Capacitor. Capacitor (Cont d)

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2 Capacitor Capacitor is a passive two-terminal component storing the energy in an electric field charged by the voltage across the dielectric. Fixed Polarized Variable Capacitance is the ratio of the change in an electric charge in a system to the corresponding change in its electric potential. q C V The unit of capacitance is Farad (F), named after the English physicist Michael Faraday. However, the Farad is a very large unit of measurement to use on its own so sub-multiples of the Farad are generally used such as F, nf and pf. 3 Capacitor (Cont d) All capacitors have a maximum voltage rating and when selecting a capacitor consideration must be given to the amount of voltage to be applied across the capacitor. The maximum amount of voltage that can be applied to the capacitor without damage to its dielectric material is generally given in the data sheets as working voltage (WV) or DC working voltage (WV DC) pf 5% + Ceramic type Film type Electrolytic type Trimmers 4 2

3 Capacitance Identification Capacitance depends on the area of the plates, the distance between the plates, the type of dielectric material, and temperature. For text marking, the third digit represents the number of zeros to add to the end of the first two digits. The resulting number is the capacitance in pf. Tolerance A B C D E F G H J K L M N 0.o5pF 0.1pF 0.25pF 0.5pF 0.5% 1% 2% 3% 5% 10% 15% 20% 30% P S W X Z 0%, +100% 20%, +50% 0%, +200% 20%, +40% 20%, +80% 5 Inductor Inductor is a passive two-terminal component storing the energy in a magnetic field induced by the electric current passing through it. core core core Inductance is the ratio of the voltage to the rate of change of current. An inductor typically consists of an insulated wire wound into a coil around a core. Inductance of a coil, Number of turns Air Iron Ferrite Variable Magnetic permeability L l 2 N A Area of coil Length of coil 6 3

4 Inductor (Cont d) The unit of inductance is the Henry (H) named for 19 th century American scientist Joseph Henry. Inductors have values that typically range from 1 µh to 20 H H 5% H 10% Air core Iron core Ferrite core Variable core Toroidal core 7 Inductance Identification For text marking, The third digit represents the number of zeros to add to the end of the first two digits. The resulting number is the inductance in H. The fourth letter or alphabet represents the tolerance value. B C S D F G H J K L M V N 0.15nH 0.2nH 0.3nH 0.5nH 1% 2% 3% 5% 10% 15% 20% 25% 30% EIA (Electronic Industries Alliance) standard for four-color-band ( H) 8 4

5 Component RLC Impedance = Potential Difference Phasor Current Phasor Impedance = Resistance + j Reactance, j = 1 Z = R + j X = Z cos + j Z sin j Im = Z e j X = Z Z where Z = R 2 + X 2 = tan 1 (X/R) Admittance, Y = 1/Z 0 R Re 9 AC Response Resistor I m sin t V m I I m V m sin t R I V Resistance V = R I = R I m sin t = V m sin t (V and I in phase) I V Phasor Diagram 10 5

6 AC Response (Cont d) V = V m sin( t) Capacitive Reactance I = dq/dt = C dv/dt V = 1/C I m sin t dt+ V 0 = 1/ C I m cos t = 1/ C I m sin( t /2) = X C I m sin( t /2) = V m sin( t /2) (V lags I by 90 ) I = C dv/dt = I m sin( t + /2) C V m Im 0 /2 V Phasor Diagram X C = 1/ C = 1/2 fc = /2 C, Higher f, Lower X C (Lowpass Filter) I I V 11 AC Response (Cont d) Note that capacitive reactance is an opposition to the change of voltage across an element. There are two choices in the literature for defining reactance for a capacitor. One is to use a uniform notion of reactance as the imaginary part of impedance, in which case the reactance of a capacitor is a negative number: X C = 1/ C and Z C = j X C = j/ C. Another choice is to define capacitive reactance as a positive number, however, one needs to remember to add a negative sign for the impedance of a capacitor: 12 X C = 1/ C and Z C = j X C = 1/j C. 6

