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OCTOBER 2019 I DESIGN007 MAGAZINE 47 sor field, which is 4.97 ns at 50 MHz and 5.03 ns at 100 MHz. These numbers are close to the 5-ns delay we assigned to the transmission line. But we still may wonder if this difference is coming from numerical calcu- lation errors or if it represents the real behavior of the circuit? As explained and illustrated in more detail [1] , what we see here is the manifestation of the fact that reflections can change the steady-state delay in a frequen- cy-dependent manner. Before we get back to why the output volt- age (Vout) drops so surprisingly little, let's look at a different example. Instead of a direct impedance mismatch, we now use a Z 0 = 50- ohm transmission line and a series 5-ohm re- sistor, which may crudely represent its con- ductive losses. As opposed to regular PCB trac- es, where the conductive losses are frequency dependent, the practical equivalent of this case could be, for instance, a thin-film transmission line. Figure 4 shows the schematics, and Fig- ure 5 demonstrates the frequency response. In this case, the result matches the simplistic expectation. Regardless of frequency, we get an approximate 5% drop in the Vout. To understand the reason for the two seem- ingly very different behaviors, we reach back to a fundamental principle: the conservation of energy. If we have a lossless (linear and time- invariant) circuit that does not lose power due to losses, radiation, or in any other way, we know that if we send a unity amount of pow- er toward the circuit, the sum of the reflect- ed and transmitted powers must equal unity. Expressed by the elements of the S-matrix of the network, this means that the sum of the squares of the S-matrix elements in each row or column must add up to one. For instance, in the case of a two-port lossless network, and as- suming that we launch the signal towards Port 1, this means (Equation 2): Equation 2 This expression tells us that if we have |G| = |S 11 | = 0.1, or 10% reflection from a lossless circuit, the magnitude of the transmitted wave Figure 4: LTSPICE simulation circuit with series loss resistance represented by a 5-ohm series resistor. Figure 5: Frequency response of circuit in Figure 4. Voltage magnitudes are solid lines referencing the left vertical axis, and phase is a dotted line referencing the right vertical axis.