Example 1: Quarter-wave transformers

Netlist: examples/transformers.cir

Goal: match 50 Ω to 100 Ω at 1 GHz, and compare a single quarter-wave section with a two-section binomial transformer. On the way you'll write a first netlist, use .param expressions, and put two independent networks in one file.

#The netlist

.freq 0.2G 1.8G 801
.ref 1G
.param ZL = 100
port a1 50      # 1
port a2 ZL      # 2
T1 a1 a2 Z0=sqrt(50*ZL) len=90deg
port b1 50      # 3
port b3 ZL      # 4
T2 b1 b2 Z0=50*(ZL/50)^0.25 len=90deg
T3 b2 b3 Z0=50*(ZL/50)^0.75 len=90deg
.plot ylim=-50,0 title=Return_loss_(dB) db(S11) db(S33)
.smith S11 S33

What to notice:

• Port impedances. Ports 2 and 4 are referenced to ZL = 100 Ω, so S11 and S33 are the input reflections when the far end is terminated in 100 Ω. A port's reference impedance is its termination.
• .ref. .ref 1G makes len=90deg a quarter wave at 1 GHz.
• T elements. A T element is a single line with both ends referenced to ground. It is shorthand for a one-line cline.
• Two networks in one file. Ports 1–2 and ports 3–4 share no nodes, so they are independent and their S-parameters don't interact. This is a convenient way to compare designs on one plot.
• Expressions. Z0 values are expressions, with no spaces inside them. The single section uses √(50·100) = 70.71 Ω. The binomial sections use 50·2^¼ = 59.46 Ω and 50·2^¾ = 84.09 Ω.

#Results

Measured on a 16 001-point sweep:

#Check against the closed form

For a single section with maximum tolerable reflection Γm (the standard result; see e.g. Pozar, Microwave Engineering, quarter-wave transformer bandwidth):

cos θm = 2 Γm √(Z0 ZL) / ( |ZL − Z0| √(1 − Γm²) )
FBW    = 2 − 4θm/π

With Γm = 0.1 (20 dB): cos θm = 2·0.1·70.71/(50·0.995) = 0.2843, so θm = 73.48° and FBW = 2 − 4·1.2825/π = 0.3671. This agrees with the simulated 36.7 %.

The Smith chart shows why the binomial version is wider. Its locus winds more tightly around the centre, because its reflection has a double zero at f0 (maximally flat) instead of a single one.

#Try

• Drag ZL (add tune 60 300 to its .param) and watch both bandwidths shrink as the ratio grows.
• Replace the binomial impedances with .param z1 = 60 tune 50 100 and .param z2 = 85 tune 50 100. Then add .goal db(S33) < -20 0.6G 1.4G and press Optimise. It stops as soon as the goal is met (z1 ≈ 64 Ω, z2 ≈ 85 Ω), giving a 20 dB band of about 83 % against the binomial's 71.6 %. That is the equal-ripple (Chebyshev) trade: less flatness at f0 for more bandwidth.