7 AC Response (Cont d) V = L di/dt = V m sin( t+ /2) I m sin( t) L V m I m /2 V I Inductive Reactance V = d(n )/dt = L d(i m sin t)/dt = L I m cos t = L I m sin( t+ /2) = X L I m sin( t+ /2) = V m sin( t+ /2) (V leads I by 90 ) X L = L = 2 fl = 2 L/ V I Phasor Diagram,, Higher f, Higher X L (Highpass Filter) 13 Frequency Response The resonance of a series RLC circuit occurs when the inductive and capacitive reactances are equal in magnitude but cancel each other because they are 180 apart in phase, ( Z = R 2 + (X L X C ) 2 = R) and 1 / 2 f 0 C = 2 f 0 L f 0 = 1 / 2 LC 14 7

8 Q-Factor and D-Factor Q-factor is to express the quality of component in ability to store and release energy or quality of L L+R, Q = Energy Stored / Power Loss = Reactance / Resistance I L = L / R = tan D-factor is for a dissipation of C C + R, D=1/Q = Power Loss / Energy Stored = R / (1/ C) = RC = tan IR IR I(1/ C) V I V I 15 Capacitor Model An ideal capacitor stores but does not dissipate energy. Because the dielectric separating the capacitor plates are not a perfect insulator, it causes a small leakage current flowing through the capacitors parallel model. D = V 2 /R / V 2 /X C = 1/ RC Plate loss due to the resistances of the plates and leads can become quite significant in higher frequency case series model. D = I 2 R / I 2 X C = RC 16 8

9 Inductor Model An ideal inductor stores but does not dissipate energy. Time-varying current in a ferromagnetic inductor, which causes a time-varying magnetic field in its core, causes energy losses in the core material that are dissipated as heat parallel model. Q=V 2 /X /V L 2 /R = R/ L Resistance of the wire series model. Q = I 2 X L / I 2 R = L/R 17 AC Bridge A I 1 B Z 1 Z 2 Detector I 2 Z 3 Z D 4 Wheatstone Bridge Balanced Bridge, I 1 Z 1 1 =I 2 Z 3 3 I 1 Z 2 2 = I 2 Z 4 4 C Z 1 1 / Z = Z 3 3 / Z R 1 +jx 1 = R 3 +jx 3 R 2 +jx 2 R 4 +jx

10 Inductance Measurement There is no pure components, e.g. an inductor can be considered to be a pure inductance (L 4 ) in series with a pure resistance (R 4 ). Maxwell-Wien Bridge (for medium Q = 1-10) R 1 C 1 D R2 Impedances, 1/Z 1 = 1/R 1 + 1/(1/j C 1 ) Z 1 = R 1 / (1+j R 1 C 1 ) R 3 R 4 L 4 Z 2 = R 2 Z 3 = R 3 Z 4 = R 4 + j L 4 19 Maxwell-Wien Bridge (Cont d) Balanced bridge, Z 1 / Z 2 = Z 3 / Z 4 Z 4 = Z 2 Z 3 / Z 1 R 4 +j L 4 = R 2 R 3 (1+j R 1 C 1 )/R 1 = R 2 R 3 /R 1 + j R 2 R 3 C 1 James Clerk Maxwell ( ), a Scottish scientist in the field of mathematical physics. Real part: R 4 = R 2 R 3 /R 1 Imagination part: L 4 = R 2R 3C 1 The balancing is independent of frequency. Adjust R 1 and R 2 to get the bridge balanced (Null) Q = L 4 /R 4 = (R 2 R 3 C 1 )/(R 2 R 3 /R 1 ) = R 1 C

11 Hay Bridge For high Q 10 C 1 R R 1 2 Impedances, Z 1 = R 1 + 1/j C 1 D = R 1 j/ C 1 Z 2 = R 2 R 3 L 4 Z 3 =R 3 R 4 Z 4 = R 4 + j L 4 21 Hay Bridge (Cont d) Balanced bridge, Z 4 = Z 2 Z 3 / Z 1 R 4 +j L 4 = R 2 R 3 / (R 1 j/ C 1 ) (R 1 R 4 + L 4 /C 1 ) + j ( R 1 L 4 R 4 / C 1 ) = R 2 R 3 Imagination part: R 1 L 4 = R 4 / C 1 L 4 = R 4 / 2 R 1 C 1 Real part: R 1 R 4 + L 4 /C 1 = R 2 R 3 R +R/ 2 1 R R 1 C 12 =RR 2 R 3 R 4 ( R 1 + 1/ 2 R 1 C 12 ) = R 2 R 3 R 4 ( 2 R 12 C )/( 2 R 1 C 12 ) = R 2 R 3 R 4 = ( 2 R 1 R 2 R 3 C 12 ) / ( 2 R 12 C ) L 4 = (R 2 R 3 C 1 ) / ( 2 R 12 C ) 22 11

