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Series and Parallel Resistor Combinations (2.5, 8.5) Dr. Holbert February 6, 2006 ECE201 Lect-6 1 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-6 2 Introduction • Complicated networks of resistors/ impedances can be replaced by a single equivalent resistor/impedance. ECE201 Lect-6 3 Equivalent Resistance i(t) i(t) + + v(t) – v(t) Req – Req is equivalent to the resistor network on the left in the sense that they have the same i-v characteristics. ECE201 Lect-6 4 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-6 5 Equivalent Impedance I I + + V V Zeq – – Zeq is equivalent to the network on the left in the sense that they have the same phasor I-V characteristics at the frequency w. ECE201 Lect-6 6 Series Resistance R1 R2 Req R3 Req = R1 + R2 + R3 ECE201 Lect-6 7 Series Two elements are in series if the current that flows through one must also flow through the other. Series R1 R2 Not Series R1 R2 ECE201 Lect-6 8 Series Impedance Z1 Z2 Zeq Z3 Zeq = Z1 + Z2 + Z3 ECE201 Lect-6 9 Example: Series Inductors • What is the equivalent impedance of two series inductors? L1 L2 ECE201 Lect-6 10 Series Inductors • The equivalent impedance is Zeq = Z1 + Z2 = jw(L1+L2) • Two inductors in series are equivalent to a single inductor whose inductance is the sum of the two inductances. Zeq = jw(L1+L2) = jwLeq ECE201 Lect-6 11 Parallel Resistance R1 R2 R3 Req 1 1 1 1 Req R1 R2 R3 ECE201 Lect-6 12 Parallel Two elements are in parallel if they are connected between (share) the same two (distinct) end nodes. R1 Parallel R2 Not Parallel R1 R2 ECE201 Lect-6 13 Parallel Impedance Z1 Z2 Z3 Zeq 1 1 1 1 Z eq Z1 Z 2 Z 3 ECE201 Lect-6 14 Example: Parallel Capacitors • What is the equivalent impedance of two parallel capacitors? C1 ECE201 Lect-6 C2 15 Parallel Capacitors • The equivalent impedance is 1 1 1 jw C1 jw C2 Z eq Z1 Z 2 1 Z eq jw C1 C2 • Two capacitors in parallel are equivalent to a single capacitor whose capacitance is the sum of the two capacitances. ECE201 Lect-6 16 Flash Example: Ladder Network Find the equivalent resistance by making combinations of series and parallel resistors until only one resistor is left. 1kW 2kW 1kW 2kW 1kW ECE201 Lect-6 1kW 17 Example #1: Bandpass Filter 10W 769pF 159mH For w = 2.86 106, find the equivalent impedance. ECE201 Lect-6 18 Compute Impedances 10W -j455W j455W Now combine series impedances ECE201 Lect-6 19 Bandpass Filter 10W j455W -j455W 455.1W 88.7 Now combine parallel impedances ECE201 Lect-6 20 Bandpass Filter j 45510 j 455 20.7kW 1.3 j 455 10 j 455 ECE201 Lect-6 21 Example #2: Loaded Bandpass Filter 10W 50kW 769pF 159mH For w = 2.86 106, find the equivalent impedance. ECE201 Lect-6 22 Class Examples • Learning Extension E2.12 • Learning Extension E2.13 • Learning Extension E8.10 ECE201 Lect-6 23