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Theoretical Information  
When two transmission lines are close together, because of the interaction of the electromagnetic fields of each line, power can be coupled between the lines. Those coupled lines are used to construct directional couplers. Generally, in design of directional couplers microstrip and stripline forms are used. Although microstrip transmission lines do not support TEM and named as quasiTEM, usually they are assumed to operate in TEM mode. It is important that whether true TEM or not, all parallel line couplers have odd and even mode, and resulting Zoe and Zoo (even and odd mode impedances respectively). In the analysis of the directional couplers we will use also evenodd mode analysis. There are many kinds of directional couplers in different forms. However
directional couplers we will mention here are single and three section
microstrip directional couplers.
1.Singlesection Microstrip Coupled Line Directional CouplerSinglesection microstrip directional couplers are constructed from the coupled
transmission microstrip lines. A singlesection microstrip directional coupler is shown in the Figure 1.
The crosssectional view of coupled microstrip lines are shown in the Figure 2. Looking at that figure, we say that in the design of coupled lines, we consider following parameters: w/h, s/h, er, eeff , Zoo, Zoe, Zo and Coupling. By using even and odd mode excitation as in the Figure 3, we find even and odd mode impedances Zoe and Zoo.
In design of coupled transmission lines when finding Zoo and Zoe we use the following approximate synthesis technique:
·
Firstly, with given Zo (single line characteristic impedance),
er (relative dielectric
constant of the substrate) and C (coupling of the coupled lines) determining
shape ratios for equivalent single microstrip lines.
Since we have in hand C (in dB) and Zo(impedance) let us find Zoe and Zoo as following:
as c is the coupling coefficient. Now, Zoo and Zoe are;
Using Zoe and Zoo if we consider Zoe and Zoo for single line as Zose and Zoso;
Now what we should do is to find (w/h)se and (w/h)so from Zose and Zoso. To do that let us use single line equations:
_{ }
where _{ }
_{ }
At that point, we are able to find (w/h)se and (w/h)so by applying Zose and Zoso (as Zo) to the single line microstrip equations. Now we can come to a point where we reach the w/h and s/h for the desired coupled microstrip line using a family of approximate equations as following:
As a result, we began from parameters C, Zo and er and at the end find w/h and s/h.
Until that point we only cared about the crosssectional parameters of the coupler. However, as it can be seen from Figure 1 directional coupler has an important parameter, length. In determining length of the directional coupler, in order to have better coupling, we should have a design frequency and arrange the length accordingly. Here for better coupling, our length consideration is l/4 (the reason of having that length will be better explained in the following analysis part). While calculating the length of the coupler we assume that the phase velocities of the coupler in even and odd mode are approximately equal. That means here is that we will only calculate effective permittivity and for even and odd mode we will consider it as equal. The important point here is to get the eeff by using the crosssectional parameters in our hand. In achieving this what we can use is the capacitances in our configuration. Although we assumed the even and odd mode phase velocities equal, in capacitance evaluations, even and odd mode permittivities are used and therefore we will use those to find effective permittivity. Capacitances of even and odd mode are shown in the Figure 4.
