IntroductionThe following discussion presents a method f o r determining the complete electrical characteristics ob a log periodic dipole antenna given only the physical parameters of the antenna. The method used is to find an integral equation relating the dri\*ng voltage to the actual currents flowing on the antenna elements and then to solve this equation for the element currents. Solving the integral equation is accomplished hy approximating the integral by a finite s u m and then solving the resulting system of simultaneous linear equations on a computer. Once the currents on the elements a r e known as a function of driving voltage, determining such antenna p a r m e t e r s as gain, input impedance and radiation patterns is just a matter of a few simple calculations.The theory is first derived for a single dipole and is then extended to the case of a complete log periodic dipole antenna.Finally, some of the numerical results obtained are presented along with some rr-easured data t o show the comparison between the mathematical model and an actual physical model. TheoryTo find an expression for the current on a dipole in terms of the applied voltage it is convenient to first derive an equation relating the voltage to the vector potential, and then find an e v r e s s i o n f o r the vector potential in terms of the current.The electric field at any point outside a dipole can be e q r e s s e d as 55 Now definingAssuming the dipole to be parallel to the Z-axis, Equation (3) reduces to where k2 = w p a . 2 F o r a perfect conductor the tangential electric field is zero on the boundary and Equation (4) becomes d 2 A Z 2 d z f k A Z = Q With a slice generator of voltage V applied at the center o f the dipole, Haltdnl found the solution of Equation (5) to be of the form where B is a constant of integration.This is the vector potential-voltage relationship we set out to find.To relate the voltage to the current flowing on the dipole, we can now use the known relationship between the vector potential, A 2 , and the dipole current, where R i s the distance between the point in space where AZ is to be determined and the point on the dipole where the current, I;, is flowing. 56Combining Equations ( 6 ) and (7) we obtain the desired current-voltsge relationship,In o r d e r to solve this integral equation, we approximate the integral by a finite sum of c u r r e n t s along the dipole. Each t e r m ir. this sum contains an unknown value of current. To solve for these unknown current values we need the same number of equations as we have increments of current along the dipole. By specifying the value of Z to a specific increment along the dipole, one equation can be written for each current segment. W i t h the dipole divided into N segments, we now have N nurnber of independent equations. A typical equation for the ith segment is (9) n = l n n where n covers all dipole segments, including the ith segment, In addition to the N number of unknown c u r r e n t s , t h e r e is the unknown constant of integration, B. Howe...
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