Block diagram model of R-L ladder network

Let $V_1$ = voltage of node 1 (across $L_1$), $I_1$ = current in $R_1$, $I_2$ = current in $R_2$ and $L_2$, and $V_L$ = output across $L_2$.

Element equations (Laplace)

$$\begin{aligned} I_1(s) &= \frac{V_{in}(s) - V_1(s)}{R_1} \\ V_1(s) &= sL_1\left[I_1(s) - I_2(s)\right] \\ I_2(s) &= \frac{V_1(s) - V_L(s)}{R_2} \\ V_L(s) &= sL_2\,I_2(s) \end{aligned}$$

Block diagram

 Vin─►(Σ)►[1/R1]─►(Σ)►[sL1]─┬─►(Σ)►[1/R2]─┬─►[sL2]─┬─►VL
       ▲-          ▲-       │   ▲-        │        │
       │           └── I2 ──┼───┼─────────┘        │
       └── V1 ──────────────┘   └── VL ────────────┘

Reduction

Each block pair forms a simple loop; let

• loop $L_a = -\dfrac{sL_1}{R_1}$ (via $V_1$),
• loop $L_b = -\dfrac{sL_1}{R_2}$ (via $I_2$),
• loop $L_c = -\dfrac{sL_2}{R_2}$ (via $V_L$).

Only $L_a$ and $L_c$ do not touch. Forward path $P = \dfrac{1}{R_1}\cdot sL_1\cdot\dfrac{1}{R_2}\cdot sL_2$, with $\Delta_1 = 1$. Using Mason's rule (same result as stepwise reduction):

$$\frac{V_L}{V_{in}} = \frac{\frac{s^2L_1L_2}{R_1R_2}}{1 + \frac{sL_1}{R_1} + \frac{sL_1}{R_2} + \frac{sL_2}{R_2} + \frac{s^2L_1L_2}{R_1R_2}}$$

Multiplying by $R_1R_2$:

$$\frac{V_L(s)}{V_{in}(s)} = \frac{L_1L_2s^2}{L_1L_2s^2 + (L_1R_1 + L_1R_2 + L_2R_1)s + R_1R_2}$$

This is a second-order high-pass network: the output is zero at DC (inductors short) and approaches $V_{in}$ at high frequency.