Now before introducing the capacitance evaluations, let us think about how we find eeff from capacitances: The evenmode and oddmode characteristic impedances are;
If we get Ce1 and Co1 from first two evenodd mode impedance equations and substitute them in above effective permittivity equations what we get is;
Writing the total capacitances for each mode;
The capacitance Cp simply relates to the parallelplate line value is given by
where c is the free space velocity. As it can be seen from Figure 4, there is oddmode fringing field capacitances for the air and dielectric regions across the coupling gap, which are Cga and Cgd, respectively. Determining Cga, it is obtained by using an equivalent coplanar strip geometry calculation,
and solve the above equation for e_{eff}. After finding the e_{eff }using capacitances, it is very easy to find the length of the directional coupler:
1.2.Analysis
The even and odd mode characteristic impedances are mentioned in the design part. At this point, in order to analyze the coupled line directional coupler we need to use an evenodd mode analysis. The schematic circuit for the coupled lines are shown in Figure 5. As it can be seen from the figure that our coupler which we analyze is a four port network. In our analysis, we terminate three of ports with the impedance Z_{0} and drive with a voltage generator of 2V and impedance of Z_{0} at port 1. Z_{0 }here is the characteristic impedance of single microstrip line. Moreover mentioning about the ports, port 1 is the input port, port 2 is the through port. port 3 is the coupled port and the port 4 is the isolated port. In addition in grounds in the Figure 5 are common. Applying the even–odd mode analysis to the coupler, we consider our coupler as in the Figure 6, in even mode feeding the port 1 and 3 with V and in odd mode feeding port 3 with –V and port 1 with V. If we use superposition technique and add all port voltages what we get is 0 for all three and 2V for port1. Due to the symmetry of the coupler voltages and currents shown in the Figure 6 are as following:
Then input impedance at port 1 of the coupler can be expressed as
At this point when we make a voltage division;
Using those and the Z_{in }equation mentioned above; that results in
Therefore
if above equations are satisfied then, port 1 and due to the symmetry
all the other ports will be matched. The above equation can be thought
as the input match condition. Writing the voltage at port 3:
now defining the midband coupling coefficient,c
and here V_{3}/V is the actual voltage coupling coefficient. Considering q=p/2 (at midband in which l=l/4 where l is the wavelength at the center frequncy) what we get is V_{3}/V=c. Moreover if we want to determine the characteristic of voltage coupling coefficient versus frequency we can use the above equation.
Similar to V_{3} consideration, we can think of V_{4 }as following:
Determining the Coupling in dB,
in midband therefore Z_{0e }and Z_{0o}can be derived easily from above equations as In the analysis we dealt with here, we made some necessary assumptions as following: · The quasistatic TEM mode of operation holds and so voltage and current are meaningful quantities on the lines, · For even and odd modes, transmission lines have the same e_{eff}. With this assumption, the even and odd modes will have the same phase velocity although in practice they are slightly different and this makes some degradation in the performance of the coupler. · The microstrip transmission lines here posses an appropriate plane of symmetry. To summarize, with the mentioned assumptions: i) Port 4 which is named also as Isolated port, always has a zero output, without being dependent of the electrical length of the coupling region. Major cause of poor isolation in practical circuits are unequal phase velocities of even and odd mode. ii) The input at each port is matched to the feed line characteristic impedance, Z_{0}, irrespective of the electrical length. iii) The total output power equals the input power. iv) The maximum coupling to port 2 occurs at the frequency that gives a quarter wave coupling length which is l/4. This is the midband frequency and due to this property, this couplers are also known as quarterwave couplers. v) At the maximum coupling frequency, there is a 90° phase difference between the voltages at port 2 and port 3. Such a coupler can also be described as quadrature coupler. vi)At the frequencies other than maximum coupling frequency (design frequency) the coupling of the coupler can be found from V_{3}(q).
2.Threesection Microstrip
Coupled Line Directional Coupler
The singlesection microstrip directional coupler is mentioned in previous section. In singlesection couplers there exists a limitation of coupling. In single section the coupling can be maximum when the length of the coupler is l/4. However sometimes better coupling is required. This can be increased using multiple sections. Since the phase characteristics are usually better, multisection coupled line couplers are generally made with an odd number of sections. In this part we will examine only threesection microstrip directional coupler since as a multiplesection, it is widely used. The Figure 7 shows the threesection microstrip directional coupler.
2.1.Design Generally, approximating the results in analysis of singlesection for multiples section the following equation is obtained:
At this point deriving the equation of threesection;
What about the length of each section. It can be seen from Figure 7 that all sections should have an equal electrical length. Here since we think the q=p/2, then the physical length of each section should be l//4. However, since all sections have different w/h and s/h ratios (section 1 and 3 are equal and section 2 is different due to symmetry), each section will have different e_{eff} (e_{eff }here can be calculated for microstrip geometry as mentioned in the singlesection part) and therefore different l for the same frequency. That results in different physical length for each section since length=l//4. 2.2.Analysis The general concepts about multisection directional coupler, and also for our focus threesection are mentioned in the previous design part. As an analysis of a threesection coupler given c_{1}=c_{3}, c_{2 }values we can calculate the coupling coefficient of the whole coupler as following: _{ } For all coupling coefficients mentioned before we always can calculate coupling in dB as following: Coupling=C=20log ( c) dB References:


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