1 Introduction

It is of utmost importance that the transmission line (TL) feeding a load transfers power efficiently from the source to the load. When a lossless TL is matched either to the source impedance or the load impedance, the incident power and the load power which are independent of the line length can be simply expressed in terms of load or source reflection coefficient (RC). However, if the line is mismatched at both ends, power, which is dependent on the line length, is not expressed in terms of load and source RCs explicitly in the literature. Instead, the necessary power calculations in a mismatched TL are always performed by relevant impedance transformations and voltage/current calculations along the line using simple circuit theory [5, 8,9,10]. This article derives a set of equations to explicitly find incident/load power in terms of the two RCs and the line length so that it can be used to describe the effect of the mismatched TL length on the load power and ways to improve power transferred in a mismatched line.

Although many text books and research/feature articles have been written on this fundamental topic of TL theory, [4, 5, 7,8,9,10,11,12,13] to name a few, the effect of TL length on the incident and load power has not been discussed in the literature. According to the derived equations, the variation of incident and load power is such that it resembles the total voltage magnitude (VM) variation along the TL with a reverse voltage wave. At certain line lengths separated by half wavelengths, incident or load power becomes maximum. Similarly, minimum incident or load power also occurs at certain total line lengths separated by half wave lengths. Expressions are derived for maximum and minimum incident and load power and corresponding total line lengths in terms of source and load RCs. Further, the use of the Smith chart to find these lengths and the ratio of maximum to minimum power is also described.

The single line complex impedance transformer [1, 2] is an example where maximum power transfer occurs when the TL is not matched to both the source and load Impedance. The derived results give an insight into this impedance transformation and the required characteristic impedance of the line and its length in the impedance transformer can be easily found from the same equations. Also, a new version of Friis transmission equation [3, 6] applicable to the scenario where the TL is not matched to both the antenna impedance and the load impedance is proposed using the derived results.

Although matching networks are used to achieve maximum power transfer from the source to the load at the design frequency, as the frequency deviates from the design frequency, since the load and source impedances can vary, the maximum power transfer condition may not be satisfied. Hence the source and the load become mismatched to the TL connecting them. As per the results derived in the paper, this mismatch causes the line length to affect the power transfer and hence the bandwidth. Thus, a TL mismatched to both the source and the load is not uncommon practically, though it is not desired by the designer. It is shown that an optimized line length could be found to maximize the bandwidth in transferring power efficiently for frequency dependent load and source impedances.

2 Power Relations

2.1 Incident Voltage

Figure 1 shows a source \(V_{S}\) with internal impedance \(Z_{S}\) connected to a load \(Z_{R}\) through a lossless TL of characteristic impedance \(Z_{0}\) and length \(l\). Let incident and reflected voltages at the source end be \(V^{ + }\) and \(V^{ - }\), respectively. It is known [5, 8] that

$$ V^{ + } = V_{S} \left( {\frac{{Z_{0} }}{{Z_{S} + Z_{0} }}} \right) + {\Gamma }_{S} V^{ - } { }, $$
(1)

where \({\Gamma }_{S}\) is the source RC given by

$$ {\Gamma }_{S} { } = \frac{{Z_{S} - Z_{0} }}{{Z_{S} + Z_{0} }}{ }. $$
(2)
Fig. 1
figure 1

TL connecting the source to the load

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The input RC at the source end is defined as

$$ {\Gamma }_{in} = \frac{{V^{ - } }}{{V^{ + } }}{ }. $$
(3)

Now using (2) and (3), \(V^{ + }\) in (1) can be found in terms of RCs as follows:

$$ V^{ + } = \frac{{V_{S} \left( {1 - {\Gamma }_{S} } \right)}}{{2\left( {1 - {\Gamma }_{S} {\Gamma }_{in} } \right)}}{ }. $$
(4)

2.2 Voltage Variation

\({\Gamma }_{in}\) can be expressed in terms of load end RC \({\Gamma }_{R}\) as follows:

$$ {\Gamma }_{in} = {\Gamma }_{R} e^{ - j2\beta l} { }, $$
(5)

where \(\beta\) is the phase constant in the line. From (4) and (5) and expressing the complex RCs in polar form, we get

$$ V^{ + } = \frac{{V_{S} \left( {1 - \left| {{\Gamma }_{S} } \right|e^{{j\phi_{S} }} } \right)}}{{2\left( {1 - \left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|e^{{ - j\left( {2\beta l - \phi_{S} - \phi_{R} } \right)}} } \right)}}{ }. $$
(6)