12 Hay Bridge (Cont d) Q = L 4 /R 4 1/ R C = 1/ R 1 C 1 Therefore, L 4 = (R 2 R 3 C 1 ) / ( (1/Q 2 ) + 1 ) R 2 R 3 C 1 if Q 10 and R 4 2 R 1 R 2 R 3 C Owen Bridge C 1 R 3 D R 2 Z 4 = R 4 + j L C 4 3 L 4 R 4 Impedances, Z 1 = 1/j C 1 Z 2 = R 2 Z 3 = R 3 j/ C

13 Owen Bridge (Cont d) Balanced bridge, Z 4 = Z 2 Z 3 / Z 1 R 4 +j L 4 = R 2 (R 3 j/ C 3 ) j C 1 = R 2 C 1 /C 3 + j R 2 R 3 C 1 Real part: R 4 = R 2 C 1 /C 3 Imagination part: L 4 = R 2 R 3 C 1 The balancing is independent of frequency. Q = L 4 /R 4 = R 2 R 3 C 1 C 3 / R 2 C 1 = R 3 C 3 25 Series Capacitance Bridge Capacitor can be considered to be a pure capacitance in series with, or sometimes in parallel with, a pure resistance. R C 2 1 Impedances, Z R 1 = R 1 2 Z D 2 = R 2 j/ C 2 Z 3 = R 3 Z 4 = R 4 j/ C 4 R 3 R 4 C

14 Series Capacitance Bridge (Cont d) Balanced bridge, Z 4 = Z 2 Z 3 / Z 1 R 4 j/ C 4 = (R 2 j/ C 2 ) R 3 / R 1 = R 2 R 3 /R 1 j(r 3 / C 2 R 1 ) Real part: R 4 = R 2 R 3 /R 1 Imagination part: C 4 = C 2 R 1 /R 3 Used for low D = D = 1/Q = R 4 C 4 = R 2 C 2 27 Parallel Capacitance Bridge Used for high D > 0.05 R Impedances, 2 Z 1 = R 1 Z D C 2 = 1 / (1/R 2 + j C 2 ) 2 = R R 2 /(1+j C 2 R 2 ) 4 Z 3 = R 3 R Z =R/(1+j C R 4 ) C 4 R

15 Parallel Capacitance Bridge (Cont d) Balanced bridge, Z 4 = Z 2 Z 3 / Z 1 R 4 / (1+j C 4 R 4 ) = R 2 R 3 / (1+j C 2 R 2 )R 1 R 1 R 4 + j C 2 R 1 R 2 R 4 = R 2 R 3 + j C 4 R 2 R 3 R 4 Real part: R 1 R 4 = R 2 R 3 R 4 = R 2 R 3 /R 1 Imagination part: C 2 R 1 R 2 R 4 = C 4 R 2 R 3 R 4 C 4 = C 2 R 1 /R 3 D = 1/ R 4 C 4 = 1/ R 2 C 2 29 R 1 Schering Bridge Used for very low D C 2 Impedances, C 1 D Z 1 = R 1 /(1+j C 1 R 1 ) Z 2 = 1/j C 2 Z 3 = R 3 R 3 C 4 R 4 Z 4 = R 4 j/ C 4 Harald Schering (25 November April 1959), a German physicist 30 15

16 Schering Bridge (Cont d) Balanced bridge, Z 4 = Z 2 Z 3 / Z 1 R 4 j/ C 4 =R(1+j C 3 1 R 1 ) /j C 2 R 1 = ( C 1 R 1 R 3 jr 3 ) / C 2 R 1 = R 3 C 1 /C 2 j(r 3 / R 1 C 2 ) Real part: R 4 = R 3 C 1 /C 2 Imagination part: 1/C 4 = R 3/R 1C 2 C 4 = C 2 R 1 /R 3 D = R 4 C 4 = R 1 C 1 31 Wien Bridge Used as frequency-dependent circuit R 1 R 2 Impedances, Z 1 = R 1 D Z 2 = R 2 R 4 Z 3 = R 3 j/ C 3 Z 4 = R 4 /(1+j C 4 R 4 ) R 3 C 4 C 3 Max Karl Werner Wien (25 December February 1938), a German physicist

