L-type Matching Community Fundamentals—Analyzing L Community Frequency Response and Bandwidth
8 min readTwo-element networks, often known as L-sections or L-networks, can be utilized to remodel one impedance to a different. Not solely that, however they’re additionally straightforward to design and might be fairly sensible.
On this article, we’ll check out the frequency response of those circuits and study a handy answer to estimate their bandwidth. We’ll additionally see that the key limitation of L-networks is their fastened bandwidth, which leads us to extra difficult matching networks, equivalent to T networks.
One Impedance Matching Downside, Many Options
For a given impedance matching drawback, there are a number of totally different options. Let’s contemplate designing a two-element lumped matching community; one can discover not less than two L-sections for any given load and enter impedances. Relying on the impedance values, it is usually attainable to have a complete of 4 totally different L-type options. For example, contemplate reworking the impedance Z_{1 }= 10 + j30 Ω to the origin of the Smith chart. With a normalizing impedance of Z_{0 }= 50 Ω, we get hold of the matching options proven under in Determine 1.
Determine 1. Instance Smith chart displaying matching options.
Utilizing the intersection factors A, B, C, and D, we are able to have 4 totally different paths from z_{1} to z_{2}. You should utilize an entire ZY Smith chart to seek out the reactance (x) and susceptance (b) of the intersection factors. These values are listed in Desk 1.
Desk 1. ZY Smith chart intersection factors, reactance, and susceptance values.
Intersection Level |
Reactance (x) |
Susceptance (b) |
A |
0.4 |
-2 |
B |
-0.4 |
2 |
C |
1 |
-0.5 |
D |
-1 |
0.5 |
Utilizing these values, we get hold of the next L-sections for this impedance-matching drawback (Determine 2).
Determine 2. L-sections for our impedance-matching instance drawback.
The L-sections in Determine 2 (a), (b), (c), and (d), respectively, correspond to paths going from z_{1} to z_{2} by the factors A, B, C, and D on the Smith chart. The element values are calculated at an working frequency of 1 GHz. The query that arises at this level is, how will we select the suitable matching community from these out there choices? To reply this query, let’s first take a look at the frequency response of those matching networks.
L-Part Frequency Response
Let’s assume that the load impedance Z_{1 }= 10 + j30 Ω within the above instance is a sequence RL circuit. At 1 GHz, we now have R = 10 Ω and L = 4.77 nH. The enter reflection coefficient of our L-sections is proven in Determine 3 under.
Determine 3. Graph displaying the enter reflection coefficient of our L-sections.
As might be seen, the reflection coefficient is nearly zero at 1 GHz, which means that every one of those matching circuits present an enter impedance of about 50 Ω at a slender band round 1 GHz (no large shock!). Subsequent, we’ll take a look at the input-output switch operate of every circuit. Determine 4 exhibits how we’ve outlined the enter and output alerts for this simulation.
Determine 4. Instance diagram of the enter and output alerts from the switch operate simulation.
A voltage supply with a supply resistance of fifty Ω is utilized to the enter of the matching community. The output sign is the voltage throughout the resistive a part of the load. The simulated switch operate of the above L-networks in decibels (20log ( fracV_outV_S)) is proven in Determine 5.
Determine 5. A graph displaying the switch operate vs frequency for the L-network sections.
The L-sections in Determine 2(a) and (c), every consisting of two capacitors, cross a slender band round 1 GHz and reject frequencies under and above this middle frequency by nearly the identical quantity. These constructions have a bandpass response ( → assuming that the load has a resistive half!). The L-network in Determine 2(b) blocks DC and severely attenuates frequencies under 1 GHz. On this case, we now have a highpass frequency response.
Lastly, the L-section in Determine 2(d) is a lowpass circuit; it passes DC and rejects larger frequencies. Because the above instance exhibits, the frequency response of an L-section might be lowpass, highpass, or bandpass. Observe that even for the lowpass and highpass circumstances, a achieve peaking happens on the frequency at which the impedance match is offered. Normally, there are eight attainable preparations of L-type matching networks, as proven in Determine 6.
Determine 6. Eight potential L-type matching community preparations.
The L-sections in Determine 6 (a), (b), (c), and (d), every consisting of both two capacitors or two inductors, have a bandpass response. Figures 6(e) and (f) present lowpass L-type matching networks, whereas Figures 6(g) and (h) are highpass constructions.
Selecting the Proper L-section
The precise alternative between various kinds of L-sections (and matching networks basically) is dependent upon the applying. For instance, the highpass response of the L-section in Determine 5(b) supplies AC coupling (or DC isolation) that is perhaps required in lots of purposes. Then again, some purposes want a lowpass matching community to offer DC feed-through between the enter and output terminations.
Moreover, relying on the parasitic reactive parts of the load and supply impedances, we would favor to make use of a selected sort of L-section community. For example, assume that the load has some parasitic parallel capacitance. On this case, it is perhaps attainable to make use of an L-section with a parallel capacitor on the load aspect to soak up the undesired capacitance of the load into the impedance-matching community. One other essential issue is the bandwidth of the matching community, however we’ll get into it in one other part.
