Delay Lines
The circuit
If a large number of simple lag networks are connected in cascade, the resulting transfer function will approach that of a dead period lag. An impulse injected into the input terminals will appear at the output with a delay time, T. As a practical example, series inductances and shunt capacitances are distributed uniformly to simulate a coaxial or two wire line in the next two circuits.
An infinite number of discrete elements, n, with infinitely small size, has a transfer function of 1 / (1 + sT / n)n ~ e-sT, which approaches that of a transport lag having an infinitely high cutoff frequency and a magnitude response of unity. The phase angle increases linearly with frequency, thus leaving the shape of delayed impulse undistorted. Delay times within the range of seconds cannot be realised with cables because the velocity of propagation is slightly below the velocity of light. A better converging approximation of dead period lag can be found, but at the expense of bandwidth. Fortunately, large delay times are usually desired with signals of lower frequency, and a satisfactory trade off between the number of lag sections and bandwidth can be achieved.
Loss Free Delay Line
The input file
discontinuous loss free delay line
.control
tran 0.5ns 100ns
plot v(1)+2 v(3)*10+1 v(4)*10 v(5)*10-1 v(6)*10-2 xlabel t[s]
+ ylabel 'v(1) v(3) v(4) v(5) v(6) [V]' title 'TRAN analysis'
.endc
v1 1 0 dc 0 pulse 0 1 0 0 0 0.5ns 1s
r1 1 2 320
r2 6 0 320
x1 2 3 loss_free_line
x2 3 4 loss_free_line
x3 4 5 loss_free_line
x4 5 6 loss_free_line
.subckt loss_free_line 1 8
l1 1 2 0.5uH
l2 2 3 1uH
l3 3 4 1uH
l4 4 5 1uH
l5 5 6 1uH
l6 6 7 1uH
l7 7 8 0.5uH
c1 2 0 10pF
c2 3 0 10pF
c3 4 0 10pF
c4 5 0 10pF
c5 6 0 10pF
c6 7 0 10pF
.ends
.end
The results
Lossy Delay Line
The input file
discontinuous lossy delay line
.control
ac dec 100 0.5gHz 5gHz
plot vdb(3) linear xlabel f[Hz] ylabel output[dB] title 'AC analysis'
tran 0.1ns 20ns
plot v(1)/2+1.5 v(3) xlabel t[s] ylabel 'input output [V]' title 'TRAN analysis'
.endc
v1 1 0 dc 0 ac 2 pulse 0 2 1ns 0 0 4ns 1us
r1 1 2 55
r2 3 0 55
x1 2 3 lossy_line
.subckt lossy_line 1 11
x1 1 2 lossy_line1
x2 2 3 lossy_line1
x3 3 4 lossy_line1
x4 4 5 lossy_line1
x5 5 6 lossy_line1
x6 6 7 lossy_line1
x7 7 8 lossy_line1
x8 8 9 lossy_line1
x9 9 10 lossy_line1
x10 10 11 lossy_line1
.ends
.subckt lossy_line1 1 11
x1 1 2 rlcg
x2 2 3 rlcg
x3 3 4 rlcg
x4 4 5 rlcg
x5 5 6 rlcg
x6 6 7 rlcg
x7 7 8 rlcg
x8 8 9 rlcg
x9 9 10 rlcg
x10 10 11 rlcg
.ends
.subckt rlcg 1 3
r1 2 3 0.04
r2 3 0 750k
l1 1 2 6.88nH
c1 3 0 2.27pF
.ends
.end
The results
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