Prof. Anyes Taffard Physics 120/220 Voltage Divider Capacitor RC circuits
Voltage Divider The figure is called a voltage divider. It s one of the most useful and important circuit elements we will encounter. It is used to generate a particular voltage for a large fixed V in. 2 Current (R 1 & R 2 ) Output voltage: I = V in R 1 + R 2 V out = IR 2 = R 2 R 1 + R 2 V in V out V in Voltage drop is proportional to the resistances V out can be used to drive a circuit that needs a voltage lower than V in.
Voltage Divider (cont.) 3 Add load resistor R L in parallel to R 2. You can model R 2 and R L as one resistor (parallel combination), then calculate V out for this new voltage divider If R L >> R 2, then the output voltage is still: V L = R 2 R 1 + R 2 V in However, if R L is comparable to R 2, V L is reduced. We say that the circuit is loaded.
Ideal voltage and current sources 4 Voltage source: provides fixed V out regardless of current/load resistance. Has zero internal resistance (perfect battery). Real voltage source supplies only finite max I. Current source: provides fixed I out regardless of voltage/load resistance. Has infinite resistance. Real current source have limit on voltage they can provide. Voltage source More common In almost every circuit Battery or Power Supply (PS)
Thevenin s theorem 5 Thevenin s theorem states that any two terminals network of R & V sources has an equivalent circuit consisting of a single voltage source V TH and a single resistor R TH. To find the Thevenin s equivalent V TH & R TH : For an open circuit (R L à ), then Voltage drops across device when disconnected from circuit no external load attached. For a short circuit (R L à0), then V Th = V open circuit R Th = V open circuit I short circuit I short circuit = current when the output is shorted directly to the ground.
Thevenin s theorem (cont) 6 Thevenin equivalent Open circuit voltage: Short circuit current: V TH = V out = V in R 2 R 1 + R 2 I short circuit = V in R 1 Lower leg of divider Total R Thevenin equivalent: Voltage source: V TH = V in R 2 R 1 + R 2 V open-circuit no external load in series with: R TH = R 1R 2 R 1 + R 2 like R 1 in parallel with R 2 R Th is called the output impedance (Z out ) of the voltage divider
Thevenin s theorem (cont) Very useful concept, especially when different circuits are connected with each other. Closely related to the concepts of input and output impedance (or resistance). 7 Circuit A, consisting of V TH and R TH, is fed to the second circuit element B, which consists of a simple load resistance R L.
Avoiding circuit loading The combined equivalent circuit (A+B) forms a voltage divider: V out = V TH R L R TH + R L = V TH ( ) 1+ R TH RL 8 R TH determines to what extent the output of the 1 st circuit behave as an ideal voltage source. To approximate ideal behavior and avoid loading the circuit, the ratio R TH /R L should be kept small. 10X rule of thumb: R TH /R L = 1/10 The output impedance of circuit A is the Thevenin equivalent resistance R TH (also called source impedance). The input impedance of circuit B is its resistance to ground from the circuit input. In this case, it is simply R L.
Example: voltage divider V in =30V, R 1 =R 2 =R load =10k 9 a) Output voltage w/ no load [Answ 15V] b) Output voltage w/ 10k load [Answ 10V]
Example (cont.) 10 c) Thevenin equivalent circuit [V TH =15V, R TH =5k] d) Same as b) but using the Thevenin equivalent circuit [Answ 10V] e) Power dissipated in each of the resistor [Answ P R1 =0.04W, P R2 =P RL =0.01W]
Example: impedance of a Voltmeter We want to measure the internal impedance of a voltmeter. Suppose that we are measuring V out of the voltage divider: 11 R TH : 2 100k in parallel, 100k/2 = 50k V TH = 20 100k 2 100k = 10V Measure voltage across R in (V out )= 8V, thus 2V drop across R TH The relative size of the two resistances are in proportion of these two voltage drops, so R in must be 4 (8/2) R TH, so R in = 200k
Terminology 12
Terminology (cont) 13 Offset = bias A DC voltage shifts an AC voltage up or down. DC bias AC signal AC signal with DC offset Gain: A V = V out V in Voltage gain Unity gain: V out =V in A I = I out I in Current gain
Terminology 14 When dealing with AC circuits we ll talk about V & I vs time or A vs f. Lower case symbols: i: AC portion of current waveform v: AC portion of voltage waveform. V(t)= V DC + v Decibels: To compare ratio of two signals: db = 20log 10 amplitude 2 amplitude 1 Often used for gain: eg ratio is 1.4 2 20 log 10 1.4 = 3 db NB: 3dB ~ power ratio of ½ ~ amplitude ratio of 0.7
Capacitors and RC circuits 15
Capacitor: reminder 16 Q = CV conducting plates Q: total charge [Coulomb] insulator C: capacitance [Farad 1F = 1C/1V] V: voltage across cap C = Q V = ε 0 A d [parallel-plate capacitor] Since I = dq dt I = C dv dt I: rate at which charge flows or rate of change of the voltage For a capacitor, no DC current flows through, but AC current does. Large capacitances take longer to charge/discharge than smaller ones. Typically, capacitances are µf (10-6 ) pf (10-12 ) C eq = C 1 + C 2 + C 3 [parallel] Same voltage drop across caps 1 C eq = 1 C 1 + 1 C 2 + 1 C 3 All caps have same Q [series]
Frequency analysis of reactive circuit 17 I(t) = C dv(t) = CωV 0 cos(ωt) dt ie the current is out of phase by 90 o to wrt voltage (leading phase) V(t) = V 0 sin(ωt) Considering the amplitude only: Frequency dependent resistance: I = V 0 1 ωc ω = 2π f R = 1 ωc = 1 2π fc Example: C=1µF 110V (rms) 60Hz power line I rms = 110 1 2π 60 10 6 ( ) = 41.5mA(rms) I rms = I 2
Impedance of a capacitor Impedance is a generalized resistance. It allows rewriting law for capacitors so that it resembles Ohm s law. Symbol is Z and is the ratio of voltage/current. 18 Recall: I = C dv dt I = C d dt V 0 e jωt I = V 0 e jωt j ωc ( ) = jωcv 0 e jωt V(t) = V 0 cos(ωt) = Re V 0 e jωt The actual current is: I = Re V 0e jωt Z Z c = j ωc c Z c is the impedance of a capacitor at frequency ω. As ω (or f) increases (decreases), Z c decreases (increases) The fact that Z c is complex and negative is related to the fact the the voltage across the cap lags the current through it by 90 o.
