EES 4 Spring 00 Lecture 0 opyright egents of University of alifornia W. G. Oldham NODAL ANALY ONTNUED Lecture 9 review: Formal nodal analysis oltage divider example Today: Nodal analysis with floating voltage sources eal voltmeters eal ammeters Parallel resistors urrent dividers Series and parallel capacitors
EES 4 Spring 00 Lecture 0 opyright egents of University of alifornia W. G. Oldham NODAL ANALY WTH FLOATNG OLTAGE SOUES A floating voltage source is a voltage source for which neither side is connected to the reference node. LL in the circuit below is an example. a LL - b 4 What is the problem? We cannot write KL at node a or b because there is no way to express the current through the voltage source in terms of a b. Solution: Define a supernode that chunk of the circuit containing nodes a and b. Express KL at this supernode.
EES 4 Spring 00 Lecture 0 opyright egents of University of alifornia W. G. Oldham FLOATNG OLTAGE SOUES (cont.) Use a Gaussian surface to enclose the floating voltage source; write KL for that surface supernode a LL - b 4 We have two unknowns: a and b. We obtain one equation from KL at supernode: a b 0 4 We obtain a second auxiliary equation from the property of the voltage source: (often called the constraint ) LL b a Equations & Unknowns 3
EES 4 Spring 00 Lecture 0 opyright egents of University of alifornia W. G. Oldham ANOTHE EXAMPLE 0K a b 3 0K 6 0K 4 0K hoose reference node (can it be chosen to avoid floating voltage source?) Label unknowns a and b a b 3 Equation at supernode: a ( ) b a 4 4 4 Auxiliary equation: b a a b - Solve: ( 4 ) a 4 4 - SOLUTON : a 0 b b a 4
5 EES 4 Spring 00 Lecture 0 opyright egents of University of alifornia W. G. Oldham ETOS N PAALLEL Define unknown node voltages X Select eference Node X Note: ss, i.e., X X ESULT EQUALENT ETANE: ESULT UENT DDE: X X
EES 4 Spring 00 Lecture 0 opyright egents of University of alifornia W. G. Oldham DENTFYNG SEES AND PAALLEL OMBNATONS Use series/parallel equivalents to simplify a circuit before starting KL/KL 3 3 eq parallel 4 5 3 6 3 0 K 4 5 6 0 K 0 K 5 K ( ) 3 eq X? ( ) ( X 5 K 3 4 5 ) 6 6
EES 4 Spring 00 Lecture 0 opyright egents of University of alifornia W. G. Oldham DENTFYNG SEES AND PAALLEL OMBNATONS (cont.) Some circuits must be analyzed (not amenable to simple inspection) 3 and are not in 4 5-4 3 and 5 are not in series Special cases: 3 0 O 3 5 Example: 3 0 ; 4 5 in series; eq 4 5 O F 3 ( 5 ) ( 4 ) 7
EES 4 Spring 00 Lecture 0 opyright egents of University of alifornia W. G. Oldham DEAL AND NON-DEAL METES DMM volts DEAL DMM volts DMM amps DEAL DMM amps in MODEL OF EAL DGTAL OLTMETE Note: in may depend on range ~ in typically > 0 MΩ in MODEL OF EAL DGTAL AMMETE Note: in usually depends on current range ~ in typically < Ω 8
EES 4 Spring 00 Lecture 0 opyright egents of University of alifornia W. G. Oldham EAL OLTMETES oncept of Loading as Application of Parallel esistors How is voltage measured? Modern answer: Digital multimeter (DMM) Problem: onnecting leads from voltmeter across two nodes changes the circuit. The voltmeter is characterized by how much current it draws at a given voltage voltmeter input resistance, in. Typical value: 0 MΩ. Lets do an example; measure in voltage divider: in in in Example: 0, 00K, 900K But if 0M, 0.99, a % error in 9
EES 4 Spring 00 Lecture 0 opyright egents of University of alifornia W. G. Oldham MEASUNG UENT nsert DMM (in current measurement mode) into circuit. But ammeters disturb the circuit. (Note: Ammeters are characterized by their ammeter input resistance, in. deally this should be very low. Typical value (in ma range) Ω.) Potential measurement error due to non-zero input resistance: meas in ammeter undisturbed circuit Example : KΩ, in Ω ma, meas 0.999 ma K Ω with ammeter meas (0.% error) in 0
EES 4 Spring 00 Lecture 0 opyright egents of University of alifornia W. G. Oldham i(t) ( ( APATOS N SEES Equivalent to i(t) eq ( eq d d i So d learly, i d, eq Equivalent capacitance defined by deq eq and i eq i, d( eq so d eq i( APATOS N SEES ) ) i eq
EES 4 Spring 00 Lecture 0 opyright egents of University of alifornia W. G. Oldham APATOS N PAALLEL i(t) ( ( d i(t) d Equivalent capacitance defined by d i eq i(t) eq ( (t) learly, eq APATOS N PAALLEL