Estimating L-section Bandwidth
The bandwidth of an L-section might be comparatively huge. For instance, with the responses proven in Determine 5, the bandwidth modifications from about 480 MHz (L-section D) to 790 MHz (L-section A). The standard issue idea Q_{L}, outlined under, is generally used to parameterize the bandwidth of L-sections:
[Q_L=fracf_0BW]
Equation 1.
Within the above equation, f_{0} is the frequency the place the match happens, and BW is the -3dB bandwidth of the frequency response. For instance, with the blue curve in Determine 5, we now have f_{0 }= 1 GHz and BW = 1.32 GHz – 0.53 GHz = 790 MHz, resulting in Q_{L }= 1.27. There’s a handy methodology of estimating the standard issue of an L-section. To this finish, we first must outline the nodal high quality issue that’s obtained from the equal impedance or admittance of various nodes of the matching circuit. If the equal impedance of a node is Z = R + jX, the node high quality issue is outlined as:
[Q_n=fracR]
Equation 2.
The node high quality issue will also be obtained from the admittance of the node. If the equal admittance of a node is Y = G + jB, we now have:
[Q_n=fracG]
Equation 3.
At every node of the L-type matching community, we are able to use both Equation 1 or 2 to seek out the standard issue of that node (each equations produce the identical worth). Every node within the circuit has its personal nodal Q. If the utmost worth of the circuit’s nodal high quality issue is Q_{n}, then we are able to use the next equation to estimate the standard issue of an L-section:
[Q_L=fracQ_n2]
Equation 4.
Observe that this equation is simply an approximation of the particular Q_{L}. Let’s take a look at an instance.
Estimating L-section Bandwidth Instance
Persevering with with the matching drawback in Determine 1, we observe that the utmost nodal Q happens at level z_{1} (equivalent to Z_{1 }= 10 + j30):
[Q_n=fracXR=frac3010=3]
You possibly can confirm that at factors A and C, we now have a smaller nodal Q of two and 1, respectively. With Q_{n }= 3, Equation 4 results in Q_{L }= 1.5. By substituting Q_{n }= 1.5 and f_{0 }= 1 GHz into Equation 1, the bandwidth is estimated to be BW = 667 MHz. The precise bandwidth modifications from about 480 MHz (L-section D) to 790 MHz (L-section A). Contemplating the simplicity of the approximation equations, the estimated bandwidth is in affordable settlement with the precise worth.
The L Community Fastened Bandwidth Limitation and a T Community Introduction
As talked about, L-sections are easy to design and fairly sensible. Nonetheless, they’ve one main drawback; their bandwidth is decided by the enter and output impedance. To raised perceive this, contemplate reworking z_{Load }= 0.2 to the middle of the Smith chart, as illustrated in Determine 7.
Determine 7. Instance Smith chart with z_{Load }= 0.2.
The intermediate impedances are the intersection factors of the r = 0.2 and g = 1 circle. As you’ll be able to see, the intersection factors are decided by the supply and cargo impedances. Due to this fact, the nodal Q and, consequently, the bandwidth of the matching community are fastened. On this instance, Q_{n} is 2. If our software wants the next or decrease Q, we are able to use two intermediate impedances (reasonably than a single intermediate level) going from z_{Load} to z_{Supply}. For example, assume that we have to have a most Q_{n} of 5. One answer for this case makes use of intermediate factors C and D, as proven in Determine 8.
Determine 8. Smith chart displaying intermediate factors C and D.
We first transfer alongside the r = 0.2 constant-resistance circle to level C which has a normalized impedance of 0.2 + j. Subsequent, we transfer alongside the g = 0.2 constant-conductance circle to level D, which has a normalized impedance of 1 – j2. Lastly, we comply with the r = 1 constant-resistance circle to reach on the middle of the Smith chart. On this case, the utmost nodal Q happens at level C, producing Q_{n }= 5. Having two intermediate factors is equal to having a three-element matching community.
Within the above instance, the primary movement from z_{Load} to C in addition to the third movement from D to z_{Supply} are alongside constant-resistance circles. These correspond to including sequence parts. The second movement, nevertheless, is alongside a constant-conductance circle, requiring a parallel element. Due to this fact, the matching community ought to have a “T community” association (Determine 9).
Determine 9. Diagram displaying a T community.
The addition of a 3rd component permits us to have management over the utmost nodal Q of the circuit. In contrast to an L-type community, it’s not straightforward to discover a easy relationship between the loaded high quality issue (Q_{L}) of the T community and the Q of the circuit nodes. Nonetheless, it’s apparent {that a} excessive worth of Q_{n} results in the next worth of Q_{L}. Within the subsequent article on this sequence, we’ll proceed this dialogue and study different impedance-matching networks.
Summarizing L Community Fundamentals
- L-sections are straightforward to design however fairly sensible.
- For a given impedance matching drawback, there is perhaps a number of totally different L-sections in addition to different extra difficult matching networks, equivalent to T-type options.
- The selection of the matching community is affected by a number of concerns, equivalent to DC biasing, the kind of frequency response (lowpass or highpass), bandwidth, and ease of implementation, to call just a few.
- If we have to have management over the bandwidth, we would want to make use of extra difficult matching networks, equivalent to a T community reasonably than an L-section.
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