Quarter-Wave Impedance Transformer (Z_T) and Off-Frequency VSWR

Match real load resistance R_L at f_0 — and audit bandwidth roll-off away from f_0

Inserting a quarter-wavelength (theta = PI()/2 at center frequency f_0) lossless transmission line of characteristic impedance Z_T = SQRT(Z_0 * R_L) between feedline Z_0 and real load resistance R_L transforms load impedance to Z_T^2 / R_L = Z_0, achieving zero reflection (|Gamma| = 0, VSWR = 1.0000)! Away from f_0 at normalized frequency ratio f / f_0, electrical length becomes theta = (PI()/2) * (f / f_0) and Pozar’s exact singularity-free reflection magnitude is |Gamma| = (ABS(R_L - Z_0) * ABS(COS(theta))) / SQRT(4 * Z_0 * R_L + ((R_L - Z_0) * COS(theta))^2).

Verify matching Z_0 = 50 Ohms (B2) to R_L = 200 Ohms (B3): required transformer impedance is Z_T = SQRT(50 * 200) = 100.0000 Ohms. At f / f_0 = 1.0000, COS(PI()/2) = 0 so |Gamma| = 0 and VSWR = 1.0000; at off-tune f / f_0 = 2/3 (B4 = 0.6666666667, theta = 60 deg), |Gamma| rises to 75 / SQRT(45,625) = 0.3511 (VSWR = 2.0823); and at DC (f / f_0 = 0), |Gamma| reverts to unmatched 150 / 250 = 0.6000 (VSWR = 4.0000)!

Quarter-Wave Matching Impedance Z_T = SQRT(Z_0 * R_L) Formula

Compute geometric-mean impedance Z_T from positive real impedances Z_0 > 0 and R_L > 0.

=IF(OR(B2 <= 0, B3 <= 0), "Invalid Impedance Input", ROUND(SQRT(B2 * B3), 4))

Singularity-Free Off-Frequency Reflection |Gamma| and VSWR Formula

Writing |Gamma| in terms of COS((PI()/2) * (f / f_0)) instead of TAN((PI()/2) * (f / f_0)) avoids division-by-zero overflow right at the center frequency f / f_0 = 1!

=LET(z_0, B2, r_ld, B3, f_rat, B4, IF(OR(z_0 <= 0, r_ld <= 0, f_rat < 0), "Invalid Impedance Input", LET(cs_th, ABS(COS((PI() / 2) * f_rat)), num_g, ABS(r_ld - z_0) * cs_th, den_g, SQRT(4 * z_0 * r_ld + (num_g^2)), gam_m, num_g / den_g, ROUND((1 + gam_m) / (1 - gam_m), 4))))

Further reading

• Arte Sheets Lesson — RF VSWR, Reflection Coefficient, and Antenna Return Loss
• Microsoft Support — COS function