S. Ross and W. G. OldhamEECS 40 Fall 2002 Lecture 8 Copyright, Regents University of California 1

S. Ross and W. G. OldhamEECS 40 Fall 2002 Lecture 8 Copyright, Regents University of California 2 SERIES ELEMENTS KCL tells us that all of the elements in a single branch carry the same current. We say these elements are in series. Current entering node = Current leaving node i 1 = i 2

S. Ross and W. G. OldhamEECS 40 Fall 2002 Lecture 8 Copyright, Regents University of California 3 RESISTORS IN SERIES Circuit with several resistors in series: Find “equivalent resistance” KCL tells us same current flows through every resistor KVL tells us Clearly, Thus, equivalent resistance of resistors in series is the sum R 2 R 1 V SS I R 3 R 4  +

S. Ross and W. G. OldhamEECS 40 Fall 2002 Lecture 8 Copyright, Regents University of California 4 VOLTAGE DIVIDER Circuit with several resistors in series R 2 R 1 V SS I R 3 R 4  +  +  + V1V1 V3V3 We know Thus, and etc…

S. Ross and W. G. OldhamEECS 40 Fall 2002 Lecture 8 Copyright, Regents University of California 5 WHEN IS VOLTAGE DIVIDER FORMULA CORRECT? R 2 R 1 V SS I R 3 R 4  +  V RRRR R V    Correct if nothing else connected to nodes 3 V SS I R 2 R 1 R 3 R 4  +  + 2 V R 5 I because R 5 removes condition of resistors in series II 3  V2V2 SS V RRRR R V   ≠

S. Ross and W. G. OldhamEECS 40 Fall 2002 Lecture 8 Copyright, Regents University of California 6 MEASURING CURRENT To measure current in a circuit, insert DMM (in current mode) into circuit, in series with measured element. But ammeters change the circuit. Ammeters are characterized by their “ammeter input resistance,” R in. Ideally this should be very low. Typical value 1 . Real Ammeter Ideal Ammeter R in ?

S. Ross and W. G. OldhamEECS 40 Fall 2002 Lecture 8 Copyright, Regents University of California 7 MEASURING CURRENT Potential measurement error due to non-zero input resistance: R in _ + V I meas R 1 R 2 ammeter with ammeter _ + V I R 1 R 2 undisturbed circuit Example: V = 1 V: R 1 = R 2 = 500 , R in = 1 

S. Ross and W. G. OldhamEECS 40 Fall 2002 Lecture 8 Copyright, Regents University of California 8 PARALLEL ELEMENTS KVL tells us that any set of elements which are connected at both ends carry the same voltage. We say these elements are in parallel. KVL clockwise, start at top: Vb – Va = 0 Va = Vb

S. Ross and W. G. OldhamEECS 40 Fall 2002 Lecture 8 Copyright, Regents University of California 9 RESISTORS IN PARALLEL KCL tells us I ss = I 1 + I 2 The two resistors are in parallel; they have the same voltage V X = I 1 R 1 = I 2 R 2 Resistors in parallel can be made into one equivalent resistor R 2 R 1 I SS I 2 I 1 x ground I ss = V X / R 1 + V X / R 2 V X = I ss R 1 R 2 / (R 1 +R 2 ) Generally, R eq = (R R R …) -1

S. Ross and W. G. OldhamEECS 40 Fall 2002 Lecture 8 Copyright, Regents University of California 10 IMPORTANT FACTS For resistors in series: –Current through R eq is equal to the current through each of the original resistors (all have same current) –Voltage over R eq is the sum of the voltages over the original resistors For resistors in parallel: –Current through R eq is equal to the sum of the currents through each of the original resistors –Voltage over R eq is equal to the voltage over the original resistors (all have same voltage)

S. Ross and W. G. OldhamEECS 40 Fall 2002 Lecture 8 Copyright, Regents University of California 11 CURRENT DIVIDER Remember V X = I 1 R 1 = I SS R eq There is a simple equation for the way current splits between two parallel resistors: R 2 R 1 I SS I 2 I 1 x ground

S. Ross and W. G. OldhamEECS 40 Fall 2002 Lecture 8 Copyright, Regents University of California 12 REAL VOLTMETERS How is voltage measured? Digital multimeter (DMM) in parallel with measured element. Connecting a real voltmeter across two nodes changes the circuit. The voltmeter may be modeled by an ideal voltmeter (open circuit) in parallel with a resistance: “voltmeter input resistance,” R in. Typical value: 10 M  Real Voltmeter Ideal Voltmeter R in

S. Ross and W. G. OldhamEECS 40 Fall 2002 Lecture 8 Copyright, Regents University of California 13 REAL VOLTMETERS Example: V SS R 1 R  V 2 But if a 1% error Computation of voltage (uses ideal voltmeter) - + V SS R 1 R 2 R in Measurement of voltage (with loading by real voltmeter)

S. Ross and W. G. OldhamEECS 40 Fall 2002 Lecture 8 Copyright, Regents University of California 14 CAPACITORS IN PARALLEL C1C1 i(t) C2C2 | ( +  V Equivalent capacitance defined by C eq i(t) | ( +  V(t)

S. Ross and W. G. OldhamEECS 40 Fall 2002 Lecture 8 Copyright, Regents University of California 15 CAPACITORS IN SERIES C1C1 V1V1 i(t) C2C2 | ( V2V2 +  +  C eq i(t) | ( V eq +  Equivalent to

S. Ross and W. G. OldhamEECS 40 Fall 2002 Lecture 8 Copyright, Regents University of California 16 DIGITAL CIRCUIT: DRAM +  initially uncharged -