\(\left| {V^{ + } } \right| \) becomes maximum or minimum for the following conditions:

$$ 2\beta l - \phi_{S} - \phi_{R} = \pm m\pi \left\{ {\begin{array}{*{20}c} {m = 0,2,4, \ldots . {\text{for }}\left| {V^{ + } } \right|_{max} { }} \\ {m = 1,3,5, \ldots . {\text{for }}\left| {V^{ + } } \right|_{min} } \\ \end{array} } \right. $$
(7)

The maximum and minimum incident VMs are given by

$$ \left| {V^{ + } } \right|_{max} = \frac{{\left| {V_{S} } \right|\left| {1 - \left| {{\Gamma }_{S} } \right|e^{{j\phi_{S} }} } \right|}}{{2\left( {1 - \left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|} \right)}}\quad {\text{and}} $$
(8)
$$ \left| {V^{ + } } \right|_{min} = \frac{{\left| {V_{S} } \right|\left| {1 - \left| {{\Gamma }_{S} } \right|e^{{j\phi_{S} }} } \right|}}{{2\left( {1 + \left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|} \right)}}{ }. $$
(9)

The ratio of maximum incident VM to its minimum, \(VR\) is

$$ VR = \frac{{1 + \left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|}}{{1 - \left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|}}{ }. $$
(10)

The incident VM \(\left| {V^{ + } } \right|\) is given by

$$ \left| {V^{ + } } \right| = \frac{{\left| {V_{S} } \right|\left| {1 - \left| {{\Gamma }_{S} } \right|e^{{j\phi_{S} }} } \right|}}{{2\sqrt {1 - 2\left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|cos\left( {2\beta l - \phi_{S} - \phi_{R} } \right) + \left| {{\Gamma }_{S} } \right|^{2} \left| {{\Gamma }_{R} } \right|^{2} } }}{ }. $$
(11)

From (7), the particular line length \(l\) which gives maximum or minimum incident voltage for given \({\Gamma }_{S}\) and \({\Gamma }_{R}\) values can be found as follows:

$$ l = \left( { \pm \frac{m}{4} + \frac{{\phi_{S} + \phi_{R} }}{4\pi }} \right)\lambda \left\{ {\begin{array}{*{20}c} {m = 0,2,4, \ldots . {\text{for }}\left| {V^{ + } } \right|_{max} { }} \\ {m = 1,3,5, \ldots . {\text{for }}\left| {V^{ + } } \right|_{min} } \\ \end{array} } \right. , $$
(12)

where \(\lambda\) is the wavelength of the line. These lengths are repeated in every half wavelength. Also, every incident voltage maximum/minimum is followed by a minimum/maximum at total line length increments of quarter wavelengths.

2.3 Power Variation

Using (11), the incident power \(P_{inc}\) and load power \(P_{L}\) can be expressed as follows:

$$ P_{inc} = \frac{{\left| {V^{ + } } \right|^{2} }}{{2Z_{0} }} = \frac{{\left| {V_{S} } \right|^{2} \left| {1 - \left| {{\Gamma }_{S} } \right|e^{{j\phi_{S} }} } \right|^{2} }}{{8Z_{0} \left( {1 - 2\left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|cos\left( {2\beta l - \phi_{S} - \phi_{R} } \right) + \left| {{\Gamma }_{S} } \right|^{2} \left| {{\Gamma }_{R} } \right|^{2} } \right)}} $$
(13)
$$ P_{L} = P_{inc} \left( {1 - \left| {{\Gamma }_{R} } \right|^{2} } \right) = \frac{{\left| {V_{S} } \right|^{2} \left| {1 - \left| {{\Gamma }_{S} } \right|e^{{j\phi_{S} }} } \right|^{2} \left( {1 - \left| {{\Gamma }_{R} } \right|^{2} } \right)}}{{8Z_{0} \left( {1 - 2\left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|cos\left( {2\beta l - \phi_{S} - \phi_{R} } \right) + \left| {{\Gamma }_{S} } \right|^{2} \left| {{\Gamma }_{R} } \right|^{2} } \right)}}. $$
(14)

