Part 2. L-Networks

Like parallel or shunt-connected loads, EZNEC L-Networks are frequency-nimble and use the same basic technique that was applied to parallel-connected loads. NEC contains an NT command for creating Y-parameter networks. Like R +/- jX loads, Y-parameter networks are frequency specific, and the user must change the command for each new frequency. Hence, a single set of NT values normally will not provide accurate results across a broad frequency sweep. EZNEC calculates a new NT command or its equivalent for each L-Network at each new frequency in a sweep. Therefore, for a given set of values for inductance and capacitance, the core has the correct data to provide accurate results at each frequency in a sweep.

Our sample networks will apply to matching a self-resonant Yagi to a 50-Ohm coaxial feedline. The root problem is only one of many possible applications for L-networks. However, by focusing in on a single exercise, we can master the steps required to use the L-network facility effectively. Once you take this step, you can easily proceed on your own to other applications.

**2-Element/Component L-Networks**

Let's begin with the self-resonant 28.5-MHz
3-element Yagi using 1.2"-diameter aluminum elements, the same one that
we used for part of the preceding episode. **Fig. 1** shows the
outline and the free-space E-plane pattern of the antenna, along with
the wire table needed to create the model. All of the matching sections
that we shall explore will be designed to convert the 27.5-Ohm
feedpoint impedance to 50 Ohms.

One way to effect the impedance conversion is to place an L-network between the antenna terminals and the feedline. L-networks are 2-component networks consisting of one series component and one shunt or parallel component. Let's call the cable impedance the source impedance, and the antenna terminal impedance will be the load impedance. If the source impedance is higher than the load impedance--as it is in our case--then the series component goes on the load side of the network, with the shunt component on the source side. If the load impedance is higher than the source impedance, then the shunt component goes on the load side with the series component on the source side. The L-network serves as the foundation for many more complex networks, which we can treat as collections of L-networks.

Since our goal is to model with EZNEC L-Networks,
we shall not cover the calculation of L-network components for given
impedance transformation situations. Instead, we shall rely on one of
many utility programs and spreadsheets to arrive at the required values
for the series and the shunt components. **Fig. 2** shows the
values of capacitance and inductance needed for two forms of the
L-network that will convert our terminal or load impedance to the
50-Ohm source or cable impedance.

Although the two forms use very different components, both circuits have some things in common. First, the reactive series component is the opposite types from the reactive shunt component. (There are special cases of load impedances that may call for components of the same type, but they involve load or antenna terminal impedances with high reactance. Our loads are almost purely resistive.) Second, the absolute value of the series and the shunt reactances are the same. In both cases, the series reactance is 26 Ohms (with a sign appropriate to the type of reactance) and the shunt reactance is 51.5 Ohms (again with a sign appropriate to the type of reactance).

To implement a model of the L-network in EZNEC,
V5, we must get used to the conventions used by the program. **Fig. 3**
provides some guidance. A network consists of two ports. Port 1 always
goes with the series branch or component. Port 2 always goes with the
shunt branch or component. Labeling the ends of the network as Port 1
and Port 2 is simply a way to differentiate them. Depending on the
application, either port may be the source and either may be the load.
Our sample case shall call for Port 1 to connect to the antenna
terminals, that is, to the source segment of the driver-element wire.
Port 2 will use a virtual wire (described in the preceding episode),
which will become the new source segment for the model. Other
applications may reverse the ports for converting low source impedances
to high load impedances.

The sketch also shows us our options in assigning component values to the branches of an L-network. For a single-frequency application, we may use R+/-jX loads. For applications that may need to cover a range of frequencies, we can use several different forms of R-L-C configurations. The most common will be the series configuration. It will apply to our sample, since the most complex entry that we shall make is to have both resistance and inductance in series. As in Part 1, we shall assign a Q of 200 to all inductors. When we choose not to have one of the R-L-C components as part of the branch, we shall enter a zero. In this case, zero is not the component value, but instead is a NEC convention for indicating a missing element in a load. The EZNEC tables will use the word "short" to indicate the missing component.

In addition to series R-L-C loads, we may also use a parallel configuration. EZNEC also makes available a trap configuration consisting of a series resistance and inductance that together are in parallel with a capacitance. Whichever configuration we select for an application, both branches of the L-network must use the same type of configuration. However, other loads that might be present in the antenna assembly can use any of the possible configurations and do not need to match the configuration used in the L-network.

We can orient ourselves to the process of
modeling an L-network by starting with a frequency-specific R +/- jX
load in each branch of the L-network. **Fig. 4** shows the model
with the designation for the network (the L in the box) plus a
designation that shows we are using a virtual wire, namely, V1 as the
source wire. The first table shows the source entry that applies to
both models. The two L-Network tables show the two versions of the
network.

