ECE 210: ZIR & ZSR Superposition (Example 17)

The Power of Superposition

Instead of solving for new integration constants at every switch event, we track the response components globally:

ZIR $i_{z i} (t)$ : The initial energy present at $t = 0$ decaying through all future circuit topologies.
ZSR $i_{z s} (t)$ : The response to the input injected later, assuming initial energy was zero.

Initial Current

i_{L} (0^{-})

2.0 A

Time Constant

τ_{1}

(for

0 \leq t < t_{s}

) 2.0 s

Switch-back Time

t_{s}

2.0 s

Time Constant

τ_{2}

(for

t \geq t_{s}

) 1.0 s

Steady State Current

i_{L} (\infty)

2.0 A

i_{L} (t) = i_{z i} (t) + i_{z s} (t)

Zero-Input Response (ZIR):

i_{z i} (t) = {\begin{cases} 2.0 e^{- t / 2.0}, & 0 \leq t < 2.0 \\ 0.736 e^{- (t - 2.0) / 1.0}, & t \geq 2.0 \end{cases}

Zero-State Response (ZSR):

i_{z s} (t) = {\begin{cases} 0, & 0 \leq t < 2.0 \\ 2.0 (1 - e^{- (t - 2.0) / 1.0}), & t \geq 2.0 \end{cases}

The total current (green) is the exact sum of the ZIR (blue) and ZSR (orange) components.