Self impedance

In Fig. P2.6 we show a single dipole with self-impedance Zn backed by a single parasitic element with self-impedance Z2,2 and load impedance Their mutual impedances are denoted Z12 = Z2,i. 

When all the mutual and the self-impedances of an assembly of stick arrays as shown in Fig. 3.8 has been determined, it is a relatively simple matter to find the currents on the elements. In fact, it was already discussed in my earlier book [62] and need therefore only to be restated here for easy reference. 

The self-impedance of a short dipole is indeed comprised of a low radiation resistance in series with a high capacitive reactance. Thus, when it is suggested to use these elements to build a broadband array, it is quite often discarded as being a bad idea. 

Fig. 6.18 Self-impedance of a short dipole is indeed capacitive. However, it contains no big bad capacitance. In fact, a dipole is comprised of an inductive wire Lo near its terminals and a capacitance Co at the tips. Thus, if Lo is reduced, Co must be increased to maintain resonance. Any conclusion to the contrary is based on voodoo physics. This figure explains why the capacitance between dipole tips must be increased if the dipoles are shortened and the resonance frequency is to be maintained.

The structure to be investigated is shown in Fig. 9.2. It is comprised of two subarrays, each with interelement spacings and D. They are interlaced into each other such that array 1 has its reference element located at the origin while the reference element for array 2 is located at (<4, 0, 0). We denote the element loads by jXii and jXi2 for arrays 1 and 2, respectively. However, we assume that the elements in both arrays have the same length 21. That simplifies our analysis by the fact that the self-impedances and for the two arrays are identical. Thus, the perturbation of e elements is obtained purely by variation of the load resistances jXii and jXii and/or dx. This corresponds approximately to variation of the element length. Furthermore, we denote the mutual impedances between the two arrays by Z and Z ... 

Inspection of (D.IO) shows that the embedded stick impedance Zemb stk consists of the stick self impedance Z° ° for the center array plus two terms representing the overcoupled impedances from the two outer arrays. 

It can be shown that the embedded stick impedance for any size finite array is structured the same way—that is, as a sum of the stick self-impedance Z° ° plus overcoupled terms associated with all the other parasitically excited stick arrays. 

We have calculated the stick self-impedance Z for an array without ground-plane with dimensions as used in Chapter 6 as obtained from the SPLAT program. This program cannot handle dielectric slabs however, it will approximate the effect of the dielectric underwear [136] by placing cylindrical dielectric shells around each element. The thickness of these dielectric coatings should be approximately equal to the thickness of the underwear. ... 

Thus, we show in Fig. D.5 an example of the stick self-impedance Z° ° with dimensions as shown in the insert and as obtained from the SPLAT program (includes a matching transmission hne). 

Finally we show in Fig. D.8 the self-impedance of a single dipole without capacitors or any other elements present but includes matching sections (see text earlier). As we would expect, the real part is low and the reactance is capacitive and high. Obviously, this is what sticks in peoples minds when they say that the type of array discussed in Chapter 6 will never have a chance for yielding a broad bandwidth (no pun intended). Note how the impedance shrinks as more elements are added and excited as observed in Fig. D.7. 

Fig. D.5 The stick self-impedance 7P ° for an array without a groundplane. Eiement dimensions identical to the case in Fig. D.3 (see insert). Obtained from the SPLAT program. The dieiectric underwear is rrxxleled by placing dielectric shells around the elements. Diameter approximately equal to total thickness of the underwear, includes a matching transformer (see text).

Fig. D.8 The self-impedance of a single dipole with total length equal to the element section used in Fig. D. 7 soe insert). A dielectric cylinder was placed around the element to model the underwear. Includes a matching section (see tex. ...

Thus, we have plotted all impedances in Smith charts normalized to the same impedance, namely 100 ohms. We can then compare the scan impedance Za as given by (D.5) in Fig. D.3 with the stick self-impedance as shown in Fig. D.5. Although some similarities are found at the middle to higher frequencies, the difference at the lower frequencies is simply enormous. One could then wonder whether that difference could be reconciled if we instead of the stick self-impedance Z would consider the embedded stick impedance Zen stk as... 

To summarize Using the embedded stick impedance Zemb stk instead of the stick self-impedance Z may lead to even greater deviation from the scan impedance Z at the middle frequency range. At the lower frequencies, Z and Zemb stk are similar but both differ substantially from the scan impedance Z. ... 

Where Zu is the self-impedance of the ith element and Z,y is the mutual impedance between the ith and jth elements. These lead to expressions for the driving point or input impedance viewed by each source connected to the elements. Thus, from these port equations one obtains... 

The term "operating impedance" is defined as the complex ratio of the voltage applied to a load to the current flowing in the load when it is operating under normal power and in its normal environment. In many cases, this impedance differs substantially from the "self-impedance" or "cold impedance" of the load. In antenna systems a radiator has a certain self-impedance when operating alone. When it is combined in an array to form a directional antenna, its operating impedance may differ substantially from its self-impedance because of coupled impedance from other radiators of the array. 

Quite often, in complex antenna systems, it is found that one or more of the elements has a negative operating impedance that is, the total of the coupled Impedance from all other elements exceeds the self-impedance of that element, and the element actually returns power to the transmitter. 

When considering the mutual impedance between an overhead conductor and an underground cable or a buried gas and/or water pipeline, the self-impedance of the overhead conductor is given by Equation 1.11, while that of the underground conductor is given by Equation 1.12. Mutual impedance will be explained in Section I.2.2.3. 

In the case of a vertical line, it should be noted that the relation of magnitudes changes as frequency changes. For example, self-impedance is present in the following relations ... 

In a simulation of an induced voltage to an overhead control cable from a counterpoise, the control cable and the counterpoise are represented as a distributed-parameter line in the EMTP [35, 36-37]. The parameters of the line models are evaluated by the EMTP Cable Parameters (CP) [37]. First, the model system is evaluated as an overhead line system by the CP with a negative sign of the depth of the counterpoise. Initially, the input data, the CP gives the self-impedance/admittance of the overhead cable and the mutual impedance to the counterpoise. Then, the self-impedance/-admittance of the counterpoise is calculated as an underground cable. Finally, the self-impedance/admittance of the counterpoise in the first calculation is replaced by those in the second. 

In fhe verfical line case, if should be noted fhaf fhe relafion of magnifudes changes as frequency changes. For example, fhe self-impedance is in fhe following relafion ... 

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