Ohm s law generalized Ohm s law for impedances: V(t) = ZI(t) Z eq = Z 1 + Z 2 + Z 3 [series]!v =! Z! I using complex notation 19 1 Z eq = 1 Z 1 + 1 Z 2 + 1 Z 3 [parallel] Resistor: Z R = R in phase with I Capacitor: Z lags I by 90 o c = j ωc = 1 jωc Inductor: Z leads I by 90 o (use mainly in RF circuits) L = jωl Can use Kirchhoff s law as before but with complex representation of V & I. Generalized voltage divider:!v out =!V in!z 2!Z 1 +!Z 2
RC circuit 20 Capacitor is uncharged. At t=0, the RC circuit is connected to the battery (DC voltage) The voltage across the capacitor increases with time according to: I = C dv dt = V i V R A is determined by the initial condition: @ t=0, V=0 thus A=-V i à V = V i + Ae t RC V = V i ( 1 e t ) RC when t=rc 1/e=0.37 V Rate of charge/discharge is determined by RC constant @ 1RC 63% of voltage @ 5RC 99% of voltage V i 0.63V i Time constant RC: For R Ohms and C in Farads, RC is in seconds For MΩ and µf, RC is seconds For kω and µf, RC is ms
RC circuit (cont.) Consider a circuit with a charge capacitor, a resistor, and a switch 21 V τ = RC = time constant V i 0.37V i Before switch is closed, V = V i and Q = Q i = CV i After switch is closed, capacitor discharges and voltage across capacitor decreases exponentially with time C dv dt V = I = à R V = V i e t/rc
Differentiator 22 Consider the series RC circuit as a voltage divider, with the output corresponding to the voltage across the resistor: V across C is V in -V I = C d ( dt V in V ) = V R If we choose R & C small enough so that dv dt << dv in dt then, V(t) = RC d dt V in (t) Thus the output differentiate the input waveform! Simple rule of thumb: differentiator works well if V out << V in Differentiators are handy for detecting leading edges & trailing edges in pulse signals.
Integrator Now flip the order of the resistor and capacitor, with the output corresponding to the voltage across the capacitor: 23 V across R is V in -V I = C dv dt = V in V R If RC is large, then V<<V in and C dv dt V in R à V(t) = 1 RC Thus the output integrate the input! Simple rule of thumb: integrator works well if Integrators are used extensively in analog computation (eg analog/digital conversion, waveform generation etc ) t V (t) in dt + cst V out << V in
High-pass filter Let s interpret the differentiator RC circuit as a frequency-dependent voltage divider ( frequency domain ): Using complex Ohm s law:!i =! V in!z total =!V in ( ) = R j ωc ( )!V in R + j ωc R 2 + 1 ω 2 C 2 24 Voltage across R is:!v out =! IR = R ( )!V in R + j ωc R 2 + 1 ω 2 C 2 Voltage divider made of R & C jz If we care only about the amplitude: Thus V out increases with increasing f Impedance of a series RC combination: V out = * (!V out!v out ) 12 R = V in!z total = R j ωc R 2 + 1 ω 2 C 2 R Z total = R 2 + 1 ω 2 C 2 φ = tan 1 1 2 1 ωc R ϕ -j/ωc Note phase of output signal V out = V in R Z total impedance of lower-leg of divider Magnitude of impedance of R & C
High-pass filter frequency response 25 3dB below unit Output ~ equal to input at high frequency when ω ~ 1/RC [rad] Goes to zero at low frequency. High-pass filter frequency response curve A high-pass filter circuit attenuates low frequency and passes the high frequencies. The frequency at which the filter turns the corner (ie V out /V in =1/ 2=0.7) is called the 3dB point: occurs when Z c =R f 3dB = 1 2π RC [Hz] Use this in lab otherwise factor 2π off NB: 3dB ~ power ratio of ½ ~ amplitude ratio of 0.7
Low-pass filter Now simply switch the order of the resistor and capacitor in the series circuit (same order as the integrator circuit earlier): 1 ωc impedance of lower-leg of divider 26 V out = V in R 2 + 1 ω 2 C 2 1 2 Magnitude of impedance of R & C A low-pass filter circuit attenuates high frequency and passes the low frequencies. 3dB below unit Low-pass filter frequency response curve