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Voltage Divider Capacitor RC Circuits

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Voltage Divider Capacitor RC Circuits Physics 120/220 Voltage Divider Capacitor RC circuits Prof. Anyes Taffard Voltage Divider 2 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 Vin. Vin Current (R1 & R2) I = R1 + R2 Output voltage: R 2 Voltage drop is Vout = IR2 = Vin ∴Vout ≤ Vin R1 + R2 proportional to the resistances Vout can be used to drive a circuit that needs a voltage lower than Vin. Voltage Divider (cont.) 3 Add load resistor RL in parallel to R2. You can model R2 and RL as one resistor (parallel combination), then calculate Vout for this new voltage divider R2 If RL >> R2, then the output voltage is still: VL = Vin R1 + R2 However, if RL is comparable to R2, VL is reduced. We say that the circuit is “loaded”. Ideal voltage and current sources 4 Voltage source: provides fixed Vout regardless of current/load resistance. Has zero internal resistance (perfect battery). Real voltage source supplies only finite max I. Current source: provides fixed Iout 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 VTH and a single resistor RTH. To find the Thevenin’s equivalent VTH & RTH: V V • For an “open circuit” (RLà∞), then Th = open circuit • Voltage drops across device when disconnected from circuit – no external load attached. V R = open circuit Th I • For a “short circuit” (RLà0), then short circuit • I short circuit = current when the output is shorted directly to the ground. Thevenin’s theorem (cont) 6 Thevenin equivalent R2 Lower leg of divider Open circuit voltage: VTH = Vout = Vin R + R V 1 2 Total R Short circuit current: I = in short circuit R 1 Thevenin equivalent: R Voltage source: V = V 2 V open-circuit – no external load TH in R + R 1 2 R R in series with: R = 1 2 TH “like” R1 in parallel with R2 R1 + R2 RTh is called the output impedance (Zout) of the voltage divider Thevenin’s theorem (cont) 7 Very useful concept, especially when different circuits are connected with each other. Closely related to the concepts of input and output impedance (or resistance). Circuit A, consisting of VTH and RTH, is fed to the second circuit element B, which consists of a simple load resistance RL. Avoiding circuit loading 8 The combined equivalent circuit (A+B) forms a voltage divider: R V V = V L = TH out TH R R R TH + L 1+ TH ( RL ) RTH determines to what extent the output of the 1st circuit behave as an ideal voltage source. To approximate ideal behavior and avoid loading the circuit, the ratio RTH/RL should be kept small. 10X rule of thumb: RTH/RL = 1/10 The output impedance of circuit A is the Thevenin equivalent resistance RTH (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 RL. Example: voltage divider 9 Vin=30V, R1=R2=Rload=10k a) Output voltage w/ no load [Answ 15V] b) Output voltage w/ 10k load [Answ 10V] Example (cont.) 10 c) Thevenin equivalent circuit [VTH=15V, RTH=5k] d) Same as b) but using the Thevenin equivalent circuit [Answ 10V] e) Power dissipated in each of the resistor [Answ PR1=0.04W, PR2=PRL=0.01W] Example: impedance of a Voltmeter 11 We want to measure the internal impedance of a voltmeter. Suppose that we are measuring Vout of the voltage divider: • RTH: 2 100k in parallel, 100k/2 = 50k 100k • V TH = 20 = 10V 2 ×100k • Measure voltage across Rin (Vout)= 8V, thus 2V drop across RTH • The relative size of the two resistances are in proportion of these two voltage drops, so Rin must be 4 (8/2) RTH , so Rin= 200k 12 Terminology 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: V out Iout AV = A V I = in Iin Voltage gain Unity gain: Vout =Vin 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)= VDC + v Decibels: amplitude 2 To compare ratio of two signals: dB = 20log 10 amplitude 1 Often used for gain: eg ratio is 1.4≈√2 20log10 1.4 = 3 dB NB: 3dB ~ power ratio of ½ ~ amplitude ratio of 0.7 15 Capacitors and RC circuits Capacitor: reminder 16 Q = CV conducting plates Q: total charge [Coulomb] insulator Q A C: capacitance [Farad 1F = 1C/1V] C = = ε [parallel-plate capacitor] V 0 d V: voltage across cap dQ dV I: rate at which charge flows Since I = I = C dt dt 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) 1 1 1 1 = + + [series] Ceq = C1 + C2 + C3 [parallel] Ceq C1 C2 C3 Same voltage drop across caps All caps have same Q Frequency analysis of reactive circuit 17 dV(t) V(t) = V sin(ωt) I(t) = C = CωV cos(ωt) 0 dt 0 ie the current is out of phase by 90o to wrt voltage (leading phase) V Considering the amplitude only: I = 0 1 ω = 2π f ωC 1 Frequency dependent resistance: R = 1 = ωC 2π fC Example: C=1µF 110V (rms) 60Hz power line 110 Irms = = 41.5mA(rms) I 1 Irms = (2π × 60 ×10−6 ) 2 Impedance of a capacitor 18 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. dV jωt Recall: I = C V(t) = V0 cos(ωt) = Re ⎣⎡V0e ⎦⎤ dt d jωt jωt I = C (V0e ) = jωCV0e dt V e jωt I = 0 − j ωC jωt ⎡V0e ⎤ j I = Re Z = − The actual current is: ⎢ ⎥ c ωC ⎣ Zc ⎦ Zc is the impedance of a capacitor at frequency ω. As ω (or f) increases (decreases), Zc decreases (increases) The fact that Zc is complex and negative is related to the fact the the voltage across the cap lags the current through it by 90o. Ohm’s law generalized 19 Ohm’s law for impedances: V(t) = ZI(t) V! = Z!I! using complex notation Zeq = Z1 + Z2 + Z3 [series] 1 1 1 1 = + + [parallel] Zeq Z1 Z2 Z3 Resistor: ZR = R in phase with I j o Capacitor: Z = − = 1 lags I by 90 c ωC jωC Inductor: leads I by 90o (use mainly in RF circuits) Z = jωL L Can use Kirchhoff’s law as before but with complex representation of V & I. Z! Generalized voltage divider: V! = V! 2 out in ! ! Z1 + Z2 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: dV V −V −t I = C = i à V = V + Ae RC dt R i A is determined by the initial condition: @ t=0, V=0 thus A=-Vi −t RC V = Vi 1− e when t=RC 1/e=0.37 ( ) V V i Rate of charge/discharge is @ 1RC 63% of voltage 0.63V i determined by RC constant @ 5RC 99% of voltage 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.) 21 Consider a circuit with a charge capacitor, a resistor, and a switch V τ = RC = time constant V i 0.37V i Before switch is closed, V = Vi and Q = Qi = CVi After switch is closed, capacitor discharges and voltage across capacitor decreases exponentially with time dV V C = I = − à V = V e−t/RC dt R i 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 Vin-V d V I = C (Vin −V ) = dt R If we choose R & C small enough so that dV dV << in dt dt d then, V ( t ) = R C V ( t ) dt in Thus the output differentiate the input waveform! Simple rule of thumb: differentiator works well if Vout << Vin Differentiators are handy for detecting leading edges & trailing edges in pulse signals. Integrator 23 Now flip the order of the resistor and capacitor, with the output corresponding to the voltage across the capacitor: V across R is Vin-V dV V −V I = C = in dt R If RC is large, then V<<Vin and 1 t dV Vin C ≅ V(t) = Vin (t)dt + cst dt R à RC ∫ Thus the output integrate the input! Simple rule of thumb: integrator works well if Vout << Vin Integrators are used extensively in analog computation (eg analog/digital conversion, waveform generation etc…) High-pass filter 24 Let’s interpret the differentiator RC circuit as a frequency-dependent voltage divider (“frequency domain”): ! ⎡ j ⎤ ! ! Vin R + Vin Vin ⎢ ( ωC)⎥ Using complex Ohm’s law: I! = = = ⎣ ⎦ Z! j R2 + 1 total R − ω 2C2 ( ωC) V! ⎡R j ⎤ in + C ! ! ⎣⎢ ( ω )⎦⎥ Voltage across R is: Vout = IR = R R2 + 1 Voltage divider made of R & C ω 2C2 1 ! ! * 2 R If we care only about the amplitude: Vout = (VoutVout ) = Vin 1 ⎡ 2 1 ⎤ 2 R + 2 2 • Thus Vout increases with increasing f ⎣⎢ ω C ⎦⎥ j Impedance of a series RC combination: Z! = R − total ωC ⎛ −1 ⎞ −1 C jZ φ = tan ⎜ ω ⎟ ⎝⎜ R ⎠⎟ R Z = R2 + 1 total ω 2C2 ϕ -j/ωC impedance of R V = V lower-leg of divider Note phase of out in Ztotal Magnitude of output signal impedance of R & C High-pass filter frequency response 25 Output ~ equal to input at high frequency 3dB below unit when ω ~ 1/RC [rad] Goes to zero at low frequency.
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