Series and Parallel Resistor Combinations (2.5, 7.5) Dr. S. M. Goodnick September 17th, 2003 ECE201 Lect-10

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-10

Introduction Complicated networks of resistors/ impedances can be replaced by a single equivalent resistor/impedance. ECE201 Lect-10

Equivalent Resistance i(t) + – v(t) i(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-10

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-10

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-10

Series Resistance R1 R3 R2 Req Req = R1 + R2 + R3 ECE201 Lect-10

Series Two elements are in series if the current that flows through one must also flow through the other. R1 R2 Series Not Series R1 R2 ECE201 Lect-10

Series Impedance Z1 Z2 Zeq Z3 Zeq = Z1 + Z2 + Z3 ECE201 Lect-10

Example: Series Inductors What is the equivalent impedance of two series inductors? L1 L2 ECE201 Lect-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-10

Parallel Resistance R2 R1 Req R3 ECE201 Lect-10

Parallel Two elements are in parallel if they are connected between (share) the same two (distinct) end nodes. R1 R2 Parallel Not Parallel R1 R2 ECE201 Lect-10

Parallel Impedance Z1 Z2 Z3 Zeq ECE201 Lect-10

Example: Parallel Capacitors What is the equivalent impedance of two parallel capacitors? C1 C2 ECE201 Lect-10

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-10

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 ECE201 Lect-10

Example #1: Bandpass Filter 10W 769pF 159mH For w = 2.86  106, find the equivalent impedance. ECE201 Lect-10

Now combine series impedances Compute Impedances 10W -j455W j455W Now combine series impedances ECE201 Lect-10

Now combine parallel impedances Bandpass Filter 10W+ j455W -j455W 455.1W  88.7 Now combine parallel impedances ECE201 Lect-10

Bandpass Filter ECE201 Lect-10

Example #2: Loaded Bandpass Filter 10W 769pF 159mH 50kW For w = 2.86  106, find the equivalent impedance. ECE201 Lect-10

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