Under conjugate matching conditions, maximum power (available power from the source \(P_{ava}\)) is transferred to the load and it is given by

$$ P_{ava} = \frac{{\left| {V_{S} } \right|^{2} }}{{8R_{S} }}{ }, $$
(15)

where \(R_{S}\) is the real part of the source impedance \(Z_{S}\). Inserting \(P_{ava}\) in (13) and (14) gives

$$ P_{inc} = = \frac{{P_{ava} R_{S} \left| {1 - \left| {{\Gamma }_{S} } \right|e^{{j\phi_{S} }} } \right|^{2} }}{{Z_{0} \left( {1 - 2\left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|cos\left( {2\beta l - \phi_{S} - \phi_{R} } \right) + \left| {{\Gamma }_{S} } \right|^{2} \left| {{\Gamma }_{R} } \right|^{2} } \right)}} $$
(16)
$$ P_{L} = = \frac{{P_{ava} R_{S} \left| {1 - \left| {{\Gamma }_{S} } \right|e^{{j\phi_{S} }} } \right|^{2} \left( {1 - \left| {{\Gamma }_{R} } \right|^{2} } \right)}}{{Z_{0} \left( {1 - 2\left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|cos\left( {2\beta l - \phi_{S} - \phi_{R} } \right) + \left| {{\Gamma }_{S} } \right|^{2} \left| {{\Gamma }_{R} } \right|^{2} } \right)}}. $$
(17)

Using (2), it can be shown that

$$ \frac{{R_{S} \left| {1 - \left| {{\Gamma }_{S} } \right|e^{{j\phi_{S} }} } \right|^{2} }}{{Z_{0} }} = \frac{{4Z_{0} R_{S} }}{{\left| {Z_{s} + Z_{0} } \right|^{2} }} = 1 - \left| {{\Gamma }_{S} } \right|^{2} . $$
(18)

Using the result in (18), \(P_{inc}\) and \(P_{L}\) in (16) and (17) can be rewritten as follows:

$$ P_{inc} = \frac{{P_{ava} \left( {1 - \left| {{\Gamma }_{S} } \right|^{2} { }} \right)}}{{\left( {1 - 2\left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|cos\left( {2\beta l - \phi_{S} - \phi_{R} } \right) + \left| {{\Gamma }_{S} } \right|^{2} \left| {{\Gamma }_{R} } \right|^{2} } \right)}} $$
(19)
$$ P_{L} = \frac{{P_{ava} \left( {1 - \left| {{\Gamma }_{S} } \right|^{2} { }} \right)\left( {1 - \left| {{\Gamma }_{R} } \right|^{2} } \right)}}{{\left( {1 - 2\left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|cos\left( {2\beta l - \phi_{S} - \phi_{R} } \right) + \left| {{\Gamma }_{S} } \right|^{2} \left| {{\Gamma }_{R} } \right|^{2} } \right)}}. $$
(20)

The maximum or minimum load power occurs at certain line lengths where incident VM becomes maximum or minimum, respectively. Therefore, corresponding line lengths are given by (12). The maximum and minimum load power are given by

$$ P_{Lmax} = \frac{{P_{ava} \left( {1 - \left| {{\Gamma }_{S} } \right|^{2} { }} \right)\left( {1 - \left| {{\Gamma }_{R} } \right|^{2} } \right)}}{{\left( {1 - \left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|} \right)^{2} }}\quad {\text{and}} $$
(21)
$$ P_{Lmin} = \frac{{P_{ava} \left( {1 - \left| {{\Gamma }_{S} } \right|^{2} { }} \right)\left( {1 - \left| {{\Gamma }_{R} } \right|^{2} } \right)}}{{\left( {1 + \left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|} \right)^{2} }}{ }. $$
(22)

3 Discussion

When the TL is matched to the source (\({\Gamma }_{S} = 0)\) or/and load (\({\Gamma }_{R} = 0\)), according to (11), (19) and (20), incident VM \(\left| {V^{ + } } \right|\), incident power \(P_{inc}\) and load power \(P_{L}\) become independent of line length \(l\). These parameters are shown in Table 1 for different combinations of \({\Gamma }_{S}\) and \({\Gamma }_{R}\).

