Series Connections of Resistors

Once the trees and simple loops are removed from a network, the remaining backbone consists primarily of long strings of resistors, one per bit pitch, with occasional cross links, as shown in Figure 74. These long sections of series resistors, with a current source at each intervening node, can be replaced by a single resistor with a current source at each end. An intuitive derivation of the equivalent circuit is given in the following section.

For the skeptical, a more formal derivation is given in Section 5.4.2.

Pad Pad

Figure 74: Power Network with Trees and Simple Loops Removed

5.4.1 Equivalent Circuit for Series Resistors

Consider the series chain in Figure 75a. There will be some voltage, VS, between the two series ends, N₁ and N_n. Conceptually, a voltage source with value V_S can be added between nodes N₁ and N_n without disturbing the network. Superposition can be applied to this network to produce the equivalent circuit shown in figure 75b. The equivalent resistance RS is just the sum of all the resistors in series. Superposition can be used to

I1 I2 I3 In-1 In

Figure 75: Series Equivalent Circuit

determine how the current from each current source divides between the two ends. All current sources except the one in question are replaced by open circuits, while the voltage source between nodes N₁ and N_n is replaced by a short circuit. The resulting system is a simple current divider, and the additional current at N₁ and N_n is just the sum of all the divided currents: An arbitrarily long series of resistors can thus be replaced by a single resistor and two current sources.

Once the network has been solved with the equivalent series circuit and the voltages

at the end nodes are known, the solver must determine the voltages at the series nodes.

Superposition can be used to calculate the intermediate values for this system:

V

Starting at one end, the voltage at each successive node is equal to the value at the previous node plus the drop in the intervening resistor. This drop has two parts. The first, labeled Voltage Divider Drop, is caused by ^VS, the voltage imposed by the rest of the network across this series section. To calculate it, ^VS is split across the resistors in proportion to their values. The second component, labeled Current Source Drop, is caused by current injected at nodes within the series. Between each pair of nodes, this drop is equal to the connecting resistor’s value times the current that would flow through this resistor if^VS was replaced by a short circuit.

Figures 76 and 77 show an example of series replacement and intermediate node voltage calculation. The three resistors in the circuit on the left are replaced by a single one, and the current at ^N₂ and ^N₃ is redistributed to the end nodes, ^N₁ and ^N₄. This equivalent circuit is connected to the rest of the network and solved.

-1mA -1mA -2mA -1mA

2Ω 1Ω 1Ω

Figure 76: Series Circuit Example

In the example, suppose that the drop ^VS between nodes ^N₁ and ^N₄ is -4mV. The next step is setting the voltages at ^N2 and ^N3. Figure 77 shows the different voltage

components for each node. The voltage divider drops are calculated by partitioning ^VS between the three resistors. The current source drop is found by calculating the current that would flow through each resistor with ^N₁ and ^N₄ grounded. Beginning at the left, the current between nodes ^N₁ and ^N₂ is ^Ie2 ^{=I e}1 ^0I1 ⁼⁰1^mA. Similarly, the current between ^N₂ and ^N₃ is ^Ie3 ^{=I e}2^0I1⁼0. The current source drop in each section is the resistance times these currents. The total drop for each node is calculated by adding the two components.

-5mV 0mV -1mV -2mV -3mV -4mV

Actual Voltage 0mV

-1mV -2mV

Current Source Drop 0mV

-1mV -2mV -3mV -4mV

Voltage Divider Drop 4V 2Ω

4Ω

-1mA 2Ω 2mA1Ω

0mA1Ω

4V1Ω 4Ω 4V1Ω

4Ω

N1 N2 N3 N4

Figure 77: Voltages for Series Circuit Example

5.4.2 Norton Equivalent Circuits for the Series Systems

The previous derivation and example were intended to be intuitive; this section gives a more formal derivation of the series equivalent circuit. Figure 78a shows a series of n resistors interspersed with current sources. R_L and I_L form the Norton equivalent circuit representing the rest of the network as seen from the left end of the series. Figure 78b shows the series equivalent circuit for this configuration, as derived in the previous

section. The following section shows that the two circuits are equivalent when viewed from the right side terminals; by symmetry, the same can be shown from the left side.

IL I1 I2 I3 In

Figure 78: Series and Equivalent Circuit

All the resistors in Figure 78a are in series, so the equivalent resistance is just their sum, labeled RT in Equation 55. When the right terminals are shorted, n+1 current dividers are formed; by superposition, the short-circuit current is just the sum of these dividers, as indicated in Equation 56.

R By inspection, the equivalent resistance of Circuit 78b is identical to that of 78a. The short circuit current (Equation 57) is the sum of the series equivalent current Ien and the fraction of IL and Ie1 supplied by the resistive divider. Combining terms under the same summation gives Equation 58. Substituting Equation 55 into Equation 58, and combining terms (Equation 59) allows the ^Pi_j⁰=¹1R_j and ^Pn_j

=iR_j summations to be combined. The combined terms cancel, giving Equation 60, identical to Equation 56. The two circuits thus have identical Norton Equivalent circuits.

I