EEGR2111 L6

Last time: Kirchoff Laws

KCL: algebraic sum of currents into node

KVL: algebraic sum of voltages around a loop

Used in conjunction with Ohms law to analyze circuits

Voltage divider

3-resistor divider

Above:
KVL: V_R1 + V_R2 + V_R3 - V1 = 0
Ohms law: V_R1 = i R1, V_R2 = i R2, V_R3 = i R3
So:
V_Rn / V1 = Rn / (R1 + R2 + R3)
where Rn is R1, R2, or R3

Equivalent series resistance

In the above example we have 3 resistors in series

From KVL we can get:
V1= i (R1 + R2 + R3)

This has the same form as Ohms law for an equivalent resistor, R_eq
V1 = i R_eq; where R_eq = R1 + R2 + R3

Thus the equivalent of N series resistors is:
R_eq = R1 + R2 + ... + RN

This is often useful for simplifying circuit analysis

For the circuit above, show thhat the power delivered to the equivalent resistor R_eq is the same as the sum of the power delivered to the three resistors
p= i²R1 +i²R2 + i²R3 = i² (R1 + R2 + R3) = i² R_eq

Current division

Three resistors in parallel:

Above:
KCL: i1 - i_R1 - i_R2 - i_R3 = 0
What is KCL for the other node?
Note - same voltage across all resistors = V
Ohms law: i_R1 = V / R1, i_R2 = V / R2, i_R3 = V / R3
i1 = V { (1/R1) + (1/R2) + (1/R3) } = V ( G1 + G2 + G3)
==> iRn / i1 = Gn / (G1 + G2 + G3)

Three resistors in parallel with voltage source:

Above:
KCL: i1 - i_R1 - i_R2 - i_R3 = 0
Ohms law: i_R1 = V1 / R1, i_R2 = V1 / R2, i_R3 = V1 / R3
i1 = V1 { (1/R1) + (1/R2) + (1/R3) } = V1 ( G1 + G2 + G3)
==> iRn / i1 = Gn / (G1 + G2 + G3)

Hence current division: total current divided by 3 resistors

Equivalent parallel resistance

In the above example we have 3 resistors in parallel

From KCL we can get:
i1 = V ( G1 + G2 + G3)

This has the same form as Ohms law for an equivalent resistor, R_eq
i1 = V / R_eq; where R_eq = 1 / (G1 + G2 + G3)

Thus the equivalent of N series resistors is:
R_eq = 1 / (G1 + G2 + ... + GN) = 1 / { (1/R1) + (1/R2) + ... + (1/RN) }

This is often useful for simplifying circuit analysis

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