Table 1 Incident VM \(\left| {V^{ + } } \right|\), incident power \(P_{inc}\) and load power \(P_{L}\) when the line is matched to the load or /and source
Full size table

When the line is neither matched to the source nor the load (\({\Gamma }_{{\text{R}}} \ne 0\) and \({\Gamma }_{{\text{S}}} \ne 0\)), according to (11), (19) and (20), \(\left| {{\text{V}}^{ + } } \right|\), \({\text{P}}_{{{\text{inc}}}}\) and \({\text{P}}_{{\text{L}}}\) vary with the line length \(l\). This resembles the total VM variation along the line created by a load equivalent to a load RC of \(\left| {\Gamma_{S} } \right|\left| {\Gamma_{R} } \right|\angle \left( {\phi_{S} + \phi_{R} } \right)\) as far as the line lengths corresponding to maximum and minimum incident VMs are concerned. This is evident from (10) and (12). Hence these lengths can also be found using the Smith chart by locating the RC \(\left| {\Gamma_{S} } \right|\left| {\Gamma_{R} } \right|\angle \left( {\phi_{S} + \phi_{R} } \right)\) in the chart and moving toward the generator until real axis is met as in the normal practice (see Fig. 2). When the sum of the angles of the two RCs is between 0 and \(180^{0}\), \(\left| {{\text{V}}^{ + } } \right|\), \({\text{P}}_{{{\text{inc}}}}\) and \({\text{P}}_{{\text{L}}}\) increase first as the line length is increased from zero. When the sum is between \(- 180^{0}\) and 0, they decrease first when the line length is increased. Further, the ratio of maximum to minimum of incident VM could be found from the chart as a VSWR calculation and its square would give the corresponding power ratio. Since the line length \(l\) for \(\left| {V^{ + } } \right|_{max}\) corresponds to an angle \(\phi_{S} + \phi_{R}\) in the Smith chart, the angle of \(\left| {\Gamma_{R} } \right|\) when transformed to the source end (angle of input RC at the source end \(\Gamma_{in}\)) must be equal to \(- \phi_{S}\) (see Fig. 3) for this line length.

Fig. 2
figure 2

Use of the smith chart to find the line lengths corresponding to maximum and minimum incident VMs

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Fig. 3
figure 3

Input RC for maximum incident VM

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In (21), since \(\left( {1 - \left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|} \right)^{2} \ge \left( {1 - \left| {{\Gamma }_{S} } \right|^{2} } \right)\left( {1 - \left| {{\Gamma }_{R} } \right|^{2} } \right), \) the maximum load power is less than the available power of the source and equal to the available power when \(\left| {{\Gamma }_{S} } \right| = \left| {{\Gamma }_{R} } \right|\). In other words, when the line is neither matched to the load nor the source impedance, if \(\left| {{\Gamma }_{S} } \right| = \left| {{\Gamma }_{R} } \right|\), the maximum power transfer condition can be satisfied by properly selecting the line length [i.e. According to (12)]. At this length as described in the previous paragraph, angle of \({\Gamma }_{in}\) (\(\phi_{in}\)) is equal to \(- \phi_{S}\). This is exactly the conjugate matching condition for maximum power transfer. This is what happens in stub matching where selected stub lengths and positions satisfy these conditions (see Fig. 4).

Fig.4
figure 4

Single stub matching for maximum power transfer

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Figure 5 shows the variation of the ratio of \(P_{L}\) to \(P_{ava}\) with the total line length \(l\) for three different sets of source and load RCs (\({\Gamma }_{S}\) and \({\Gamma }_{R}\)) specified in Table 2. In case (i), both RCs are real positive (\(\phi_{S} + \phi_{R} = 0)\) and therefore according to (12), the first \(P_{Lmax}\) occurs when \(l = 0\) (negligible line length) as can be seen from the figure. In case (ii), \(\left| {{\Gamma }_{S} } \right| = \left| {{\Gamma }_{R} } \right|\) and therefore, \(P_{Lmax} = P_{ava}\). The line length corresponding to \(P_{Lmax}\) is determined by angles of the RCs. In case (iii), according to the RC angles,\( P_{Lmin}\) occurs at a shorter line length than \(P_{Lmax}\). Thus, the variation of load power (also incident power and incident VM) with the TL length can be explained by using source and load RCs.