17 Wien Bridge (Cont d) Balanced bridge, Z 4 = Z 2 Z 3 / Z 1 R 4 / (1+j C 4 R 4 ) = R 2 (R 3 j/ C 3 ) / R 1 R 1 R 4 /R 2 =R 3 +RC 4 4 /C 3 +j( C 4 R 3 R 4 1/ C 3 ) Imagination part: C 4 R 3 R 4 = 1/ C 3 C 4 R 4 = 1/ 2 C 3 R 3 Real part: R 1 R 4 /R 2 = R 3 + R 4 C 4 /C 3 R 4 = (R 3 R 2 C 3 + R 2 R 4 C 4 )/ R 1 C 3 = (R 3 R 2 C 3 + R 2 / 2 C 3 R 3 ) / R 1 C 3 = R 2 ( 2 C 32 R ) / ( 2 C 32 R 1 R 3 ) and C 4 = 1/ 2 C 3 R 3 R 4 = C 3 R 1 / R 2 ( 2 C 32 R ) D = 1/ R 4 C 4 = R 3 C 3 33 Wien Bridge Oscillator A type of electronic oscillator that generates sine waves with a large range of frequencies. At balance, Imagination part: C 2 4 R 4 = 1/ C 3 R 3 2 = 1 / R 3 R 4 C 3 C 4 f = 1/ 2 R 3 R 4 C 3 C 4 If R 3 = R 4 = R and C 3 = C 4 = C Then f = 1/ 2 RC 34 17

18 Stray Impedance Because capacitance can B conduct AC by charging Z 1 Z and discharging, g there 2 are stray A D capacitances C stray between the various element and the earth Z 3 Z 4 potential (ground). D C stray C C stray C stray They form stray current paths to the AC voltage source and become significant when performing measurements at high frequency which may effect 35 the bridge balanced condition. Stray Impedance (Cont d) One way to reduce the effect is to connect the null potential ti A at ground potential, so that there will be no AC potential between the two. Thus no current through stray capacitances. B Z 1 Z 2 D Z 3 Z D 4 C stray C C stray 36 18

19 Stray Impedance (Cont d) Directly connecting to ground potential is not possible, the voltage divider circuit called Wagner earth is used to A minimize stray capacitances between the detector terminals and earth. To ensuring that the points B and D of a balanced bridge are at ground potential B Z 1 Z 2 D Z 3 Z 4 D Z 5 Z 6 C 37 Ex1. Maxwell-Wein Bridge Impedances, = 2 f = 2 (1k) = 2000 rad/s D Z 1 = R 1 / (1+j R 1 C 1 ) = 1k 1+j(2000 )(1k)(0.5 ) L 4 = 92 j289 R 4 Z 2 = R 2 = 600 f = 1kHz Z 3 = R 3 = 400 Z 4 = R 4 + j L 4 = R 4 + j2000 L

20 Ex1. (Cont d) at balance, Z 4 = Z 2 Z 3 / Z 1 R 4 +j2000 L 4 =(600)(400) /(92 j289) = j754 R 4 = 240 L 4 = 754 / 2000 = 120 mh Q = L 4 / R 4 = (2000 )(120m)/(240) = Ex1. (Cont d) check, R 4 = R 2 R 3 / R 1 =(600)(400) /(1k) = 240 CORRECT L 4 = R 2 R 3 C 1 = (600)(400)(0.5 ) =120 mh CORRECT Q = R 1 C 1 = (2000 )(1k)(0.5 ) = 3.14 CORRECT 40 20

21 Ex1. (Cont d) L 4 = 120 mh R 4 = 240 Q = 3.14 Q =? L 4 =? R 4 =? Parallel model, Z 4 = (R 4 )(j L 4 ) = 2 (R 4 )(L 4 ) 2 + j (R 4 ) 2 (L 4 ) R 4 + j L 4 (R 4 ) (L 4 ) 2 Series model, Z 4 = R 4 + j L 4 = (240) + j (120m) 41 = Z 4 4 Ex2. R 4 = 12 k C 4 = 10 pf Equivalent Series Resistance (ESR) D =? D = Parallel model at 10 khz source, Z 4 = R 4 /(1+j C 4 R 4 ) = (12k) / (1 + j2 (10k)(10p)(12k)) R 4 =? = (12000) j(90.47) Series model, Z 4 = R 4 j/ C 4 = Z 4 therefore R 4 = 12 k, C 4 = 0.18 F, D = C 4 =? 42 21

