Lecture 8 Periodic Structures Image Parameter Method Insertion Loss Method Filter Transformation EE 41139 Microwave Technique

Periodic Structures periodic structures have passband and stopband characteristics and can be employed as filters EE 41139 Microwave Technique

Periodic Structures consider a microstrip transmission line periodically loaded with a shunt susceptance b normalized to the characteristic impedance Zo: EE 41139 Microwave Technique

Periodic Structures the ABCD matrix is composed by cascading three matrices, two for the transmission lines of length d/2 each and one for the shunt susceptance, EE 41139 Microwave Technique

Periodic Structures i.e. EE 41139 Microwave Technique

Periodic Structures q = kd, and k is the propagation constant of the unloaded line AD-BC = 1 for reciprocal networks assuming the the propagation constant of the loaded line is denoted by g, then EE 41139 Microwave Technique

Periodic Structures therefore, or EE 41139 Microwave Technique

Periodic Structures for a nontrivial solution, the determinant of the matrix must vanish leading to recall that AD-CB = 0 for a reciprocal network, then Or EE 41139 Microwave Technique

Periodic Structures Knowing that, the above equation can be written as since the right-hand side is always real, therefore, either a or b is zero, but not both EE 41139 Microwave Technique

Periodic Structures if a=0, we have a passband, b can be obtained from the solution to if the the magnitude of the rhs is less than 1 EE 41139 Microwave Technique

Periodic Structures if b=0, we have a stopband, a can be obtained from the solution to as cosh function is always larger than 1, a is positive for forward going wave and is negative for the backward going wave EE 41139 Microwave Technique

Periodic Structures therefore, depending on the frequency, the periodic structure will exhibit either a passband or a stopband EE 41139 Microwave Technique

Periodic Structures the characteristic impedance of the load line is given by , + for forward wave and - for backward wavehere the unit cell is symmetric so that A = D ZB is real for the passband and imaginary for the stopband EE 41139 Microwave Technique

Periodic Structures when the periodic structure is terminated with a load ZL , the reflection coefficient at the load can be determined easily EE 41139 Microwave Technique

Periodic Structures Which is the usual result EE 41139 Microwave Technique

Periodic Structures it is useful to look at the k-b diagram (Brillouin) of the periodic structure EE 41139 Microwave Technique

Periodic Structures in the region where b < k, it is a slow wave structure, the phase velocity is slow down in certain device so that microwave signal can interacts with electron beam more efficiently when b = k, we have a TEM line EE 41139 Microwave Technique

Filter Design by the Image Parameter Method let us first define image impedance by considering the following two-port network EE 41139 Microwave Technique

Filter Design by the Image Parameter Method if Port 2 is terminated with Zi2, the input impedance at Port 1 is Zi1 if Port 1 is terminated with Zi1, the input impedance at Port 2 is Zi2 both ports are terminated with matched loads EE 41139 Microwave Technique

Filter Design by the Image Parameter Method at Port 1, the port voltage and current are related as the input impedance at Port 1, with Port 2 terminated in , is EE 41139 Microwave Technique

Filter Design by the Image Parameter Method similarly, at Port 2, we have these are obtained by taking the inverse of the ABCD matrix knowing that AB-CD=1 the input impedance at Port 2, with Port 1 terminated in , is EE 41139 Microwave Technique

Filter Design by the Image Parameter Method Given and , we have , , if the network is symmetric, i.e., A = D, then EE 41139 Microwave Technique

Filter Design by the Image Parameter Method if the two-port network is driven by a voltage source EE 41139 Microwave Technique

Filter Design by the Image Parameter Method Similarly we have, , A = D for symmetric network Define , EE 41139 Microwave Technique

Filter Design by the Image Parameter Method consider the low-pass filter EE 41139 Microwave Technique

Filter Design by the Image Parameter Method the series inductors and shunt capacitor will block high-frequency signals a high-pass filter can be obtained by replacing L/2 by 2C and C by L in T-network EE 41139 Microwave Technique

Filter Design by the Image Parameter Method the ABCD matrix is given by Image impedance EE 41139 Microwave Technique

Filter Design by the Image Parameter Method Propagation constant For the above T-network, EE 41139 Microwave Technique

Filter Design by the Image Parameter Method Define a cutoff frequency as, a nominal characteristic impedance Ro , k is a constant EE 41139 Microwave Technique

Filter Design by the Image Parameter Method the image impedance is then written as the propagation factor is given as EE 41139 Microwave Technique

Filter Design by the Image Parameter Method For , is real and which imply a passband For , is imaginary and we have a stopband EE 41139 Microwave Technique