Fig. 5
figure 5

Variation of the ratio of load power \(P_{L}\) to available power \(P_{ava}\) with the total line length

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Table 2 Different combinations of load and source RCs
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4 Applications

As per the derived results, if the condition \(\left| {{\Gamma }_{S} } \right| = \left| {{\Gamma }_{R} } \right|\) can be satisfied, maximum power (\(P_{ava} )\) is transferred when the line length is chosen appropriately as in (12). For any combination of load and source impedances, this condition can be satisfied for a particular line characteristic impedance \(Z_{0}\) given by

$$ Z_{0} = \sqrt {\frac{{R_{S} \left| {Z_{R} } \right|^{2} - R_{R} \left| {Z_{S} } \right|^{2} }}{{R_{R} - R_{S} }}} , $$
(23)

where \(R_{S} = Re\left( {Z_{S} } \right)\) and \(R_{R} = Re\left( {Z_{R} } \right)\). This is the design method of a single line complex impedance transformer [1, 2].

The Friis transmission equation (FRE) [3, 6], which is used to find the power delivered to the load in a receiving antenna (Rx) has two versions. The first version applies when the Rx antenna’s input impedance and the load satisfy the maximum power transfer condition and it is given by [3]:

$$ \frac{{P_{L} }}{{P_{in} }} = \left( {\frac{\lambda }{4\pi R}} \right)^{2} G_{t} G_{r} , $$
(24)

where \(P_{in}\) is the input power to the transmitting (Tx) antenna, \(R\) is the distance between the Tx and Rx antennas and \(G_{t} \) and \(G_{r}\) are the gains of Tx and Rx antennas, respectively. The second version applies when the Rx antenna is not matched to the line but the load is matched to the line. When polarization losses are also included, it takes the following form [3]:

$$ \frac{{P_{L} }}{{P_{in} }} = \left( {\frac{\lambda }{4\pi R}} \right)^{2} G_{t} G_{r} \left| {\hat{\rho }_{t} .\hat{\rho }_{r} } \right|^{2} \left( {1 - \left| {{\Gamma }_{S} } \right|^{2} { }} \right), $$
(25)

where \({\Gamma }_{S}\) is the RC of the Rx antenna and \(\left| {\hat{\rho }_{t} .\hat{\rho }_{r} } \right|^{2}\) is the polarization loss factor between the two antennas. However, when the line is neither matched to the load nor the Rx antenna, there is no version of the FRE available to express the load power. As per the load power given in (20), a third version for the FRE can be obtained to calculate the load power when both the Rx antenna and the load are not matched to the line:

$$ \frac{{P_{L} }}{{P_{in} }} = \left( {\frac{\lambda }{4\pi R}} \right)^{2} . \frac{{G_{t} G_{r} \left| {\hat{\rho }_{t} .\hat{\rho }_{r} } \right|^{2} \left( {1 - \left| {{\Gamma }_{S} } \right|^{2} { }} \right)\left( {1 - \left| {{\Gamma }_{R} } \right|^{2} } \right)}}{{\left( {1 - 2\left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|cos\left( {2\beta l - \phi_{S} - \phi_{R} } \right) + \left| {{\Gamma }_{S} } \right|^{2} \left| {{\Gamma }_{R} } \right|^{2} } \right)}}. $$
(26)

Here, the antenna received power varies with the line length as depicted in Fig. 5. For certain line lengths given by (12), the antenna received power becomes maximum and minimum, respectively:

$$ \frac{{P_{Lmax} }}{{P_{in} }} = \left( {\frac{\lambda }{4\pi R}} \right)^{2} \frac{{G_{t} G_{r} \left| {\hat{\rho }_{t} .\hat{\rho }_{r} } \right|^{2} \left( {1 - \left| {{\Gamma }_{S} } \right|^{2} { }} \right)\left( {1 - \left| {{\Gamma }_{R} } \right|^{2} } \right)}}{{\left( {1 - \left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|} \right)^{2} }} , $$
(27)
$$ \frac{{P_{Lmin} }}{{P_{in} }} = \left( {\frac{\lambda }{4\pi R}} \right)^{2} \frac{{G_{t} G_{r} \left| {\hat{\rho }_{t} .\hat{\rho }_{r} } \right|^{2} \left( {1 - \left| {{\Gamma }_{S} } \right|^{2} { }} \right)\left( {1 - \left| {{\Gamma }_{R} } \right|^{2} } \right)}}{{\left( {1 + \left| {{\Gamma }_{S} } \right|\left| {{\Gamma }_{R} } \right|} \right)^{2} }}. $$
(28)

Hence, the received power can be maximized by selecting appropriate TL length.

