This paper investigates the problem of model-order reduction for linear systems arising from minimum-phase circuits and filters, where stability and frequency characteristics must be preserved. The MPPBT framework uses a Riccati–Lyapunov Gramian pair and constructs a balancing transformation with the sequence of MPPBT singular values, from which a relative error bound in the $H_{\infty }$ norm and a scoring function, $S_{\beta }$, are derived to select the model order. We apply the algorithm to a fourth-order Butterworth low-pass filter with a full-order state dimension, $n=4$, and reduced-order models with $r=1, 2, 3$ are examined. The results show that the model with $r=3$ yields an $H_{\infty }$ error of approximately $6.3\times {10}^{-3}$ and an $H_2$ error of approximately $2.2\times {10}^{-3}$. The model with $r=1$ gives an $H_{\infty }$ error of approximately $1.46$ and an $H_2$ error of approximately $4.8\times {10}^{-1}$. The model with $r=2$ attains a composite score of $S_{\beta }\approx 0.16$, preserves stability and the minimum-phase property, and is regarded as a balanced choice between accuracy and complexity. A further comparison on an RLC ladder circuit of order $n=15$ shows that at $r=3$, MPPBT achieves the lowest $H_{\infty }$ error among BT, PRBT, and MPPBT, while retaining an $H_2$ error close to that of BT.