22 43 Transformer Ratio Bridges Not only varying the impedances, but bridge can be also balanced by varying the turns ratio of a transformer. There is a small number of standard resistors and capacitors and no effect of temperature changes. I 2 N 2 V 2 Z 2 V s D N 1 V 1 Z 1 I

23 Single Ratio Transformer Bridge Tap a transformer voltage divider of V s V 1 = kn 1 = I 1 Z 1 I 1 = kn 1 / Z 1 V 2 = kn 2 = I 2 Z 2 I 2 = kn 2 / Z 2 To balance the bridge or no current through the detector, t D = Null I 1 = I 2 Z 1 / Z 2 = N 1 / N 2 Impedance Ratio Turn Ratio 45 Single Ratio Transformer Bridge (Cont d) Resistance Measurement Z 1 = Unknown resistor R x Z 2 = Standard resistor R s R x = R s N 1 N

24 Single Ratio Transformer Bridge (Cont d) Capacitance Measurement Z 1 = Unknown C x R x (leakage resistance) = 1/(1/R x + j C x ) = R x / (1+j R x C x ) Z 2 = Standard C s R s = R s / (1+j R s C s ) 47 Single Ratio Transformer Bridge (Cont d) Capacitance Measurement (Cont d) Balanced, Z 1 / Z 2 = N 1 / N 2 1/Z 1 = (N 2 /N 1 ) 1/Z 2 (1+j R x C x )/R x = (N 2 /N 1 ) (1+j R s C s )/R s 1/R x + j C x = (N 2 / N 1 R s ) + j C s N 2 /N 1 Real part: R x = R s N 1 /N 2 Imagination part: C x = C s N 2 /N

25 Single Ratio Transformer Bridge (Cont d) Inductance Measurement Z 1 = Unknown L x R x = 1/(1/R x + 1/j L x ) = 1/(1/R x j/ L x ) Z 2 = Standard C s R s = 1/(1/R s + j C s ) 49 Single Ratio Transformer Bridge (Cont d) Inductance Measurement (Cont d) Balanced, 1/Z 1 = (N 2 /N 1 ) 1/Z 2 1/R x j/ L x = (N 2 /N 1 ) (1/R s + j C s ) Real part: R x = R s N 1 /N 2 Imagination part: 1/ L x = (N 2 /N 1 ) C s L x = N 1 / N 2 2 C s, Reversed Current 50 25

26 Double Ratio Transformer Bridge To measure the impedance of components in Situ. Z 1 I 1 V s N 2 N 1 V 2 V 1 n 1 Z 2 n 2 D I 2 I 1 = V 1 / Z 1 = k(n 1 +N 2 )/Z 1 I 2 = V 2 / Z 2 = kn 2 /Z 2 51 Double Ratio Transformer Bridge (Cont d) Balanced, null current or zero magnetic flux, n 1 I 1 =ni 2 2 n 1 (N 1 +N 2 )/Z 1 = n 2 N 2 /Z 2 Z 1 = Z 2 n 1 (N 1 +N 2 ) n 2 N

27 Q-Meter RLC Series Resonance Z = R + j L + 1/j C R L = R + j( L 1/ C) V s C V C Resonant frequency (When the voltage across C is a maximum.) 0 L = 1/ 0 C 2 0 = 1/LC 0 = 1/ LC Im X L = L L R Re X C = 1/ C 53 Q-Meter (Cont d) I 0 = V s /R V C = I 0 X C = (V s /R)(1/ 0 C) = (1/ 0 RC) V s = Q V s Q where Q = Reactance/Resistance = 0 L / R = 0 (1/ 02 C) / R Q = 1 / 0 RC (Unloaded) 54 27