A scenario where the line is neither matched to the source nor the load can arise when the source and the load are frequency dependent. Though the line could be matched to both the load and the source at the design frequency, as the frequency deviates from the design frequency, the line gets mismatched to both the load and the source. Hence the line length affects the power transfer as the frequency deviates from the design frequency, which subsequently affect the bandwidth. This implies that there could be an optimized length to maximize the bandwidth of power transfer from the source to the load for frequency dependent source and load impedances. In order to verify this, an antenna (source) connected to a load of \(100 + j30 {\Omega }\) through a 75 \({\Omega }\) lossless TL at a design frequency of 2 GHz is considered. The input impedance \(Z_{S}\) of the antenna is assumed to be

$$ Z_{S} = 12.5f + 50 + j50, $$
(29)

where \(f\) is the frequency in GHz. Accordingly, the antenna has an input impedance of \(75 + j50 {\Omega }\) at 2 GHz and it is made self-resonant by connecting a series capacitance of 1.59 pF so that the antenna is matched to the line at 2 GHz. The load is matched to the line separately at 2 GHz by using a complex to real impedance transformer (\(100 + j30 {\Omega }\) to 75 \({\Omega }\)) realized using a single TL section [1, 2]. The required characteristic impedance and electrical length of this single TL section, which can be found using (23) and (12), are \(101 {\Omega }\) and \(131.7^{ \circ }\) (see Fig. 6).

Fig. 6
figure 6

Antenna feeding a frequency dependent load through a TL

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Although the self-resonant antenna and the combined load comprising the load and the single line transformer are both matched to the line at 2 GHz, since they are frequency dependent, the line becomes mismatched as the frequency deviates from 2 GHz. Both impedances can be calculated analytically to find the frequency response of the system defined as

$$ \left| {S_{11} } \right|_{dB} = 10log_{10} \left( {1 - \frac{{P_{L} }}{{P_{ava} }}} \right) . $$
(30)

The ratio \(\frac{{P_{L} }}{{P_{ava} }}\) can be found from (20). Figure 7 shows the frequency response of the system for 4 different line lengths: \(l1 = 0.25\lambda ,l2 = 0.062\lambda , l3 = 0.0997\lambda \) and \( l4 = 0, \) where \(\lambda\) is the wavelength at 2 GHz. There is perfect matching at 2 GHz for all lengths. However, they have different bandwidths as the length affect the transferred power at frequencies other than the design frequency. \(l2\) is selected as per (12) to get a maximum power transfer at 3.5 GHz and hence there is an additional resonance in the frequency response at 3.5 GHz. \(l3\) is the optimized length to obtain maximum − 20 dB bandwidth where the optimization is done using genetic algorithms. The optimized length gives a − 20 dB fractional bandwidth, which is about 5 times that of the design with \(l4 = 0,\) where the compound load is directly connected to the antenna without a TL. This clearly shows the importance of proper selection of the length of a mismatched TL for maximum power transfer over a wide bandwidth.

Fig. 7
figure 7

Effect of the TL length on the frequency response for frequency dependent source and load impedances

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5 Conclusion

When the TL is matched neither to the load nor the source impedance, incident VM, incident power and load power vary with the line length. This can often arise when the frequency deviates from the design frequency if both the source and load impedances are frequency dependent. Expressions were derived to show that these parameters vary periodically with a period of half wavelength. Ratio of maximum to minimum of these parameters and corresponding line lengths for these maxima and minima were derived in terms of source and load RCs. It is possible to deliver available power of the source to the load when the magnitudes of the two RCs are equal and the line length is selected for the maximum power. The line lengths corresponding to maxima/minima and voltage ratio of maximum to minimum can be graphically found using a Smith chart by locating an equivalent RC equal to the product of the source and load RCs.

Based on these results, a new version of the FRE, when neither the receiving antenna nor the load is matched to the TL, is presented. Also, it is shown that an optimum TL length could be found to maximize the bandwidth of efficient power transfer for frequency dependent source and load impedances.