Filter Design by the Image Parameter Method this is a constant-k low pass filter, there are two parameters to choose (L and C) which are determined by wc and Ro when , the attenuation is slow, furthermore, the image impedance is not a constant when frequency changes EE 41139 Microwave Technique

Filter Design by the Image Parameter Method the m-derived filter section is designed to alleviate these difficulties let us replace the impedances Z1 with EE 41139 Microwave Technique

Filter Design by the Image Parameter Method we choose Z2 so that ZiT remains the same therefore, Z2 is given by EE 41139 Microwave Technique

Filter Design by the Image Parameter Method recall that Z1 = jwL and Z2 = 1/jwC, the m-derived components are EE 41139 Microwave Technique

Filter Design by the Image Parameter Method the propagation factor for the m-derived section is EE 41139 Microwave Technique

Filter Design by the Image Parameter Method if we restrict 0 < m < 1, is real and >1 , for w > the stopband begins at w = as for the constant-k section When w = , where e becomes infinity and the filter has an infinite attenuation EE 41139 Microwave Technique

Filter Design by the Image Parameter Method when w > , the attenuation will be reduced; in order to have an infinite attenuation when , we can cascade a the m-derived section with a constant-k section to give the following response EE 41139 Microwave Technique

Filter Design by the Image Parameter Method the image impedance method cannot incorporate arbitrary frequency response; filter design by the insertion loss method allows a high degree of control over the passband and stopband amplitude and phase characteristics EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method if a minimum insertion loss is most important, a binomial response can be used if a sharp cutoff is needed, a Chebyshev response is better in the insertion loss method a filter response is defined by its insertion loss or power loss ratio EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method , IL = 10 log , , M and N are real polynomials EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method for a filter to be physically realizable, its power loss ratio must be of the form shown above maximally flat (binomial or Butterworth response) provides the flattest possible passband response for a given filter order N EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method The passband goes from to , beyond , the attenuation increases with frequency the first (2N-1) derivatives are zero for and for , the insertion loss increases at a rate of 20N dB/decade EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method equal ripple can be achieved by using a Chebyshev polynomial to specify the insertion loss of an N-order low-pass filter as EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method a sharper cutoff will result; (x) oscillates between -1 and 1 for |x| < 1, the passband response will have a ripple of 1+ in the amplitude For large x, and therefore for EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method therefore, the insertion loss of the Chebyshev case is times of the binomial response for linear phase response is sometime necessary to avoid signal distortion, there is usually a tradeoff between the sharp-cutoff response and linear phase response EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method a linear phase characteristic can be achieved with the phase response EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method a group delay is given by this is also a maximally flat function, therefore, signal distortion is reduced in the passband EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method it is convenient to design the filter prototypes which are normalized in terms of impedance and frequency the designed prototypes will be scaled in frequency and impedance lumped-elements will be replaced by distributive elements for microwave frequency operations EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method consider the low-pass filter prototype, N=2 EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method assume a source impedance of 1 W and a cutoff frequency the input impedance is given by EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method the reflection coefficient at the source impedance is given by the power loss ratio is given by EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method compare this equation with the maximally flat equation, we have R=1, which implies C = L as R = 1 which implies C = L = EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method for equal-ripple prototype, we have the power loss ratio Since Compare this with EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method we have or note that R is not unity, a mismatch will result if the load is R=1; a quarter-wave transformer can be used to match the load EE 41139 Microwave Technique

Filter Design by the Insertion Loss Method if N is odd, R = 1 as there is a unity power loss ratio at w = 0 of N being odd Table 9.4 can be used for equal-ripple low-pass filter prototypes Table 9.5 can be used for maximally flat time delay low-pass filter prototypes after the filter prototypes have been designed, we need to perform impedance and frequency scaling EE 41139 Microwave Technique

Filter Transformations impedance and frequency scaling the source impedance is , the impedance scaled quantities are: EE 41139 Microwave Technique

Filter Transformations both impedance and frequency scaling low-pass to high-pass transformation , EE 41139 Microwave Technique

Filter Transformations Bandpass transmission As a series indicator , is transformed into a series LC with element values A shunt capacitor, , is transformed into a shunt LC with element values EE 41139 Microwave Technique

Filter Transformations bandstop transformation A series indicator, , is transformed into a parallel LC with element values A shunt capacitor, , is transformed into a series LC with element values EE 41139 Microwave Technique

Filter Implementation we need to replace lumped-elements by distributive elements: EE 41139 Microwave Technique

Filter Implementation there are four Kuroda identities to perform any of the following operations: physically separate transmission line stubs transform series stubs into shunt stubs, or vice versa change impractical characteristic impedances into more realizable ones EE 41139 Microwave Technique

Filter Implementation let us concentrate on the first two a shunt capacitor can be converted to a series inductor EE 41139 Microwave Technique