Because the actual source impedance has several decimal places, as do the calculated components, the rounded numbers in the tables do not return identical source impedance values. The first version reports an impedance of 50.5 - j0.0 Ohms, while the second reports 53.6 - j0.0 Ohms. In practice, the difference does not make a difference, since every model will depart slightly from physical reality that includes small variations from the model.

If we wish to employ series R-L-C branches in the
L-network--the more normal case--the L-Network table becomes more
complex. **Fig. 5** shows the tables for the two varieties of
L-networks pictured in **Fig. 1**. The first version provides a
series inductor with a Q of 200 and a shunt branch holding the
capacitor. (In a series R-L-C load, you may ignore the Frequency entry.
It applies to trap-type loads. Traps are very often designed to be
self-resonant at the bottom or just below the bottom of an operating
passband.)

The second version of the L-network uses a series capacitor with a shunt inductor--again with a Q of 200. Both versions connect Port 1 to the load wire, in this case, segment 11 of wire 2, the Yagi driver. Port 2 for both networks goes to wire V1, the remote short virtual wire, which also serves as the source segment.

**Table 1** lists the results of the modeling
with each L-network. Each version lists a variant of the model that
omits the series resistance in the inductor branch, this providing a
lossless model, except for the aluminum element material, of course.
These entries also supply the performance data (excluding the feedpoint
impedance) for the basic or pre-match model.

With a normal or lossy coil, both version of the L-network result in the same gain value and the same efficiency. Version 1 shows an impedance about as much below 50 Ohms as the impedance of version 2 is above 50 Ohms. The variance results from rounding the values, beginning with the original source impedance applied to the external L-network calculator.

The modeler can be as creative as he or she wishes in the development of models that use a single L-network. For example, one might model a multi-band center-fed doublet, perhaps about 125' long overall. From the center segment of the doublet, one may insert a transmission line of choice, including the loss factor, and set its length to approximate the length of a practical line. Initially, one can place the source on the virtual wire that terminates the transmission line. From a record of source impedance values for the various bands, one can derive from an external L-network calculator the type of network and the component values needed to transform the impedance to 50 Ohms. (The type of network refers to whether the shunt or the series branch connects to the load.) Then one may go back and insert the prescribed L-network for each band into a separate model to confirm the results. If the modeler is dissatisfied with the results, perhaps due to the need for extreme component values in one or the other leg of the network, one might try different doublet lengths, different transmission-line characteristic impedances (with adjustments to the velocity factor and the loss factor), and even different line lengths. Since the line length is not a part of the overall antenna geometry, the procedure cannot account for disruptive influences on the line that a casual physical installation might encounter. Nevertheless, the exercise can give the modeler with an L-network tuner a good idea--well in advance of purchasing materials--what approximate setting an L-network tuner may need--not to mention the best line to use for a given installation.

**3-Element/Component Networks**

We can connect the individual ports of an
L-Network in EZNEC to any wire, real or virtual. Therefore, we might
place two (or more) L-Networks back-to-back to form a 3-component
network, such as a PI or a T. Although rarely used at the terminals of
an antenna, they are often the network forms used in antenna tuners.
For our samples, we can dispense with the transmission line and connect
our new networks directly to the terminals of our self-resonant
3-element Yagi. For the impedance of the driver (about 27.6 Ohms) and
the cable or source impedance (50 Ohms), we might use any of the
networks shown in **Fig. 6**. Indeed, there is a remaining option,
the high-pass PI, but I have never seen it used in this type of
application.

Creating such networks will require that we use 2
L-Networks. In each case, the center component is shared by both
networks, so we shall have to connect together either a pair of Port 1s
or a pair of Port 2s, as shown in **Fig. 7**. For a PI network, the
Port 1s go together (on a suitable virtual wire) so that we have two
series branches connected in series. A T network requires that we bond
the two Port 2s together on a wire. The result is two shunt branches
connected in parallel. We also need to keep track of the ultimate ends
of the system. For convenience, I shall adopt the convention of
treating the first L-Network as connected to the load, that is, the
antenna terminals (or, in the model, the proper driver segment). A new
virtual wire, V2, which also receives the model source, terminates the
far end of the second L-Network. Some convention of this sort is
necessary to ensure model-to-model consistency and to thereby minimize
the chances for misconnection errors.

Let's begin by forming the low-pass PI network. **Fig.
8** shows the model. but lists only one network and one virtual wire.
On the right, we have the network to compare with the L-Network table
that follows. In the table, we can readily identify the shunt
capacitors at the extreme ends of the assembly.

The series branches divide the inductor. In this case, the inductance and its series resistance for a Q of 200 form two equal parts. Since inductances and resistances simply add when in series, the sum of the resistive and the inductive values represent the total that forms the PI network. It is not necessary to divide the values in half. Other partitions will result in accurate results. However, for greatest accuracy, the smaller of the two parts should be above about 1% of the total. If you place all of the inductance in one L-Network and simply set the other network's series branch to zero, you will end up with a missing component, and the results may not be accurate. For reasons that will become clear as we proceed, the division in half is a convenient convention to adopt.

