 GHz?

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**EXAMPLE 5.3 (Cont) RS RL L1 L3 L5 C2 C3**

Solution: The high pass filter (HPF) prototype ladder diagram is shown as follow: N = 5 = 1 GHz At c = 0.6 GHz, ; referring back to Fig 4.12 The attenuation for N = 5, is about 41 dB RS RL L1 L3 L5 C2 C3

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EXAMPLE 5.3 (Cont) From the equal ripple filter table (with 3.0 dB ripple), the filter elements are as follow;

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EXAMPLE 5.3 (Cont) Impedance and frequency scaling:

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EXAMPLE 5.3 (Cont)

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EXAMPLE 5.4 Design a 4th order Butterworth Low-Pass Filter. Rs = RL= 50Ohm, fc = 1.5GHz. L1=0.7654H L2=1.8478H C1=1.8478F C2=0.7654F RL= 1 g0= 1 Step 1&2: LPP Step 3: Frequency scaling and impedance denormalization L1=4.061nH L2=9.803nH C1=3.921pF C2=1.624pF RL= 50 g0=1/50

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EXAMPLE 5.5 Design a 4th order Chebyshev Low-Pass Filter, 0.5dB ripple factor. Rs = 50Ohm, fc = 1.5GHz. L1=1.6703H L2=2.3661H C1=1.1926F C2=0.8419F RL= g0= 1 Step 1&2: LPP Step 3: Frequency scaling and impedance denormalization L1=8.861nH L2=12.55nH C1=2.531pF C2=1.787pF RL= 99.2 g0=1/50

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EXAMPLE 5.6 Design a bandpass filter with Butterworth (maximally flat) response. N = 3. Center frequency fo = 1.5GHz. 3dB Bandwidth = 200MHz or f1=1.4GHz, f2=1.6GHz.

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EXAMPLE 5.6 (cont) From table, design the Low-Pass prototype (LPP) for 3rd order Butterworth response, c=1. Zo=1 g1 1.000F g3 g2 2.000H g4 1 2<0o Step 1&2: LPP

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**EXAMPLE 5.6 (cont) LPP to bandpass transformation.**

Impedance denormalization. Step 3: Frequency scaling and impedance denormalization 50 Vs 15.916pF 0.1414pF 79.58nH 0.7072nH RL

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