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Series and Parallel Resistor Combinations (2.5, 7.5). Dr. Holbert September 11, 2001. 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. Introduction.
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Series and Parallel Resistor Combinations (2.5, 7.5) Dr. Holbert September 11, 2001 ECE201 Lect-7
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-7
Introduction • Complicated networks of resistors/ impedances can be replaced by a single equivalent resistor/impedance. ECE201 Lect-7
i(t) + v(t) – Equivalent Resistance i(t) Req is equivalent to the resistor network on the left in the sense that they have the same i-v characteristics. + v(t) Req – ECE201 Lect-7
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-7
Equivalent Impedance I I 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. + + V V Zeq – – ECE201 Lect-7
R1 Req R2 R3 Series Resistance Req = R1 + R2 + R3 ECE201 Lect-7
Series R1 R2 Series Two elements are in series if the current that flows through one must also flow through the other. Not Series R1 R2 ECE201 Lect-7
Series Impedance Zeq = Z1 + Z2 + Z3 Z1 Z2 Zeq Z3 ECE201 Lect-7
Example: Series Inductors • What is the equivalent impedance of two series inductors? L1 L2 ECE201 Lect-7
Series Inductors • The equivalent impedance is jw(L1+L2) • Two inductors in series are equivalent to a single inductor whose inductance is the sum of the two inductances. ECE201 Lect-7
R1 R2 Req Parallel Resistance 1/Req = 1/R1 + 1/R2 + 1/R3 R3 ECE201 Lect-7
R1 R2 Parallel Two elements are in parallel if they are connected between (share) the same two end nodes. Parallel Not Parallel R1 R2 ECE201 Lect-7
Z1 Z2 Parallel Impedance 1/Zeq = 1/Z1 + 1/Z2 + 1/Z3 Z3 Zeq ECE201 Lect-7
Example: Parallel Capacitors • What is the equivalent impedance of two parallel capacitors? C1 C2 ECE201 Lect-7
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-7
Example: Ladder Network • Ladder networks are used in analog-to-digital converters to provide reference voltages that are 1/2, 1/4, 1/8, etc. of a source voltage. ECE201 Lect-7
1kW 1kW 1kW 2kW 2kW Ladder Network Find the equivalent resistance. ECE201 Lect-7
Ladder Network Find the equivalent resistance by making combinations of series and parallel resistors until you have only one resistor left. ECE201 Lect-7
1kW 2kW 2kW 2kW Ladder Network ECE201 Lect-7
1kW 2kW 1kW Ladder Network ECE201 Lect-7
Ladder Network 2kW 2kW ECE201 Lect-7
Ladder Network The equivalent resistance of the ladder network is 1kW. 1kW ECE201 Lect-7
10W 769pF 159mH Example: Bandpass Filter For w = 2.86 106, find the equivalent impedance. ECE201 Lect-7
Compute Impedances Now combine series impedances 10W -j455W j455W ECE201 Lect-7
Bandpass Filter Now combine parallel impedances 10W+ j455W -j455W 455.1W 88.7 ECE201 Lect-7
Bandpass Filter ECE201 Lect-7
10W 50kW 769pF 159mH Loaded Bandpass Filter For w = 2.86 106, find the equivalent impedance. ECE201 Lect-7
Class Examples • Learning Extension E2.12 • Learning Extension E2.13 • Learning Extension E7.10 ECE201 Lect-7