Let's next form a low-pass T network from 2
L-Networks. **Fig. 9** shows the ultimate network and the formation
tables within EZNEC. The series components are clear. The two shunt
capacitors each carry half the value of the total capacitance required
by the T-network, since capacitances in parallel simply add. (Note:
these exercises have presumed perfect capacitors with an indefinitely
high Q. However, you may find occasion to assign a series-equivalent
resistor to a capacitance to simulate a Q.) Once more, other splits in
the total capacitance will work equally well so long as the lower value
is at least 1% of the total. Do not place the entire capacitance within
one L-network and treat the other shunt value as zero or a short.

Our final sample uses a high-pass T network. As
shown in **Fig. 10**, this situation produces two capacitors as the
series elements. The inductance falls into the shunt branches of the 2
L-Networks. Here, equal parts for each shunt simplifies the arithmetic
to an easy mental exercise. If we assign to each shunt branch twice the
inductance and twice the resistance relative to the externally
calculated total, the parallel combination will be correct. There are
other combinations that will do the job, but they might require a
calculator.

The question that follows from these formation
drills is whether results are accurate. The schematics reflect (with
rounding) the network values derived from an external program, which we
shall presume to be correct. Ideally, the source impedance for each
model should by 50 Ohms. The data in **Table 2** provides the
reports from running the NEC models. Each entry in the table provides
reports for perfect or lossless inductors and for inductors with a Q of
200.

The low-pass PI network is interesting because it exhibits the lowest efficiency of the group with an inductor Q of 200. The high-pass and low-pass T networks are about equal with respect to efficiency. However, as the gain values suggest, none of the matching network types yields performance reductions that one could notice in operation. As well, the efficiencies of the 3-component networks is only 1% to 2% lower than the values associated with 2-component L-networks.

A more direct measure of the EZNEC L-Network system is the source impedance reports. Using components calculated externally, the network systems produce the results expected within very close tolerances. Like the parallel-connected components examined in the preceding episode, the L-networks convert--at each frequency within a sweep range--into Y-parameter networks or their equivalents. Hence, the L-network system used in EZNEC provides accurate results across a significant span of frequencies.

Nothing prevents us from using sequential L-Networks rather than back-to-back configurations. Although we would gain nothing from the process in many circumstances, let's create the situation to establish that they work. In the preceding episode, we examined a 1/4-wavelength shortened dipole for 14.175 MHz composed of AWG #12 copper wire. It used a 9.44-uH center-loading coil to bring it to resonance with an impedance of about 15.4 Ohms. Normally, we might use a single L-network with a series inductor on the load or antenna side of 0.26 uH, with a shunt capacitor on the source side of 337 pF. This arrangement would yield a source impedance of 49.7 - j0.3 Ohms.

For our sequential system, let's install two
L-networks arranged as shown in **Fig. 11**. The first network will
convert the 15.5-Ohm load impedance to 27 Ohms. The second will convert
the 27-Ohm impedance to 50 Ohms. The tables show the basic dipole,
although I have omitted the inductive load. The network tables show the
series and shunt branch values necessary to effect the 2-stage
impedance transformation. As always, the inductors have a Q of 200.

The 2-stage L-network system produces a source impedance on wire V2 of 49.8 - j0.2 Ohms. The dipole gain is 0.84, indicating a small network loss. The efficiency is 79.3%, considering the effects of both the network and the loading coil. A single-stage L-network shows an efficiency of 79.4%. Even though the 2-stage L-network system shows no advantage over a single-stage network--and indeed requires unnecessary component complexity--it does illustrate well enough the accuracy of EZNEC L-networks in the sequential mode. The SWR graph is identical to the one shown for the dipole in Part 1 of this episode pair.

**Conclusion**

The goal of these episodes has been to show the steps needed to model effectively using new facilities within the latest version of EZNEC. These facilities include transmission-line losses, parallel-connected loads, ideal transformers, and L-networks. As applicable, each facility shares the frequency-nimble properties of R-L-C loads in NEC. The program achieves this ability by recalculating an NT command or its equivalent for each frequency step within a defined sweep. Hence, the new facilities are highly useful in evaluating potential antenna performance across a band of frequencies.

Although the examples have focused on antenna impedance matching, this application is but one of many possible uses to which we might put the facilities. For example, L-networks--and more complex networks that we might construct from them--are useful not only for impedance transformation, but as well for phase-shifting a signal. Learning the required modeling steps and developing personal conventions that make them consistent from one model to the next is crucial to error-free and confident modeling with the new facilities in this program.