Network Loading and Power Audits

Loaded Divider Load Dependence

branch split and reduced output

A loaded divider is checked as a topology calculation: the load changes the lower branch before the output voltage is read. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A load is another resistor on the output node

The load is not a caption on the divider. It is a parallel path from the output node to the return wire, so it changes the lower equivalent resistance before the divider voltage is computed.

Rlow=Rb∥RL,Vo=IsRlowR_{\text{low}} = R_b \parallel R_L,\quad V_o = I_s R_{\text{low}}
Loaded dividerThe load shares the same output node as the bottom resistor.18 V3 ohm3 ohm3 ohm4 A2 A2 A+-12 V+-6 V+-6 Voutput

The loaded lower branch sets the output voltage

Read each row left to right. First combine the bottom resistor and load as a parallel resistance; then the source current sets the output voltage across that combined branch.

VsRtRb∥RLIsVo12 V2 ohm1 ohm4 A4 V18 V3 ohm32 ohm4 A6 V12 V2 ohm2 ohm3 A6 V\begin{array}{c|c|c|c|c}V_s&R_t&R_b\parallel R_L&I_s&V_o\\12\ \text{V}&2\ \text{ohm}&1\ \text{ohm}&4\ \text{A}&4\ \text{V}\\18\ \text{V}&3\ \text{ohm}&\tfrac{3}{2}\ \text{ohm}&4\ \text{A}&6\ \text{V}\\12\ \text{V}&2\ \text{ohm}&2\ \text{ohm}&3\ \text{A}&6\ \text{V}\\\end{array}
Loaded dividerThe middle row is the rendered loaded-divider case.18 V3 ohm3 ohm3 ohm4 A2 A2 A+-12 V+-6 V+-6 Voutput

The output current splits into two exact branches

After the output voltage is known, the bottom resistor current and load current are separate Ohm-law reads. Their sum must return to the source current.

VoRbRLIbILIs4 V2 ohm2 ohm2 A2 A4 A6 V3 ohm3 ohm2 A2 A4 A6 V4 ohm4 ohm32 A32 A3 A\begin{array}{c|c|c|c|c|c}V_o&R_b&R_L&I_b&I_L&I_s\\4\ \text{V}&2\ \text{ohm}&2\ \text{ohm}&2\ \text{A}&2\ \text{A}&4\ \text{A}\\6\ \text{V}&3\ \text{ohm}&3\ \text{ohm}&2\ \text{A}&2\ \text{A}&4\ \text{A}\\6\ \text{V}&4\ \text{ohm}&4\ \text{ohm}&\tfrac{3}{2}\ \text{A}&\tfrac{3}{2}\ \text{A}&3\ \text{A}\\\end{array}
Loaded divider branch splitThe middle row shows equal branch currents.18 V3 ohm3 ohm3 ohm4 A2 A2 A+-12 V+-6 V+-6 Voutput

The rendered row closes both voltage and current

In the rendered row, the top resistor drops 12 volts and the output branch drops 6 volts. The two branch currents are 2 amperes and 2 amperes.

12 V+6 V=18 V,2 A+2 A=4 A12\ \text{V}+6\ \text{V}=18\ \text{V},\quad 2\ \text{A}+2\ \text{A}=4\ \text{A}
Loaded divider closureKVL and KCL close on the same rendered row.18 V3 ohm3 ohm3 ohm4 A2 A2 A+-12 V+-6 V+-6 Voutput

Loading is a network calculation, not a label

The output is smaller than an unloaded equal-resistor divider would suggest because the load is part of the network. The table, diagram, and current split all cite the same topology.

Rb∥RL<Rb,Ib+IL=Is,Vt+Vo=VsR_b\parallel R_L < R_b,\quad I_b + I_L = I_s,\quad V_t + V_o = V_s
Loaded divider auditThe topology itself causes the output reduction.18 V3 ohm3 ohm3 ohm4 A2 A2 A+-12 V+-6 V+-6 Voutput