28 Q-Meter (Cont d) Unknown Inductor L x R x Oscillator V s across very low resistance Tuning C Capacitor V C = QV s V Tuning to resonance, 0 = 1/ LC L x = 1 / 02 C R x = 1 / Q 0 C 55 Q-Meter (Cont d) The circuit can be adjusted to the resonance by Preset the source frequency, and then vary the tuning capacitor for maximum value under this condition Preset the tuning capacitor and then adjust the source frequency This measures valus of Q in commonly regarded as the Q of the coil under test 56 28

29 Q-Meter: Low Impedance Measurement Unknown Large Capacitor L R x Oscillator V s C x C V C = QV s Short circuit tuning C = C 1, L 1 =L L, R 1 Q 1 = 1/ 0 R 1 C 1 R 1 = 1/ 0 C 1 Q 1 57 Q-Meter: Low Impedance Measurement (Cont d) Remove short circuit then tuning C = C 2 C 2 & C x = C 1, L 2 = L, R 2 = R 1 & R x Series capacitors, C 1 = 1 / (1/C x + 1/C 2 ) = C 2 C x / (C x +C 2 ) C 1 C x + C 1 C 2 = C 2 C x C x = C 1 C 2 / (C 2 C 1 ) Q 2 = 1/ 0 R 2 C 2 R 2 = 1/ 0 C 2 Q

30 Q-Meter: Low Impedance Measurement (Cont d) R 2 = R 1 +R x R x = R 2 R 1, Leakage resistance = 1/ 0 C 2 Q 2 1/ 0 C 1 Q 1 R x = (C 1 Q 1 C 2 Q 2 ) / ( 0 C 1 C 2 Q 1 Q 2 ) Q x = 1/ 0 R x C x = ( 0 C 1 C 2 Q 1 Q 2 )(C 2 C 1 ) 0 (C 1 Q 1 C 2 Q 2 )C 1 C 2 Q x = Q 1 Q 2 (C 2 C 1 ) / (C 1 Q 1 C 2 Q 2 ) 59 Q-Meter: Low Impedance Measurement (Cont d) If the unknown component is an inductor, L x = 1/ 02 C x L x = (C 2 C 1 )/ 02 C 1 C 2 If the unknown component is a pure resistor (no reactance), R x = (C 1 Q 1 C 2 Q 2 ) / ( 0 C 1 C 2 Q 1 Q 2 ) R x = (Q 1 Q 2 ) / 0 C 1 Q 1 Q 2, C 1 = C

31 Q-Meter: High Impedance Measurement L Oscillator V s R x C x C V C For high resistance, inductance > 100mH, or capacitance < 400 pf Open circuit and tune C = C 1, L 1 = L, R 1 Then short circuit and tune C = C 2 C 2 // C x = C 1, L 2 = L, R 2 = R x // R 1 C x + C 2 = C 1 C x = C 1 C 2 R 2 = R x R 1 = R x R 1 /(R x +R 1 ) R x = R 1 R 2 /(R 1 R 2 ) 61 Q-Meter: High Impedance Measurement (Cont d) Parallel RLC Q = 0 RC (loaded) R 1 = Q 1 / 0 C 1 and R 2 = Q 2 / 0 C 2 Therefore R x = Q 1 Q 2 0 C 1 C 2 / 02 C 1 C 2 (Q 1 C 2 Q 2 C 1 ) R x = Q 1 Q 2 / 0 (Q 1 C 2 Q 2 C 1 ) Q x = 0 R x C x = 0 Q 1 Q 2 (C 1 C 2 ) / 0 (Q 1 C 2 Q 2 C 1 ) Q x = Q 1 Q 2 (C 1 C 2 ) / (Q 1 C 2 Q 2 C 1 ) 62 31

32 Q-Meter: High Impedance Measurement (Cont d) For unknown inductance, L x = 1/ 02 C x L 2 x = 1/ 02 (C 1 C 2 ) For pure resistance, R x = Q 1 Q 2 / 0 (Q 1 C 2 Q 2 C 1 ) R x = Q 1 Q 2 / 0 C 1 (Q 1 Q 2 ), C 1 = C 2 63 References an_handbook_general/10-74.html wisc com/objects/viewobject aspx?id=dce7104 Hotek Technologies, Inc webpage : Yokogawa webpage: MAGNET LAB Wheatstone Bridge webpage: heatstonebridge/index.html Electronics Demonstrations webpage:

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