ECE201 Lect-61 Series and Parallel Resistor Combinations (2.5, 8.5) Dr. Holbert February 6, 2006

ECE201 Lect-62 Introduction For analysis, series resistors/impedances can be replaced by an equivalent resistor/ impedance. Parallel resistors/impedances can be replaced by an equivalent resistor/ impedance.

ECE201 Lect-63 Introduction Complicated networks of resistors/ impedances can be replaced by a single equivalent resistor/impedance.

ECE201 Lect-64 Equivalent Resistance i(t) + – v(t) i(t) + – v(t) R eq R eq is equivalent to the resistor network on the left in the sense that they have the same i-v characteristics.

ECE201 Lect-65 Equivalent Resistance The rest of the circuit cannot tell whether the resistor network or the equivalent resistor is connected to it. The equivalent resistance cannot be used to find voltages or currents internal to the resistor network.

ECE201 Lect-66 Equivalent Impedance I + – V I + – V Z eq Z eq is equivalent to the network on the left in the sense that they have the same phasor I-V characteristics at the frequency .

ECE201 Lect-67 Series Resistance R1R1 R3R3 R2R2 R eq R eq = R 1 + R 2 + R 3

ECE201 Lect-68 Series Two elements are in series if the current that flows through one must also flow through the other. R1R1 R2R2 Series R1R1 R2R2 Not Series

ECE201 Lect-69 Z eq Series Impedance Z1Z1 Z eq = Z 1 + Z 2 + Z 3 Z3Z3 Z2Z2

ECE201 Lect-610 Example: Series Inductors What is the equivalent impedance of two series inductors? L2L2 L1L1

ECE201 Lect-611 Series Inductors The equivalent impedance is Z eq = Z 1 + Z 2 = j  (L 1 +L 2 ) Two inductors in series are equivalent to a single inductor whose inductance is the sum of the two inductances. Z eq = j  (L 1 +L 2 ) = j  L eq

ECE201 Lect-612 Parallel Resistance R eq R3R3 R2R2 R1R1

ECE201 Lect-613 Parallel Two elements are in parallel if they are connected between (share) the same two (distinct) end nodes. Parallel Not Parallel R1R1 R2R2 R1R1 R2R2

ECE201 Lect-614 Parallel Impedance Z3Z3 Z1Z1 Z2Z2 Z eq

ECE201 Lect-615 Example: Parallel Capacitors What is the equivalent impedance of two parallel capacitors? C1C1 C2C2

ECE201 Lect-616 Parallel Capacitors The equivalent impedance is Two capacitors in parallel are equivalent to a single capacitor whose capacitance is the sum of the two capacitances.

ECE201 Lect-617 Flash Example: Ladder Network Find the equivalent resistance by making combinations of series and parallel resistors until only one resistor is left. 1k  2k  1k  2k  1k 

ECE201 Lect  769pF 159  H Example #1: Bandpass Filter For  = 2.86  10 6, find the equivalent impedance.

ECE201 Lect  -j455  j455  Compute Impedances Now combine series impedances

ECE201 Lect-620 -j455  10  j455  Bandpass Filter Now combine parallel impedances   88.7 

ECE201 Lect-621 Bandpass Filter

ECE201 Lect-622 Example #2: Loaded Bandpass Filter For  = 2.86  10 6, find the equivalent impedance. 10  769pF 159  H 50k 

ECE201 Lect-623 Class Examples Learning Extension E2.12 Learning Extension E2.13 Learning Extension E8.10